D.J. Staziker

Water Wave Scattering by Undulating Bed Topography.

February 1995

Abstract

The purpose of this thesis is to investigate the scattering of a train of small amplitude harmonic surface waves on water by undulating one-dimensional bed topography.

The computational efficiency of an integral equation procedure that has been used to solve the mild-slope equation, an approximation to wave scattering, is improved by using a new choice of trial function. The coefficients of the scattered waves given by the mild-slope equation satisfy a set of relations. These coefficients are also shown to satisfy the set of relations when they are given by any approximation to the solution of the mild-slope equation.

A new approximation to wave scattering is derived that includes both progressive and decaying wave mode terms and its accuracy is tested. In particular, this approximation is compared with older approximations that only contain progressive wave mode terms such as the mild-slope approximation. The results given by the new approximation are shown to agree much more closely with known test results over steep topography, where decaying wave modes are significant. During this analysis, a new set of boundary conditions is found for the mild-slope equation and the subsequent results give much better agreement with established test results.

Finally, the full wave scattering problem over a hump, that is, a local elevation in an otherwise flat uniform bed, is examined. Green's theory is used to convert the problem into an integral equation and a variational approach is then used to obtain approximations to the coefficients of the scattered waves. The results are used to further test the accuracy of previous approximations to wave scattering.

Contents

Intro duction

Background Fluid Dynamics

The linearised b oundaryvalue problem

Timeindep endent solutions

Separation solutions

Approximations to time indep endentvelo city p otential

The dimensional mildslop e equation

Integral equations and variational principles

Further development of Chamb erlains theory

Intro duction to approximation metho ds

A onedimensional trial space approximation

Outline of metho d

Results

Atwodimensional trial space approximation

Outline of metho d

Results

Extra computational saving

Results

Eckarts equation

Symmetry Prop erties

A new approximation to wave scattering

A Galerkin approximation metho d

Scaling ii

Boundary conditions

A system of Fredholm integral equations

Solution pro cedure

Numerical results

Graphical results

Obliquely incidentwaves

The full linear problem for a hump

A secondkind integral equation

An approximation

A rstkind integral equation

A direct derivation of the rstkind integral equation

Nondimensionalisation

Avariational principle

Numerical solution metho d

Results

Summary and Further Work

Bibliography iii

Chapter

Intro duction

A longstanding but p ersistent problem in the area of water wave theory is the

determination of the eect of b ed top ography and obstacles on a given wave

eld An example of a practical problem faced by coastal engineers is to predict

the amplitude of waves in harb ours where b oth manmade breakwaters and the

shap e of the sea b ed aect the wave b ehaviour Such problems involve the scat

tering diraction and refraction of waves and are mathematically formidable for

linearised theoryeven with relatively simple b ed andor obstacle geometries

The work presented in this thesis is solely concerned with the eect of b ed

top ography on an incidentwave train We do not address problems where an

obstacle such as a barrier aects an incidentwave train except for mentioning

them in this intro duction and noting the solution metho ds used The eect of

variations in the stillwater depth on an incidentwave train is examined using

linearised theoryWe prescrib e the incidentwave train and the deviation in the

stillwater depth and seek the additional waves the scattered waves caused by

this deviation A typical problem requires the determination of a velo city p oten

tial satisfying Laplaces equation within the uid a mixed b oundary condition

on the free surface and a given normal velo city on rigid b oundaries If the uid

domain extends to innity a radiation condition is required to ensure uniqueness

This b oundaryvalue problem is well known and is formally presented in Chap

ter We shall refer to the problem of nding a solution of the b oundaryvalue

problem as the full linear problem Analytic solutions of the full linear problem

are rare for any deviation from the constantwater depth case that is for any

deviation from a at sea b ed Problems where analytic solutions exist are usual

ly for a limited selection of straightforward geometries which include horizontal

andor vertical b oundaries

In Chapters and we are faced with solving secondkind and rstkind

integral equations resp ectivelyMany problems in water wave theory where an

incidentwave train is inuenced by b ed top ography andor obstacles can b e

reducedtointegral equations It is useful to review some of these twodimensional

and quasitwodimensional wave scattering problems here

The twodimensional scattering of surface waves by xed vertical barriers has

seen much attention over recent decades In water of innite depth that is water

deep enough that the b ed has no signicant eect on surface waves Dean

considered the case of a submerged semiinnite barrier whose upp er edge was a

nite depth b elow the surface His solution pro cedure used the reduction metho d

which consists of replacing the velo city p otential by another function whichis

chosen to simplify the b oundary conditions Dean showed that the reected wave

is small unless the edge of the barrier is very near the edge of the surface when

the wave motion is no longer oscillatory and there is a rapid owover the barrier

when a crest arrives Ursell solved this problem and its complementinwhich

the barrier extends from ab ove the free surface to a nite depth byanintegral

equation pro cedure for which exact solutions were found In water of nite depth

wo problems b ecome more dicult and have b een solved numerically by these t

Mei and Black using a variational metho d employed by Miles when he

considered wave scattering due to an innite step

The problem of two submerged semiinnite barriers placed an arbitrary dis

tance apart was addressed by Jarvis He found expressions for the co e

cients of the reected and transmitted waves in terms of four denite integrals

Jarvis evaluated these integrals bynumerical approximation and showed that to

tal transmission could b e achieved at certain wavelengths Evans and Morris

considered the complementary problem of two surface piercing barriers immersed

to a given depth by a quite dierent integral equation pro cedure Approximate

solutions were found by using Ursells exact solution for a single barrier as a

trial function in the integral equation Their analysis revealed that total trans

mission or total reection is p ossible This approximation metho d proved less

accurate when the ratio of barrier separation to barrier length b ecame small

Newman addressed this case by a dierent approachwhichinvolved match

ing solutions b oth near the two obstacles and in the far eld Newman also found

that b oth total transmission or total reection is p ossible

Another problem of muchinterest concerns the ow around vertical barriers

containing gaps Tuck considered the problem of a thin semiinnite imp er

meable barrier extending downwards from the uid surface and having at some

depth a small gap He constructed an approximation to the transmission co ef

cient under the assumption that the gap was small Porter considered the

same problem using a reduction metho d and an integral equation pro cedure and

his analysis showed that b oth approaches relied on the same basic step Porter

calculated several reection and transmission co ecients without needing to make

any restrictions on the gap width A more general problem of this form where an

arbitrary numb er of thin barriers one ab ove the next with the upp ermost barrier

allowed to intersect the free surface and the lowest allowed to extend innitely

far into the uid are allowed to oscillate with the same frequency as the incident

wave was considered byPorter The resulting b oundaryvalue problem was

converted into an integral equation from which the amplitudes of the scattered

waves could b e readily determined The case of two parallel semiinnite barriers

that pierce the free surface and extend downwards throughout the uid having at

tegral equation some depth a small gap was addressed byEvans using an in

approach The numerical results again showed the p ossibility of total transmis

sion or total reection o ccurring for an innite numb er of congurations of barrier

spacing gap width and incidentwavelength

The more general problem for surface waves in deep water incidentonan

arbitrary number of evenly spaced identical thin vertical barriers each contain

ing arbitrarily p ositioned gaps was addressed byPorter Expressions for the

transmission and reection co ecients were derived by approximation metho ds

for two sp ecial cases The rst was for each barrier containing a single smal

l gap and the second was for the separation b etween adjacent barriers greatly

exceeding the wavelength In b oth cases it was found that there existed band

sofwavelengths for which total transmission of the incidentwave is p ossible

and the insertion of an additional barrier resulted in an innityofwavelengths

corresp onding to zero reection

These problems with barriers that contain gaps b ecome more dicult when

the water is not assumed to b e deep as the motion is then also aected by the b ed

Macaskill considered the reection of water waves by a thin vertical barrier

of arbitrary p ermeabilityinwater of nite depth An integral equation for the

horizontal uid velo citywas derived by an application of Greens theorem and

was solved by collo cation metho ds A serious problem for the solution pro cess to

overcome involved the numerical dicultyofevaluating a Greens function given

by an innite series We are faced with the same problem in Chapter

In Porter there is a general discussion ab out the refractiondiraction

problem for verticalsided breakwaters of nite depth in relation to Greens the

ory and integral equation metho ds Examination of several sp ecial cases for the

class of problems where the breakwaters are straight parallel walls containing

gaps showed that the resulting integral equations are conducive to straightforward

numerical solution techniques Indeed a very ecient computational metho d

which solves the problem of diraction of a plane wave train through a gap in an

innite straight breakwater and the complementary problem of diraction bya

nite strip was given byChu and Porter The solution pro cedure involved

the conversion of known rstkind equations for these problems into secondkind

equations which are much more amenable to numerical techniques This was an

early example of the use of the technique of invariantimb edding in water wave

theory

For all the problems discussed so far the geometries haveinvolved vertical

b oundaries in water of innite or constant depth Allowing the water depth to

vary increases the diculty of the problem and consequently exact solutions are

exceedingly rare for such problems The exceptions to this are limiting cases such

as shallowwater where the wavelength is assumed to b e much larger than the

water depth

Lamb derived an exact expression for the reection co ecient for the

problem of waves incidentona vertical step in shallowwater Bartholomeusz

considered the same problem without the shallowwater assumption By matching

eigenfunction expansions Bartholomeusz derived a rstkind integral equation for

the horizontal uid velo city After transforming this equation into a secondkind

integral equation for which extensive theory exists Bartholomeusz showed that

the solution reduced to that obtained byLamb in the shallowwater limit so

giving a verication of this theory

Newman considered the propagation of water waves over an innite step

where the vertical part of the step extends down to innity By matching so

lutions for the two uid regions at the cut ab ove the step Newman derived an

integral equation for the horizontal velo city comp onent on the cut Newman

recast this integral equation as an innite system of equations and solved it by

truncation Miles considered the more dicult problem of a nite step which

Bartholomeusz solved in the shallowwater limit Miles used a variational

technique to obtain approximate solutions of the rstkind integral equation for

the horizontal uid velo city By taking the limit of the vertical part of the step

to innity Miles found that his results showed excellent agreement with those

obtained by Newman

Allowing the depth prole to vary smo othly rather than changing abruptly

as in the case of a step creates further diculties in the wave scattering problem

As far as is known the only analytic expression for the velo city p otential in a

en by Roseau He obtained an explicit problem of this typ e has b een giv

solution for the reection and transmission of waves across a shelf of a sp ecial

prole which asymptotically joins two regions of constant but unequal depth

According to Wehausen and Laitone a pro of of the existence and uniqueness

of a velo city p otential in a general problem of this typ e had yet to b e given

Evans addressed this problem in considering the transmission of water waves

over a shelf of arbitrary prole He used Greens theorem to derive a second

kind integral equation for the velo citypotential The asso ciated Greens function

whichEvans derived is very complicated and the kernel of the integral equation

was given in terms of the normal derivative on the shelf of this function By

suitably restricting the shelf prole Evans was able to prove that the velo city

p otential exists and is unique except p ossibly for certain discrete values of the

parameters of the problem corresp onding to trapping mo des over the shelf The

complexity of this integral equation is well illustrated by the fact that no one has

so far found a suitable numerical metho d to solveit

FitzGerald investigated wave scattering by a region of varying depth

He used a Fourier transform pro cedure to convert the b oundary value problem

satised by the velo city p otential intoapairofintegrodierential equations

The required solution was then given by a linear combination of the solutions

of the integrodierential equations FitzGerald was able to prove the existence

and uniqueness of the velo city p otential in two limiting cases and presented the

asso ciated asymptotic results

The propagation of long waves over water of slowly varying depth for the two

and threedimensional cases was treated by Harband Asymptotic expressions

were found for the scattered long waves under the assumption that the water is

shallow compared with the wavelength

Finding an accurate solution of the full linear problem of water wave scatter

ing due to an arbitrary varying b ed has proved so dicult and computationally

exp ensivethatvarious typ es of approximation to this problem have b een pro

p osed which can b e solved accurately One of the earliest and most widely used

approximations is that of linearised shallowwater theoryinwhic h the vertical

structure of the uid motion is ignored A derivationofthisapproximation from

rst principles can b e found in Stoker and one example of its application is

given by Lautenbacher who considered the runup of tsunamis on Hawaiian

islands Lautenbacher used Greens theorem to convert the governing dierential

equation into an integral equation and then found approximate solutions of the

integral equation by using a numerical nite dierence technique

In water where the depth is not assumed to b e shallow one of the earliest

approximations to the full linear problem was given byEckart He derived

his approximation byconverting the full linear problem for the velo city p oten

tial into an integrodierential equation Eckart showed that this equation was

approximated by a partial dierential equation by discarding a presumably s

mall integral without examining the justication of this approximation Eckart

showed that his approximate equation reduces to the linearised shallowwater e

quation in the shallowwater limit Through this work Eckart also discovered a

useful approximation to the ro ot of the well known disp ersion relation a tran

scendental equation which connects the deepwater wavenumber to the wave

numb er in nite water depth Eckart did not pursue his approximation to the

full linear problem further p ossibly according to Miles b ecause he had ob

tained a rather unsatisfactory approximation to the group velo city in his

lecture notes

A more recent and very p opular approximation of the full linear problem was

given by Berkho whose combined refractiondiraction equation has

now b ecome known as the mildslop e equation There havebeenmany subse

quent derivations of the mildslop e equation whichtypically approximate the

vertical structure of the motion and restrict the b ed slop e to b e small or mild

in a sense to b e describ ed later The derivation given by Berkho is a clari

cation of the original derivation given in Berkho However the mathematical

approach used in Berkho which is a p erturbation pro cedure in terms of two

small parameters is still not rigorous Smith and Sprinks gave a more math

ematically sound derivation of the mildslop e equation by expanding the vertical

dep endence of the velo city p otential in terms of an orthogonal set of functions

and removing the dep endence on the vertical coordinate byintegration over the

depth The nal step of their derivation still requires physical intuition to justify

the neglect of terms on the basis that the b ed slop e is small or mild enough

ations of the mildslop e equation have b een given by Lozano and Mey Other deriv

er and Massel but p erhaps the most elegantwas given by Miles

He uses a variational approachwhich through a suitable choice of trial func

tion can b e used to generate the linearised shallowwater the Eckart and the

mildslop e equations Miles go es on to compare the mildslop e and Eckart

equations through the calculation of reection from a gently sloping b eech and

of edgewave eigenvalues for a uniform slop e He nds that Eckarts equation is

inferior to the mildslop e equation for the amplitude in the reection problem

but is sup erior in the edgewave problem

Another test of the accuracy of the mildslop e approximation was given by

Bo oij when he considered the problem of wave scattering due to a talud as its

steepness was varied Bo oij compared the amplitude of the reected wavegiven by

the mildslop e approximation with values he computed using full linearised theory

From his results Bo oij suggested that the mildslop e approximation gave results

in go o d agreement with those he had computed using full linearised theory

for talud gradients up to one third The results Bo oij computed using full

linearised theory have b een well used by authors prop osing new approximations

to the velo city p otential b ecause as far as is known these are the only published

estimates of the reection co ecient for the full linear wave scattering problem

over a b ed of varying depth

The numerical metho ds that have b een used to solve the b oundaryvalue prob

lem resulting from the mildslop e approximation are mainly nite dierence for

example Eb ersole Li and Anastasiou and nite element for example

Berkho andBooijmethods An alternative solution metho d was

given by Chamb erlain who converted the mildslop e b oundaryvalue problem

into a secondkind Fredholm integral equation Through a p owerful variational

metho d which used problemdep endent trial functions Chamb erlain was able to

nd solutions that were machine accurate The maximum trial space dimension

required to achieve this accuracy was only six for all the problems he considered

Wave tank exp eriments have also b een carried out as another means of testing

the accuracy of approximations such as the mildslop e to the velo city p otential

e tank exp eriments was satisfying the full linear problem One such set of wav

conducted byDavies and Heathershaw who measured the scattering of water

waves by a nite patch of small amplitude ripples set in an otherwise horizontal

b ed It was found that the mildslop e approximation gave a p o or estimate of

the p eaks in the reected amplitudes that arise in ripple b ed problems when

the water wavenumb er is approximately twice the ripple wavenumber Such

p eaks in the amplitude are known as Bragg resonance p eaks Improvements

have therefore b een sought to the mildslop e approximation by attempting to

improve the mildslop e approximation itself and by nding new approximations

to the velo city p otential

Toovercome this deciency in the mildslop e approximation Kirby p

resented a mo del in which the depth prole was expressed in terms of a slowly

varying mildslop e comp onentonto which a rapidly varying comp onent of small

amplitude is sup erimp osed Kirby then used a vertical integration pro cedure to

derive what is now called the extended mildslop e equation He veried that it

gavemuch b etter agreement with the exp erimental data at the Bragg p eaks for

ripple b ed problems than the mildslop e approximation However this improve

ment to the mildslop e approximation is only valid for ripple b ed problems

A further improvement to the mildslop e equation whichisvalid over arbi

trary depth proles was given byChamb erlain He followed the same pro ce

dure which Lozano and Meyer used to derive the mildslop e equation In other

words Chamb erlain approximated the vertical structure of the velo city p otential

and removed the dep endence on the vertical coordinate byintegration over the

depth Chamb erlain do es not make the further assumption that the b ed slop e

is mild and consequently nds a new approximation to the velo city p otential

Chamb erlain and Porter formalised the derivation of this new approximation

giving derivations using a variational approach and a Galerkin approach They

named the resulting equation the mo died mildslop e equation and showed that

it reduced to the mildslop e equation when the b ed slop e is assumed to b e mild

Chamb erlain and Porter also show that for ripple b ed problems the mo died

mildslop e equation subsumes the extended mildslop e equation to o They found

that the results given by the mo died mildslop e equation are in b etter agree

en ment with the full linear results for Bo oijs talud problem than those giv

by the mildslop e equation Chamb erlain and Porter also found that in ripple

b ed problems the results from the mo died mildslop e equation gave excellent

agreement with the exp erimental data at the Bragg p eaks Indeed Chamb erlain

and Porter use the wellknown symmetry prop erties b etween the co ecients

of the reected and transmitted waves to obtain results for the mo died mild

slop e equation over anynumb er of ripples from knowing the results for just one

ripple These symmetry prop erties were rst derived for the full linear problem

by Kreisel and Newman and Chamb erlain derived them for the mild

slop e approximation Chamb erlain and Porter show that these symmetry

prop erties are an intrinsic part of the problem rather than of its exact solution

in the sense that they are automatically satised regardless of the accuracy of the

solution

Massel also uses a Galerkin approach to deriveanapproximation to the

full linear velo city p otential that included decaying wave mo de terms to o He

named the resulting system of secondorder dierential equations the extended

refractiondiraction equation In this pap er Massel only gives solutions when

the decaying wave mo de terms are neglected leaving the solution of the full

problem incorp orating decaying mo des to a subsequent pap er which has yet to

app ear The system of secondorder dierential equations derived by Massel

reduces to a single equation when the decaying wave mo de terms are omitted

which turns out to b e the mo died mildslop e equation Massel noticed that this

equation reduced to the mildslop e equation on the assumption that the slop e of

the b ed is mild and also showed it was sup erior to the mildslop e equation for

Bo oijs talud problem and for ripple b ed problems

Other approximations of the full linear problem that include decaying wave

mo de terms have b een given by OHare and Davies and Rey Both sets

of authors use a similar approach to derive the approximation to the velo city

p otential The common approachinvolves approximating the b ed prole as a se

ries of horizontal shelves separated by abrupt vertical steps The wellknown at

b ed innite series representation of the velo city p otential is used to represent the

p otential over each horizontal shelf Continuity of the velo city p otential and its

horizontal derivative is imp osed throughout the depth at the ends of each shelf

which gives a matrix system to b e solved This approximation can b e compu

tationally exp ensive as a large numb er of steps are required to obtain reliable

results In OHare and Davies the approximation assumes that the steps are

suciently far apart that the decaying mo des generated at one step are negligible

at neighb ouring steps OHare and Davies applied their approximation to ripple

b ed problems and noted that the results were in go o d agreement with known

wave tank data In a subsequent pap er OHare and Davies compared their

approximation against the extended mildslop e equation Kirby for ripple

b ed problems They found that their approximation provides a more explicit

formulation of the problem and gave b etter agreementwiththewave tank data

than the extended mildslop e equation but was much more computationally ex

p ensive In Rey the series of steps whichapproximates the depth prole is

sub divided into smaller subsystems called patches In each patch the decaying

mo des generated at one step are not assumed negligible at neighb ouring steps

in the patch However the decaying mo des generated in one patch are assumed

negligible at neighb ouring patches This seems to b e a sup erior approximation to

that used by OHare and Davies Rey go es on to test his approximation

on Bo oijs talud problem nding go o d agreement with the full linear results

computed by Bo oij He also found that his results were quite dierent to those

given by the mildslop e approximation even for taluds with gradient less than

one third for which Bo oij had claimed were in go o d agreement with the full

linear mo del Rey also tested his approximation on ripple b eds nding go o d

agreement with wave tank data

After reestablishing the wellknown full linearised equations for the scattering

of waves byvarying top ography in Chapter we go on to state the mildslop e

equation and briey review the integral equation pro cedure that Chamb erlain

used to solveit

Noting that coastal engineers require only ab out three decimal place accuracy

in solutions of water wave problems we go on in Chapter to improvethecom

putational eciency of Chamb erlains integral equation solution metho d This

is done by using a new choice of trial function and seeking solution accuracy to

three decimal places rather than the machine accuracy achieved byChamb erlain

We also reinvestigate Eckarts approximation and the symmetry prop erties

of the co ecients of the reected and transmitted waves

The purp ose of Chapter is to derive a new approximation to the full linear

velo city p otential that includes b oth progressive and decaying wave mo de terms

and test its accuracyWe compare this new approximation with older approxima

tions that only contain progressivewave mo de terms such as the mildslop e and

the mo died mildslop e approximations as the steepness of the depth prole is

varied Weinvestigate if the new approximation is b etter for steep er b ed proles

as the steep er the b ed prole the more signicant the decaying wave mo des b e

come In the course of this analysis a new set of b oundary conditions is found for

the mo died mildslop e and mildslop e equations The subsequent results give

much b etter agreement with the results that have b een obtained using full linear

theory and those found by Rey than the results obtained using the original

b oundary conditions

In Chapter we consider the full linear wave scattering problem over an

arbitrary hump The term hump is used to describ e a lo cal elevation in an

otherwise at uniform b ed Using Greens theory the b oundaryvalue problem for

the velo city p otential is converted into a secondkind integral equation Initially

an approximation in this equation is tried but it proves to b e rather inaccurate

It is found however that the secondkind integral equation for the p otential can

be converted into a rstkind integral equation for the tangential uid velo city

The kernel of the rstkind equation is much easier to evaluate numerically than

that of the secondkind equation A variational approach is then used to obtain

approximations to the co ecients of the reected and transmitted waves which

are secondorder accurate compared to the approximation of the solution of the

integral equation These results are then used to test the accuracy of the new

decaying mo de approximation derived in Chapter and also to test the accuracy

of the mo died mildslop e and mildslop e approximations

A summary of the work presented together with conclusions and suggestions

for future research concludes this thesis

Chapter

Background Fluid Dynamics

In this chapter we present the linearised equations satised by the velo city p oten

tial for the irrotational ow of an incompressible homogeneous uid over a b ed

of varying depth It is assumed that the uid o ccupies a region which extends to

innityinevery horizontal direction The uid is also b ounded b elowbyabed

of given p ermanent shape and aboveby a free surface whose shap e is sought A

harmonic time dep endence is removed from the velo city p otential Separation

solutions of the b oundaryvalue problem satised by the time indep endentpart

of are examined in the sp ecial case where the depth is constant

However analytic solutions are rare when there is any departure from the

constant depth case Therefore three vertically integrated approximations of

these equations are then considered the wellknown mildslop e and shallowwater

approximations and the less familiar Eckart approximation

Some of the subsequentwork in this thesis revolves around solving integral

equation forms of the onedimensional mildslop e Eckart and shallowwater e

quations This chapter therefore concludes with a review of an integral equation

pro cedure which requires only or dimensional trial spaces to give excellent

approximations to the solutions of these vertically integrated mo dels

The linearised b oundaryvalue problem

Let x and y b e horizontal cartesian coordinates and z avertical coordinate

ely upwards with the undisturb ed free surface at z Let measured p ositiv

the uid velo city at time t and a given p ointx y z in the uid b e denoted

by q x y z t Assuming the uid motion starts from rest with gravity the

is necessarily irrotational It follows that there only external force acting then q

exists a velo city p otential x y z t suchthatq r where r

x y z

The assumption of the uid b eing homogeneous and incompressible reduces the

continuity equation to

r q

and hence satises Laplaces equation

in the uid

The bed is assumed to be xed and imp ermeable and is dened by

z hx y as depicted in Fig As the uid cannot ow through the b ed

the normal derivative of the velo citypotential on the b ed must b e zero giving

rise to the b oundary condition

on z hx y

where denotes the outward normal derivativeon z hx y n z y Free surface of the fluid x η(x,y,t) z=0

h(x,y)

-n

Figure Vertical cross section of the uid domain

By considering the Stokes derivative r which denotes dierenti q

Dt t

ation following the motion of the uid this b oundary condition may b e rewritten

as a socalled kinematic b oundary condition as follows On any uid b oundary

given by f x y z t wehave

as otherwise there would b e a nite ow of uid across the b oundary Lamb

The b ed is dened by z hx y so substituting f z hx y into

delivers

rhr on z hx y

is the gradient op erator with resp ect to the horizontal co where r

x y

ordinates only Comparing equations and shows that

rhr

where the denominator app ears on the right hand side so that the outward unit

given by normal n

h h n

x y

satises the requirement jn j

Similarly equation must hold at the free surface given by z x y t

Substituting f z into gives

r r on z

t z

A further condition at the free surface is delivered by considering Bernoullis

equation for unsteady incompressible homogeneous ows namely

q gz f t

where p is the pressure the constant densityand g the acceleration due to

gravityNow as is dened by q r f t can b e absorb ed into and

similarly assuming that surface pressure is constant then on z the lefthand

side of can also b e absorb ed into reducing to

g q on z

Equations and are linearised by expanding ab out z and ne

glecting secondorder terms yielding resp ectively

on z

z t

and

g on z

On eliminating from and we obtain the nal b oundary condition

of the linear b oundaryvalue problem whichisgiven by

r hz

on z

z g t

rhr on z hx y

together with a radiation condition imp osed as x y If wewrite

where represents the incidentwave eld then the radiation

i s i

condition causes to represent only outgoing waves as x y Evans

has shown that when the b ed is given by a shelf of arbitrary prole then

the solution of the ab ove problem is unique except p ossibly for certain discrete

values of the parameters of the problem which corresp ond to trapp ed mo des over

the shelf

Timeindep endent solutions

A harmonic time dep endence can b e removed from the velo citypotential by

setting

x y z t x y z cost x y z sint

i t

x y z e Re

where x y z x y z ix y z i and is an assigned

angular frequency

Dening g it follows that if satises together with a radiation

condition then satises

r hz

on z

rhr on z hx y

together with a radiation condition

The free surface elevation may b e recovered bycombining equations

and to give

i t

x y tRe x y e

Separation solutions

Wehave already noted that analytic solutions of are rare However assum

ing that is indep endentof y then in the at b ed case where h the undisturb ed

uid depth is the constant h separation solutions of are easily found

These can b e combined to give the general solution

x z xw z

n n

where

ik x ik x B x B x

n n

n IN x C e D e xC e D e

n n n

z h cos B cosh k z h

wz n IN w z

cosh k h cos B h

and where k is the real p ositive ro ot of the relation

k tanh k h

and B n IN are the real p ositive ro ots of the relation

B tan B h

n n

arranged in ascending order of magnitude

Once an incidentwave has b een assigned the arbitrary constants C and

D n IN are sp ecied As already mentioned the radiation condition

enforces that is b ounded as jxj causing the constants to b e chosen so

that n IN j jas jxj Thus the terms w n IN are called

n n n

the decaying or evanescent mo des see for example Mei Therefore w is

the only mo de which propagates as a wave throughout the domain and is thus

called the progressivewave mo de

Denoting the wavelength by then using the fact that k allows

to b e rewritten as

tanh h

where k is the wave sp eed An interpretation of this equation is that waves

of dierent lengths travel at dierent sp eeds In other words they are disp ersive

and hence is known as a disp ersion relation

The following approximations to the full linear problem for use the assump

tion that the evanescent mo des are negligibly small x whenh is

suitably restricted

Approximations to time indep endent ve

lo citypotential

As previously stated analytic solutions for the timeindep endentvelo city p oten

tial satisfying for any departure from the constant depth are rare This

naturally leads to the pursuit of an approximation to As discussed in Chapter

the mildslop e approximation to is very wellknown with many authors using

various pro cedures to establish the approximation The mildslop e approxima

tion to suitably restricts the depth prole h so that the decaying mo de parts of

are assumed to b e negligibly small Thus the mildslop e approximation to seeks

ximating its to approximate the propagating mo de of Thisisachieved byappro

dep endence on the z coordinate and discarding terms of O r h jrhj onthe

basis of the mildslop e assumption jrhj kh Meyer where h hx y

is the undisturb ed uid depth and k k x y is the lo cal wavenumb er satisfying

the lo cal disp ersion relation

k tanh kh

Omitting the details of a derivation here the mildslop e approximation to

is given by

x y z x y w z h

where

cosh k z h

w z h

cosh kh

satises the dierential equation

rur k u

where

uh w dz tanh kh

k sinh kh

and the dierential equation satised by is known as the mildslop e

equation

In Chapter a variational approach similar to Miles is employed in

whicha variational principle and an n term trial function are used to obtain an

approximation to that includes the decaying wave mo de terms to o It will b e

shown that if a oneterm trial function is used in this variational principle and

terms of O r h jrhj are neglected then the mildslop e equation is derived

It should b e noted that under the further assumption that kh so that

tanh kh and sinhkh are approximated by their arguments the mildslop e

equation reduces to

rhr

This is another wellknown equation the linearised shallowwater equation

which can of course b e derived from rst principles using the shallowwater ap

proximation Stoker However forming as a limit of is justied

as the assumptions of the shallowwater approximation imply mildness of slop e

Jonsson et al give some detail as to what constitutes small kh and show

that is valid if kh

Another vertically integrated approximation to was prop osed some forty

years ago by Carl Eckart in which the linearised b oundaryvalue problem for

given by is transformed into an integrodierential equation Here the

integral term is assumed small without justication in some cases and is thus

discarded After neglecting third and higher derivatives of the depth function h

an approximation for linear gravitywaves in water of variable depth h hx y

is arrived at in the form

rur k u

i t

where Re e isanapproximation to the free surface shap e

k coth h

is the deep water wavenumber and

Eckarts equation also reduces to the linearised shallowwater equa

tion under the same assumption as in the mildslop e case Eckart notes that

actually approximates the ro ot of the disp ersion relation to within

for all values of khHowever he seemed to b e discouraged from further de

velopment of his approximation due to the unsatisfactory approximation to the

group velo city he obtained Recently Miles has shed new lightonEckarts

approximation deriving as well as and via an elegantvari

ational pro cedure Miles notes that the direct calculation of the group velo city

from Eckarts disp ersion relation gives an approximation to the ratio of

group velo city to phase velo city within of the exact value for all values of

kh Miles also notes that while the mildslop e approximation conserves wave

energyEckarts approximation do es not except in uniform depth Also on

a gently sloping b each Eckarts approximation is inferior to the mildslop e ap

proximation in predicting the amplitude of the reected wave However in the

calculation of edge waveeigenvalues over a b each of uniform and not necessarily

small slop e Eckarts approximation is sup erior to the mildslop e approximation

We will return to Eckarts approximation in Chapter where it will b e compared

to the mildslop e approximation Some ideas will then b e prop osed to improve

the Eckart prediction of the amplitude of the reected wave

Most numerical metho ds used to solve the mildslop e equation have

b een based on either nite dierence for example Eb ersole Li and Anas

tasiou or nite element metho ds for example Berkho Bo oij

Eciency of metho ds and accuracy of solution are obviously of considerable

imp ortance Chamb erlain has used a dierent approachinwhichtheone

dimensional mildslop e equation is converted into twointegral equations the

solutions of which are approximated byvariational techniques For a sp ecied

incidentwave this pro cedure can calculate highly accurate approximations of the

co ecients of the resulting transmitted and reected waves and of the free sur

face shap e with only or dimensional problemdep endent trial spaces As

work in the following chapters uses this integral approach it is convenient to give

a brief description of Chamb erlains solution metho d for dimensional scattering

problems

The dimensional mildslop e equation

Let x and y b e horizontal cartesian coordinates as dened in section The

class of depth proles now considered is suchthath is indep endentof y and varies

only in some nite interval of xsothat

h x

h x l

where h and h are constants for a given problem and hx is assumed to b e

h are continuous on In the case of a lo calised hump in the b ed h and

equalthey ma yhowever b e unequal when the b ed prole is that of a lo calised

h 1

h 0

x=0 x=1

Figure Vertical cross section of the uid domain

talud as depicted in Fig In the wave motion it is assumed that the crests

are parallel to the y axis this implying that x

The mildslop e equation is now scaled using the nondimensionalisation pro

cess outlined byChamb erlain whichmay b e summarised as follows Let

ul x U x

H x hl x

l x x

The following account discards the accents from these denitions in the pursuit

of a simple notation In the ab ove circumstances the mildslop e equation

may b e written as

d d

U U

dx dx

where

tanh H H

U x

sinh H

and x k xl the dimensionless lo calised wavenumb er is the p ositive

real ro ot of

tanh H

and where and are two dimensionless parameters given by

h l

This scaling pro cess will b e used throughout this text

The pro cess of converting the dierential equation and its b oundary

conditions into an integral equation is made much simpler byconverting

into normal form This is done byintro ducing a new variable dened by

U x

x x

Then satises

where for the rest of this chapter the prime denotes dierentiation with resp ect

to x

U x U x

x x

U x U x

and where the notation and is used

The co ecients app earing in the dierential equation are uniquely de

ned once H and are assigned The only remaining information necessary

is the choice of incidentwaves On the at b ed for x the nondimensional

Similarlyontheatbed mildslop e equation reduces to

Therefore we supp ose in general for x reduces to

that there are two incidentwaves with known co ecients A propagating from

x resp ectively This will result in two outgoing waves with unknown

co ecients B propagating towards x resp ectively Hence on the at

b eds we take

i x i x

A e B e x

i x i x

A e B e x

and using we see that the corresp onding form for is

i x i x

A e B e x

i x i x

x A e B e

We use jump conditions at x and x as H is allowed to b e discontinuous

at the ends of the varying b ed to eliminate B between and yielding

i i a

i i e b

where

a c c c

b c c c

The complex num bers c j are given by

c A

c A e

c e

where the terms involving U are the consequences of allowing slop e discontinuities

inHatx Greater detail of the derivation of these b oundary conditions and

the merit of writing them in the form of can b e found in Chamb erlain

Linearityallows sup erp osition of solutions corresp onding to waves incident

from the left with solutions corresp onding to waves incident from the right This

removes the need to solve the problem with two incidentwaves By removing one

incidentwave the amplitude of the remaining incidentwavemay b e set equal to

unity without loss of generality Accordingly the two reection and transmission

co ecients denoted by R and T for this problem are dened as follows

B B

If A then R and T

A A

B B

and T If A then R

A A

The subscripts distinguish b etween waves incident from the left or the right

Nowwe dene to b e the solution of and for an incidentwave from

the left requiring A in is dened to b e the solution for an

incidentwave from the right requiring A in Using the equations

the reection and transmission co ecients can nowbedenedintermsof

j in the following way

If A

U e

A U

If A

U e

e R

A U

The outgoing wave co ecient B comprises of two parts that part of A

transmitted b eyond the talud and that part of A reected back from the talud

We can make a similar statementabout B and these resulting relationships can

b e summarised as

B T R A

B B C C B C

A A A

B R T A

The matrix on the righthand side of this equation is called the scattering matrix

Clearly as so on as the reection and transmission co ecients have b een found

then B can b e determined for any A

Integral equations and variational princi

ples

Avariation of parameters pro cedure shows that the b oundaryvalue problem for

given by and is equivalenttotheintegral equation

i x i x i jxtj

xae be e t t dt

With some simple algebra this integral equation can b e rewritten in terms of the

integral equation with a realvalued kernel

i x i x

sin jx tj t t dt x Me Ne

where

M a b b b

N b b b b

c c b

b c c

b c c e

c c e b

i i

b c c e e

and where

i t

b b b e t t dt

i t

b b b e t t dt

Full details of this derivation can b e found in Chamb erlain

The equation can b e written as an op erator equation in the Hilb ert

space L with inner pro duct

g t dt f g f t

on dening two selfadjointoperators L and P by

L x sin jx tj t dt

and

Px x x

Then L satises

Mf Nf LP

where

i x

f e

and since is b ounded and L is a compact op erator then LP is also compact

It follows that if there exists a solution L of this op erator equation

it determines the desired function x of This issue and others which arise

from using Hilb ert space metho ds can b e found in Porter and Stirling

A summary is now presented of the approximation metho d employed by

Chamb erlain for integral equations of the form where the function is

sought The apparent diculty p osed b y the free term where endp ointvalues of

the unknown function are contained in the constants M and N iscircumvented

by exploiting linearity It can b e shown that

xM N x iM N x

where the functions j are the solutions of realvalued integral equa

tions that can b e expressed in op erator form as

A f j

j j

where A I LP I b eing the identity op erator and the free terms f j

are given by

x f x cos

and

f x sin x

It follows that once and are determined the unknown function can

also b e determined through once the endp ointvalues and and

thus the co ecients in have b een found Also on nding and

then the reection and transmission co ecients are known through and

On substituting the ab ove equation for into the right hand side of

equations and and rearranging them it can b e shown that and

are determined by solving the rank system of equations

b b B B c b c b

BB C B C B C C B C

A A A A A

b b c b c b B B

c b B B b

C B B C C B

A A A

c b b B B

in which

A A and B A iA A B

and

A ttf t dt Pf j k

jk j k j k

Stationary principles are used to generate approximations to the inner pro d

ucts A j k with upp er and lower b ounds by rstly ensuring that the

function is entirely onesigned This allows the nonselfadjoint op erator A in

the integral equations to b e replaced by a selfadjoint one In general

do es not p ossess this prop erty but can b e made to do so by adding to it or sub

tracting from it a known quantity This pro cess requires a slightchange in the

b oundary conditions which causes the denitions of the c j in

to b e amended Full details of this device are in Chamb erlain With

onesigned a new selfadjoint op erator S can b e dened by

Sx sxx

where

x sx

and

sgn that is

Then the integral equations for j given by can b e rewritten as

A Sf j

j j

where

S j

j j

and

A I S LS

Clearly A is selfadjoint and as S is b ounded SLS is a compact op erator

It is easy to show that the functionals

J L IR k

J L L IR

given by

p S f Ap p k J p

k k

J p p Sf p p Sf Ap p

have stationary values

Sf Pf k

k k k k

Sf Pf

resp ectively Problemdep endent N dimensional trial functionsp andp given

k n

p a LP f k

k k

IRn N are used to approximate and resp ectively for some a

Substitutingp andp in the ab ove functionals generates approximations to the

inner pro ducts A j k which are secondorder accurate compared with

the approximations to the unknown functions and The unknown con

stants a n N are chosen so as to make the functionals and

stationary within the N dimensional trial spaces see Chamb erlain for

details To clarify the notation we shall denote the functionsp k deter

mined by the rst functional as k resp ectively and the functions

p k determined by the second functional as k resp ec

k k

tively

The further assumption that there exists b suchthat p L

b kpk Ap p a kpk

where the existence of an a is guaranteed since A is a b ounded op erator

establishes the following upp er and lower b ounds on the inner pro ducts of interest

J A Sf A Sf Sf J k

k k k k k k k k k k

a b

and

G R Sf G R

where the functionals G and R are given by

A Sf A Sf G J

b a

and

A Sf A Sf R

b a

An excellent derivation of these upp er and lower b ounds may b e found in Porter

and Stirling pp Approximations to a and b can b e found

in Chamb erlain Disapp ointingly the approximation to b can b e negativein

certain cases resulting in just a stationary approximation to the inner pro ducts

with no upp er and lower b ounds

The implementation of the solution pro cess follows by rstly assigning H

and the direction of the incidentwave Then after ensuring is onesigned by

adjusting it to make it so if necessary and cho osing the dimension of the trial

space the trial functions given by are generated and the approximations

J J andJ to the inner pro ducts Pf j k are

j k

calculated Upp er and lower b ounds to the inner pro ducts are also calculated if

b Then the rank system is solved to give the approximations to

and Finally and are used to deliver approximations to the

reection and transmission co ecients and when b upp er and lower b ounds

on the reected and transmitted amplitudes are also found The approximation

to and thus the free surface shap e can then b e found through

Chamb erlain has shown that or dimensional trial spaces can result in

the determination of approximations to the reection and transmission co ecients

to machine accuracy

This integral approach can also b e used with the linearised shallowwater e

quation and Eckarts equation with certain mo dications Chamb erlain has

done this for the linearised shallowwater equation and the necessary mo dica

tions required to use this integral approachtosolveEckarts equation are given

in Chapter

Chapter

Further development of

Chamb erlains theory

In this chapter some extensions to the work app earing in Chamb erlain

are presented A new computationally cheap integral equation solution metho d

is develop ed for the three mo del equations mentioned in Chapter namely the

mildslop e equation Eckarts equation and the linearised shallowwater equation

over a range of parameter values This metho d uses the approximation metho ds

discussed in section but with a new choice of trial functions Eckarts equation

is further investigated and improvements to it are suggested Finally the sym

metry prop erties of the solutions of the three mo del equations are studied and

an unexp ected prop ertyisdiscovered that any approximations of the solutions

still p ossess the symmetry prop erties

Intro duction to approximation metho ds

This section b egins by illustrating the interest in solving the mo del equations

over a continuous range of their parameters for a sp ecied b ed shap e Bo oij

provided some exp erimental evidence concerning the accuracy of the mildslop e

approximation to the velo city p otential satisfying As a part of that

pap er a talud problem was considered and a graph see Fig was presented

of reected amplitude jRjgiven by the mildslop e equation against W apa

rameter which denotes the length of Bo oijs talud In terms of the notation used

in Chapter the dimensionless parameters of the mildslop e equation are given

in terms of W by W and W See Chamb erlain p

s s s

for details Bo oij computed jRj using full linearised theory and sup erimp osed

it onto his graph observing that the two sets of results coincide for talud slop es

with gradient that is for values of W s

−1 10 | 1 |R

Mild−slope approximation Full linear

−2 10 −1 0 10 10

Figure Reected amplitude over the depth prole H x x x

In the absence of any analytical estimates of the accuracy of the mildslop e

approximation exp erimental evidence such as this provided by Bo oij has b ecome

invaluable

Another prop erty coastal engineers are interested in is the wavenumber at

which for a given shap e of b ed a signicantpartofthewave is transmitted

An easy way to nd this information is to solve the mo del equations over a wide

range of wavenumb ers and present the results graphically as in Fig

So it is obvious that there exists muchinterest in nding solutions of our

three mo del equations over a range of their parameters From Chapter we

know that in a uid of undisturb ed depth H H x the nondimensional mild

slop e equation is given by

d d

U U

dx dx

where

H tanh H

U x

sinh H

and x is the p ositive real ro ot of the disp ersion relation

tanh H

The integral equation metho d given in section can b e used to solve the mild

slop e equation for a waveincident from x on a given depth prole for

some and and gives extremely accurate approximations to the co ecients

of the resulting reected and transmitted waves These co ecients are dened

in Chapter and are denoted by R and T k where the subscripts

k k

distinguish b etween waves incidentfromx and x The integral

equation metho d involves using variational principles to generate approximations

to the inner pro ducts Pf j k and approximations to the solutions

j k

k L of the integral equations

I LP f k

k k

Here the op erators L and P are dened by

Lx sin jx tj t dt and Pxxx

with the solution of the disp ersion relation at x and x

given by

U x U x

x x

U x U x

where for the rest of this chapter the prime denotes dierentiation with resp ect

to x and the functions U and are dened by and resp ectively The

free terms f k in are dened by

f x cos x and f x sin x

The results presented in Fig were obtained by using the integral equation

metho d to nd extremely accurate approximations to R and T k given

k k

by the mildslop e equation at values of and

In Chapter we used maximum principles to generate approximations to

Pf k and approximations k to the unknown functions

k k k

k resp ectively A stationary principle is also used to generate an

approximation to Pf and approximations k to k

k k

resp ectivelyWealsosaw that when the op erator A is b ounded ab ove and b elow

that is p L

b kpk Ap p a kpk

where b then we can nd upp er and lower b ounds on our approximation

sto Pf j k The maximum error in the estimates of jR j and

j k k

jT j k can then b e calculated Wehave also shown in Chapter that

the approximations to the inner pro ducts Pf j k and therefore

j k

the approximations to R and T k are secondorder accurate compared

k k

with the approximations to the unknown functions and In other words

requiring the maximum error in jR j and jT j k to b e O is approx

k k

imately equivalent to requiring the maximum norm of the residual errors that is

max fkI LP f k kI LP f kgtobeO Examples illus

k k k k

trating this can b e found in Porter and Stirling Chap

For the results presented in Fig at values of and where our approx

imation to b is greater than zero extremely accurate approximations to R and

T k are such that the maximum error in jR j and jT j k is

k k k

O for all values of and In the case where our approximation to b is

negative then wedonothaveany upp er and lower b ounds on our approxima

tionsto Pf j k However we still know that the approximations

j k

to these inner pro ducts and hence to R and T k are secondorder

k k

accurate compared with the approximations of and Therefore at values

of and where no upp er and lower b ounds exist we use the fact that the er

ror in jR j and jT j k is approximately O max kI LP f k

k k k k

kI LP f k Hence for the results presented in Fig where no

k k

upp er and lower b ounds exist extremely accurate approximations to R and

T k are such that max fkI LP f k kI LP f kg is

k k k k k

O forallvalues of and

In future whenever we sp ecify the accuracy in the approximations to the re

ection and transmission co ecients of any of the mo del equations we shall just

give the maximum error in jR j and jT j k In the case where the upp er

k k

and lower b ounds do not exist this will imply that max kI LP f k

k k

is the same order of magnitude as the maximum allow kI LP f k

k k

able error in jR j and jT j k This will not b e mentioned again in the

k k

next sections but it is the p olicy employed whenever the approximation to b is

negative

Many coastal engineers would consider approximations to the reection and

transmission co ecients generated byChamb erlains metho d as to o accurate for

applications Instead approximations that are correct to two signicant gures

sf would b e considered quite adequate So wenow arrive at the following

question can the accuracy of the Chamb erlain solutions b e relaxed enabling less

accurate solutions of the mo del equations to b e generated over a parameter range

at a much reduced computational cost

Sections and show that this is indeed p ossible for all three mo del

equations of interest We shall concentrate on developing the new approximation

metho d to solve the mildslop e equation Once this has b een completed the

minor changes required for the metho d to encompass the other mo del equations

the Eckart equation and the shallowwater equation will b e given

A onedimensional trial space approxima

tion

Outline of metho d

es the mildslop e equation Chamb erlains metho d summarised in Chapter solv

byconverting it into an integral equation The solution of the integral equa

tion was found through a rank two system of equations once approximations

to the solutions and of the realvalued integral equations and to

the inner pro ducts Pf j k had b een obtained For convenience

j k

we shall recap the variational metho d used to obtain approximations to

and Pf j k as this is where we shall seek to make computational

j k

savings

Now with the function onesigned we dene a new selfadjoint op erator S

Sx xx

where sgn that is The integral equations for k

given by can b e rewritten as

A Sf k

k k

where S and A I S LS

k k

The functionals J L IRk and J L L IR

given by

p S f Ap p k J p

k k

J p p Sf p p Sf Ap p

have stationary values

Sf Pf k

k k k k

Sf Pf

resp ectively Problemdep endent Ndimensional trial functionsp andp given

k n

p a LP f k

k k

are used as approximations to and resp ectively for some a n N

We generate approximations to the inner pro ducts Pfj k bysub

j k

stitutingp andp into the functionals given by and The approxi

mations to Pfj k are secondorder accurate compared with the

j k

approximationsp andp to the unknown functions and The unknown

are chosen so as to make the functionals and constants a n N

stationary within the Ndimensional trial spaces We shall denote the func

tionsp k determined by as k resp ectively and the

k k

functionsp k determined byas k resp ectively

k k

As already noted the Chamb erlain solutions and k are very

k k

accurate approximations to the solutions and of the integral equations

but they also require considerable computer time to determine The integral

equation solution metho d wehave used to solve the mildslop e equation at each

value of and employs this exp ensive pro cess of generating the Chamb erlain

solutions Instead of this we shall use the Chamb erlain solutions at a chosen

and to approximate and in the neighbourhood of and In the

present circumstances we only need to consider problems where is either xed

or is a function of as in the problem considered by Bo oij given in section

where In the following we shall only refer to the value of at

whichwe are solving the problem and we shall not mention as we automatically

know its value once is assigned

A sup erscript is nowintro duced into our established notation to denote the

value of at whicheach op erator function and functional is evaluated

Weintro duce the dimensional trial functions

p r k

k k

as approximations to and for some r IRk determined so as

to make the functional stationary Therefore substituting into

we see that

p J J k A r f S r r

k k k

k k k k k k k

regarded as a function of r is stationary where

Hence the constants r k are given by

f S

k k

k r

k k

f k are and the approximations to the inner pro ducts P

k k

S f

k k

p J k

k k

resp ectively

To nd an approximation to the inner pro duct f P we use the

dimensional trial functions

q k

k k

for some IRk determined so as and as approximations to

to make the functional stationary Therefore substituting into

we see that

J J q q

S S f f A

is stationary where

Hence the constants j are given by

f S

and

S f

and the approximation to the inner pro duct f P is

f S f S

q q J

The approximations to and given by and are very much

quicker to compute than the Ndimensional Chamb erlain solutions given by

when N It is clear that the approximate solutions and

will not b e accurate enough if j j is to o large So where the error exceed

s a given tolerance wechooseanew nd the Chamb erlain solutions there

and then the approximate solutions and in its neighb ourho o d This

pro cess of solving the mildslop e equation over a range of values of canbe

continued in sucha wayastominimise the number of Chamb erlain solutions

required Rememb er from section that the tolerance in the error will b e

given by the maximum allowable error in the amplitudes of the reection and

transmission co ecients

We note that after this new choice of trial functions has b een used to generate

the approximations to and P f j k we revert back

to the same stage in the integral equation pro cedure given in Chapter to nd

the resulting reection and transmission co ecients

Nowwe are in a p osition to use this metho d It is implementedby rstly

assigning the depth prole H the tolerance in the error the initial and nal

values of and the increment to b e added to to give the next at which

a solution is to b e found Chamb erlains metho d outlined in Chapter is used

to solve the mildslop e equation at the initial togive the extremely accurate

solutions remarked ab out in section The increment is then added to and

wenow use the new dimensional metho d to generate the solutions at this

The solutions are then checked to make sure the error is within the user sp ecied

tolerance We carry on in this manner until the error exceeds the given tolerance

and at this we use Chamb erlains metho d again to generate extremely accurate

solutions Then at the next we revert back to the new dimensional trial

functions to generate the solutions again and we carry on in this manner until

solutions of the mildslop e equation have b een found over the desired range

Results

Solutions of the mildslop e equation can b e obtained using this new dimensional

trial space metho d in a greatly reduced computational runtime compared with

using solely Chamb erlains pro cedure at each successive However the

range over which these cheap solutions can b e obtained has an upp er limit

This means that whatever the tolerance sp ecied in the error of the cheap

max

solutions the new metho d still has to use Chamb erlains pro cedure to generate

the solutions at all the desired solution p oints where Therefore no

max

more computational savings are p ossible at these The value of decreases

max

as we increase the tolerance in the error and so increase the accuracy of the

cheap solutions Also for a xed tolerance in the error we nd that the value

of varies from one depth prole to another

max

Consider for example the solution of the mildslop e equation MSE for the

test problem of Bo oij mentioned in section for an incidentwaveofunit

amplitude from the left Here the depth prole is given by

H x x x

In section Chamb erlains metho d was used in generating extremely accurate

solutions of the MSE for this problem with taking values b etween and

at intervals of with given at eachvalue of by to pro duce

the results seen in Fig This required the use of a dimensional trial space

for and a dimensional trial space for The total CPU runtime

required to generate all these results was m s It should b e noted that the

same Sun workstation was used to generate all the CPU runtimes for all the

metho ds used in this chapter

The new metho d is now used to solve this problem for the mildslop e equation

over the same range Wecho ose the tolerance in the error to b e a minimum

of sf accuracy in jR j and jT j k The new metho d pro duces cheap

k k

solutions at the values of in the range For all

max

our new metho d has to use Chamb erlains metho d to generate the solutions

Fig depicts the amplitude of the reected wave generated by b oth metho ds

over the range from to against W the parameter used byBooij

in his corresp onding graph where W As one would exp ect with

these prescrib ed tolerances in the error there is practically no dierence in the

two sets of results The new metho d uses Chamb erlain solutions at values of

which are

and

to generate results over the range from to Thus the number of

Chamb erlain solutions required to generate the results is reduced from to

that is by a factor of This signicant decrease in the number of Chamb erlain

solutions required is reected in the decrease of the CPU runtimes The total e k P er a k v es no er an hiev x v etobe v hiev x b erlain solutions e need to understand x H This happ ens b ecause b erlains metho d is m s b er of Cham um b erlain solutions are not used the yCham e this problem w increases then clearly the new metho d ac max y the new metho d is m s and so the total y This large reduction in CPU runtimes where Cham alue is still small v alues of y the new onedimensional trial functions still ha ere found in tests with other b ed proles So the new metho d t as the reduction in the n vings In order to resolv el of accuracy rises as y the new metho d cannot generate accurate enough solutions at where the initial er if solutions of the MSE that are accurate to sf are desired o t lev b erlain notes that the dimension of trial space required to ac ev max w Similar results w Cham Ho range where the largest Figure Reected amplitude for the depth prole CPU runtime to generateThe these total results CPU b runtime required b used b ecause at the v CPU runtime has b een reduced b is not as signican solutions generated b pro duces solutions accurate to sf in a greatly reduced CPU runtime o calculated there computational sa the reason wh range of a constan 10 |R 1 | − 10 1 − 1 MSE: Chamberlainmethod MSE: New1 − D method W s 10 0

and hence kLP k increase with making more terms in the trial functions

p k given by and consequently larger trial spaces necessary

to b e assured of the desired accuracy Corresp ondinglythenewapproximation

metho d has to resolve the problem via Chamb erlains metho d at more frequent

intervals for larger This o ccurs b ecause as increases kP k increases and

hence x and the solutions and oftheintegral equations change

more rapidly from one value of to the next Fig depicts the approximations

and to the solutions and of from the previous example at

appro two dierentvalues of At the smaller value of the ximation has

a zero whilst at the larger b oth approximations have zeros and the p osition

approximation is dierent from that in the smaller case of the zero in the

Thus it is easy to understand why there is a maximum value of for

max

whichatany the trial approximations C and C where

max

and C j are some constants will b e a p o or approximation to

Hence the minimum practical trial space dimension is two

Atwodimensional trial space approxima

tion

Outline of metho d

The onedimensional trial space metho d describ ed in section used Chamb er

tegral equations lain solutions at a chosen to approximate the solutions of the in

in the neighbourhood of The twodimensional trial space metho d uses

with Chamb erlain solutions at two particular choices of denoted by and

at the intermediate values that is at and to approximate

So we use the twodimensional trial functions

p r s k

k k k

k k

as approximations to and for some r and s IRk determined

k k

so as to make the functionals stationary Therefore substituting into k alues of ov w i k f S k b erlain solutions at t s k r k s k k f k e see that S k k A s k k r k k r r J h h k J k C A and k Figure Comparison of Cham p i k ξ ξ − 1 − 1 k k k 0.5 1.5 0.5 1.5 0.5 0.5 f f − − J 0 1 0 1 1 1 the functional w 0 0 k o system S S k w k k r A k ξ ξ 1 s 1 B for for 0.5 0.5 α x x α k 0 0 =0.5 C A =2 k y the rank t k k r s en b k B C A is stationary where A k 1 1 k s are giv k

ξ ξ − − 2 2 0.5 0.5 0.5 0.5 k 0 1 0 1 s 0 0 k k k and k s k r k k ξ ξ 2 and 2 for for k r A A 0.5 0.5 k α x x α 0 0 =0.5 ts =2 J k k k k 1 1 A A B regarded as a function of Hence the constan

f k are therefore The approximation to the inner pro ducts P

k k

p k resp ectively given by J

To nd an approximation to the inner pro duct P f we use the

twodimensional trial functions

q k

k k k

k k

for some and IRk determined as approximations to and

k k

so as to make the functional stationary Therefore substituting into

the functional we see that

q q J J

S f S f

f S f S

A A

regarded as a function of and k is stationary where

k k

J J

and k

k k

Hence the constants and are given by the rank two system

S A f A

C B B C C B

A A A

S f A A

and the constants and are given by the rank two system

A A S f

B C B C B C

A A A

A A S f

The approximation to the inner pro duct P f is therefore given by

P f j k J q q Once approximations to and

have b een obtained we revert back to the same stage in the integral equation

pro cedure given in Chapter to nd the resulting reection and transmission

co ecients

Clearly the approximations to and given by and are

very muchquicker to calculate than the Ndimensional Chamb erlain solutions

given by when N We shall refer to the solutions generated by the

trial functions and as the cheap solutions It is obvious that the

approximations and to will not b e accurate enough for all

if b ecomes to o large We recall from section that as

increases kP k increases and so x and the solutions of the integral equations

vary more rapidly from one value of to the next Therefore the length

of the interval where the error in the cheap solutions found at the

intermediate is within the sp ecied tolerance will decrease as

increases

and the inner pro ducts The worst approximations to the functions

j k will o ccur when has a value close to the midp oint f P

of the interval Therefore a check is made to see whether the error in

is within the sp ecied tolerance If the the cheap solution at

error at is within the given tolerance then the trial functions

and are used to generate the solutions at all the In

the unlikely case that the error in one of these cheap solutions do es exceed the

tolerance we pro ceed as if the error in the cheap solution at the midp ointof

that is at has exceeded the given tolerance

If the error in the cheap solution at exceeds the given

tolerance the Chamb erlain solutions at are saved and a new set of Chamb erlain

and wecheck the error in the cheap solutions are found at

solution at the midp oint of the interval This pro cess is rep eated

until an interval has b een found in which the error of the cheap solution

at all the intermediate is within the sp ecied tolerance Wenowmoveonto the

next interval where the Chamb erlain solutions are known at

and the value of has to b e chosen If we supp ose weknow the Chamb erlain

solutions at values of given by

b erlain then wecho ose and the whole pro cess starts again If no Cham

solutions are known at values of then is chosen according to the size

of the maximum error in all the cheap solutions found in the previous interval

We use the following rule of thumb to cho ose If the maximum

error in the cheap solutions found in the interval is or more orders

of magnitude smaller than the sp ecied tolerance then is chosen so that the

length of the interval is greater than the length of the interval

If the maximum error is the same order of magnitude as the sp ecied tolerance

then is chosen so that the length of the interval is less than the

length of the interval Otherwise is chosen so that the length of the

interval is equal to the length of the interval The Chamb erlain

solutions are then found at and the whole pro cess starts again

We clarify this situation in the following example Fig depicts a typical

situation In this case the errors in all the cheap solutions found in Interval

Interval 1 Interval 2 Interval 3 Interval 4

Error in ‘cheap’ solution at mid-point exceeds the tolerance

Interval 2A Interval 2B

Maximum error in the Maximum Maximum Maximum error in ‘cheap’ solutions is error in the error in the the ‘cheap’ solutions 1 order of magnitude ‘cheap’ ‘cheap’ is the same order solutions is solutions is less than the tolerance of magnitude as the 3 orders of 3 orders of tolerance magnitude magnitude less than the less than the tolerance tolerance

α 0 (1) (2) (3) (4) (5) (6)

α0 ααα00 0 α00 α

Figure An example situation depicting the values of at whichChamb erlain

solutions are found

are within the sp ecied tolerance No Chamb erlain solutions are known at

and the maximum error of all these cheap solutions is order of

magnitude less than the tolerance Therefore the next value of at which

Chamb erlain solutions are found is chosen so that the length of Interval is

in Fig equal to the length of Interval This value of is denoted by

The error in the cheap solution at the midp ointofInterval is not within the

tolerance Consequently Chamb erlain solutions at midp ointofInterval

in Fig The errors in the are found This value of is denoted by

cheap solutions in Interval A are within the tolerance and as the Chamb erlain

solutions are known at the next interval is B Here the errors

in the cheap solutions are also within the tolerance No Chamb erlain solutions

and the maximum error of all the cheap solutions found are known at

in Interval B is orders of magnitude less than the tolerance Therefore the

next value of at which Chamb erlain solutions are found which is denoted by

in Fig is chosen so that

length of Interval B length of Interval length of Interval

Here the upp er b ound arises b ecause suciently accurate cheap solutions could

not b e obtained at all values of in Interval and therefore could not b e

obtained in Interval if it had the same length as Interval The errors in the

cheap solutions found in Interval are within the tolerance with the maximum

error b eing the same order of magnitude as the tolerance As no Chamb erlain

the next Chamb erlain solutions are found at solutions are known at

where is chosen to make the length of Interval less than the

length of Interval The pro cess continues likethisuntil solutions have b een

found for the required range

We are now in a p osition to use this metho d It is implemented by rstly

assigning the depth prole H the tolerance in the error the initial range

the nal value of the relationship b etween and and the incre

ment to b e added to to give the next at which a solution is to b e found

Chamb erlain solutions are then found at and at and the metho d

outlined in the last few pages is used to generate solutions of the mildslop e

equation over the desired range

Results

The twodimensional trial space metho d w orks excellently with the mildslop e

equation in the sense that solutions satisfying a minimum accuracy requirement

can b e obtained over an range using a small numb er of Chamb erlain solutions

in much less CPU time than just using Chamb erlains pro cedure at each successive

The problem of the onedimensional trial space metho d concerning the small

range of values of over whichcheap solutions could b e obtained do es not

arise

Let us consider the example we tested the onedimensional trial space metho d

on Here we seek the solution of the MSE for an incidentwave of unit amplitude

from x over the depth prole given by

H x x x

We shall use the twodimensional metho d to solve this problem for the values

of between and at intervals of with given at eachvalue of

by We again cho ose the tolerance in the error to b e a minimum of

sf accuracy in jR j and jT j k The twodimensional metho d pro duces

k k

cheap solutions over the range and uses Chamb erlain solutions at

values of given by

and

Thus the number of Chamb erlain solutions required to generate the results has

b een reduced from to that is by a factor of This signicant decrease in

the number of Chamb erlain solutions required to generate the results is reected

in the large decrease of the CPU runtimes The total CPU runtime to gener

ate these results byChamb erlains metho d is m s The total CPU runtime

required by the new metho d is m s which represents a saving of This

reduction in CPU runtimes is not as signicant as the reduction in the number

of Chamb erlain solutions used b ecause at the values of where Chamb erlain

solutions are not used the solutions generated bythenewtwodimensional trial

functions still have to b e calculated The solid higher line in Fig depicts the

reected amplitude given by the mildslop e equation plotted against the param

eter W as used by Bo oij generated by b oth metho ds As one would exp ect

with these prescrib ed tolerances in the error there is practically no dierence

between the two sets of results

Nowthatwehave a metho d that works well with the mildslop e equation we

are in a p osition to make slight amendments so that it encompasses the other two ater and x ww U x art and shallo k x H H euse e y the corresp onding Ec er the depth prole v x arts equation w U k y and b e en b x us in the case of Ec H x Figure Reected amplitude o functions giv mo del equations In fact all that is required is to replace the mildslop e ones Th U 10 10

e shall also solv |R 1 | − − 10 2 1 and als of with − 1 y terv en b e seek the solution of H W x art results w EE: Chamberlainmethod EE: New2 MSE: Chamberlainmethod MSE: New2 k x euse coth x − D method − D method een and at in tfrom w x bet ater equation w W x s ww H 10 er the talud with depth prole giv v 0 alues of q x e of unit amplitude inciden v arts equation o In order to compare the mildslop e and Ec a k In the case of the shallo Ec arts equation for v k for a w Ec and

given at eachvalue of by Using Chamb erlains metho d to gen

erate extremely accurate solutions of Eckarts equation with threedimensional

trial spaces for and sixdimensional trial spaces for has a total CPU

runtime of m s The twodimensional trial space metho d with the tolerance

in the error set to b e a minimum of sf accuracy in jR j and jT j k

k k

uses Chamb erlain solutions at values of given by

and

Therefore the number of Chamb erlain solutions required has b een again reduced

by a factor of The total CPU runtime required is m s which represents a

saving of in CPU time compared with using Chamb erlain solutions at each

value of The dashed lower line in Fig depicts the reected amplitude given

by the Eckarts equation plotted against the parameter W as used by Bo oij

generated by b oth metho ds Again with these prescrib ed tolerances in the error

there is practically no dierence in the two sets of results The dierence in the

Eckart and mildslop e results are discussed in section

As a dierent example we shall solve the shallowwater equation SWE over

the trench whose depth prole is given by

H x x x x

for a wave of unit amplitude incidentfromx We seek the solution of

the SWE for the values of between and at intervals of

Using Chamb erlains metho d with threedimensional trial spaces for and

sixdimensional trial spaces for has a total CPU runtime of m s

The twodimensional trial space metho d with the tolerance in the error set to

b e a minimum of sf accuracy in jR j and jT j k uses Chamb erlain

k k

solutions at values of given by

and

Now the numb er of Chamb erlain solutions required has b een reduced by a factor

of and the total CPU runtime is m s This represents a saving of in

CPU time compared with using Chamb erlain solutions at eachvalue of which

is slightly higher than that achieved in the mildslop e and Eckart examples due to

the larger decrease in the number of Chamb erlain solutions used From Fig

we notice that there is no observable dierence in the results as exp ected with

0.14

0.12

0.1 New 2−D method Chamberlain method

| 0.08 1 |R

0.06

0.04

0.02

0 0 1 2 3 4 5 6 7 8 9 10

α 0

Figure SWE reected amplitude for depth prole H xxx x

the choice in the tolerance of the errors Similar p ercentage savings have b een

found for all three mo del equations on all b ed proles tested with the tolerance

in the error as sp ecied ab ove

This section has shown that the twodimensional trial space metho d signif

icantly reduces CPU runtimes by approximately one third for all three mo del

equations on all depth proles tested Although this is an excellentimprovement

we can do even b etter as shown in the next subsection

Extra computational saving

A further source of computational saving can b e eected without compromising

the approximations already describ ed As already mentioned in this chapter by

substituting trial functions k into the functionals J k given

k k

J S f A k

k k k k k k

we generate approximations to the inner pro ducts

P f k

k k

Similarlyby substituting trial functions k into the functional J

given by

f f J S S A

we generate an approximation to the inner pro duct

f P

The trial functions k are determined by making the functionals

J k stationary within their resp ectful trial spaces Similarly the trial

stationary k are determined by making the functional J functions

within its trial space It was noticed that if k were used in the

functional J to estimate P f this estimate was the same to sf

as that obtained when k are used in J The previous tests that

wehave used to check the accuracy of the reection and transmission co ecients

revealed that the maximum error in jR j and jT j k using b oth choices of

k k

trial function in J was the same order of magnitude with neither choice always

giving the smallest error Porter and Stirling p showhow to obtain

approximations to the inner pro ducts of the form and and the errors

incurred However they did not investigate whether the trial functions generated

P f k gave as go o d an estimate to by the functionals J

when substituted into the functional J as the trial functions generated by the

functional J

It turns out that J and J corresp ond to at least sf

in all problems undertaken so far for the mildslop e Eckart and shallowwater

equations However this result still remains to b e proved in full generality As only

very approximate solutions are b eing lo oked for here then the approach of only

P j k k to estimate f using the trial functions

k k

is quite justied Computationally this device gives a great saving b ecause it

reduces the work load in the cheap solution metho d by nearly one third We

nowtestthetwodimensional metho d with this device incorp orated whichwe

shall refer to as the streamlined twodimensional metho d

Results

We return to the solution of the MSE for an incidentwav e of unit amplitude from

x over the depth prole given by

x x H x

We use the streamlined twodimensional metho d to solve this problem with

taking values b etween and at intervalsofwith given at each

We again cho ose the tolerance in the error to value of by

b e a minimum of sf accuracy in jR j and jT j k The streamlined

k k

twodimensional metho d uses Chamb erlain solutions at the same values of as

the original twodimensional metho d However the total CPU runtime is now

reduced to m s which represents a saving of in CPU time compared

with using Chamb erlain solutions at eachvalue of The reected amplitude

given by the MSE is depicted in Fig and again there is no dierence in the

results

Similarlyifwenowsolve the ab ove problem for Eckarts equation instead of

the MSE then the streamlined twodimensional metho d reduces the total CPU

runtime to m s which represents a saving of in CPU time compared

with using Chamb erlain solutions at eachvalue of Finally if wenowsolve the

previous shallowwater problem given in subsection using the streamlined

twodimensional metho d the total CPU runtime is reduced to m s This

represents a saving of in CPU time compared to using Chamb erlain solutions er v heap range x e do not y further e the error x odimensional w er the v ving in CPU runtime en in this section the Therefore an x ys of solutions W H b erlain solutions found o decrease in the previously b erlain solutions to nd solutions alues of tage ber of Cham In the examples giv um b erlain solutions used in the t b erlain solutions to b e found o er Cham ber of Cham e rise to a small p ercen um y reducing the total n y a factor of or ed b alue of heap solutions noticeable in graphical displa range b hiev hv e note that further reduction of the tolerance in the errors of the c W odimensional metho ds use or less Cham as ac w solutions enabling few will only eect a smallsf reduction tolerance in in CPU the runtimes error already This obtained is with b ecause the the ma jor sa w an t of the mo del equations atreduction in the or n more v Figure MSE reected amplitude for depth prole metho ds will only giv at eac in the c obtained CPU runtimes Also a reduction in the tolerance will mak 10 10 |R 1 | − − 10 2 1 − 1 Chamberlain method Streamlined 2 − D method W s 10 0

therefore pursue this issue further

In conclusion we can see that this new approach has b een very successful

with very large savings in CPU runtimes b eing achieved without lo osing any

observable accuracy in the solutions

Wenow turn our attention to Eckarts equation

Eckarts equation

Eckarts equation which gives an alternative approximation to owover top og

raphy has nondimensional form

d d

U U

dx dx

i t

where Re xe is an approximation to the free surface elevation

U x

and xisgiven by

x coth H

with the dimensionless parameters and given by

h l l

and

g l

The same scaling pro cess used in section with the mildslop e equation has

b een employed here to convert Eckarts equation given in section to the ab ove

nondimensional form

We notice that the terms in Eckarts equation are all given explicitly and

so is much more computationally attractive than the mildslop e equation

in whichthewavenumber is given implicitlyby the disp ersion relation

An advantage of the explicit nature of is reected in the CPU run

times for solving the mildslop e and Eckart equations Consider the example

given in section where the mildslop e and Eckart equations are solved over

x x Using Chamb erlains the talud with depth prole H x

metho d to solve b oth equations at values of and the mildslop e equation

had a runtime of ms and Eckarts equation had one of ms Miles also

revived Eckarts equation when he derived it from a new variational approach

Miles notes that Eckarts equation conserves wave action but unlike the mild

slop e equation do es not conservewave energy except for uniform depth Miles

compares the two approximations through the calculation of reection from a

gently sloping b each of nite oshore depth and nds that Eckarts equation

is inferior to the mildslop e equation in its prediction of the amplitude in the

reection problem if the oshore depth is neither shallow or deep This agrees

with the evidence app earing in Fig where the reected amplitudes of the mild

slop e and Eckart approximations are compared over a talud with depth prole

H x x x The similarity of the mildslop e and Eckart

solutions depicted in Fig for the depth prole H x x x

encouraged attempts to improveEckarts approximation without compromising

its advantageous explicit form

The depth proles of concern here are the ones whichvary only in the interval

for x Thus at and have at b ed depths H for x and H

eachvalue of and in Eckarts equation the travelling waves on the leftright

e e

of the undulating region havewavenumbers dened as

q q

e e

coth H coth H

Eckart notes that the p ositive real ro ot x of the disp ersion relation

a x tanhx

is approximated by

x a cotha

with a maximum dierence of for anyvalue of a Thus at each and the

e e

wavenumb er of a left right incidentwave is not quite the same as the

m m m m

correct wavenumb er used in the mildslop e equation where

is the p ositive real ro ot of the disp ersion relation

m m m m

tanh H tanh H

So in eect at eachvalue of and the mildslop e and Eck art equation are

solving dierent problems The dierence in the mildslop e and Eckart wave

numb ers could b e resolved by simply using the correct wavenumb er given by

the p ositive real ro ot of the disp ersion relation in Eckarts equation instead

of Eckarts approximation to it whichisgiven by However this device

defeats the advantage oered byEckarts equation that each term in the equa

tion was explicit So the issue is whether a new explicit approximation to the

p ositive real ro ot of can b e generated which is more accurate than

This is indeed p ossible

A direct approach is used in which the solution x of is approximated

by adding a small correction term to Thus x is approximated in the form

x x x

where x a cotha and x is a small correction term Substituting for x in

gives

a x x tanhx x

that is

tanhx tanhx

a x x

tanh x tanhx

Then using the expansion tanh x x O x

a tanhx tanhx

a a tanh x tanh x

tanh x tanhx

tanhx tanhx a x x

tanh x tanh x

tanhx x O x

a x x tanh x x O x

x O x tanh x

a x tanh x x O x

and

x x tanhx tanh x x tanh x x x tanhx O x

Thus b ecomes

a x tanh x x x tanh x x x tanhx O x

and neglecting second and higher order terms in x as x is assumed to b e small

gives

a x tanhx

x tanh x x tanh x

Thus the new twoterm explicit approximation to is given by

x x x

a x tanhx

x tanhx x tanh x

which simplies to

x sech x a

x sech x tanh x

Computations have shown that gives the solution x of to machine

accuracy for a and for a the maximum dierence is Thus

is a new explicit approximation to which is a signicant improvement

We shall now replace the function x dened byby

v sechv H

v sech v tanhv

coth H in Eckarts equation and call the resulting where v v x H

equation the new Eckart equation Wenow nd the solution of the new Eckart

equation for an incidentwave of unit amplitude from x over the depth

prole given by

H x x x

We use the streamlined twodimensional metho d to solve this problem with

taking values b etween and at intervalsofwith given at each

Wecho ose again the tolerance in the error to b e value of by

a minimum of sf accuracy in jR j and jT j k The amplitude of

k k

the reected wavegiven bythenewEckart equation is depicted in Fig along

with the corresp onding results from the mildslop e and Eckarts equations We see

that the p eaks and troughs of the reected amplitudes predicted by the mildslop e

and new Eckart equations are almost in line The size of the reected amplitude

predicted by the new Eckart equation is also an increase on that given byEckarts

equation However the reected amplitude given by the new Eckart equation is

still not as large as that given by the mildslop e equation The dierence in results

from b oth is now due to the dierence of the mildslop e and Eckart U functions

As yet no approximation has b een found such as the one used in conjunction

with the wavenumb er functions that can rectify this dierence The new Eckart h en x et b een y in the re x The approac is to use the er a depth prole giv x v H ximations ercome the inferiorit v o appro w h a trial function has not as y en to o er suc er the depth prole ev v w corresp onding to the new U x arts equation for the problem o e b een undertak H k v y Ho en b ely to pro duce the Other attempts ha y ariational principle of Miles with the trial function that generates the equation is therefore an amalgamation of t most lik v Figure Reected amplitude o ected amplitude of Ec b function as giv found

10 |R | 10 ying 1 b ers are the − − 10 2 1 − um 1 en v a New Eckart MSE function as it dep ends y of articial b ed shap es An U x − Eckart ariet art k art and mildslop e w k x y using a v W s 10 0 ould also aect the Ec as replaced b w yp e w and h of this t arts original equation the mano euvre of emplo k instead of to ensure that the Ec In Ec same at approac

on the b ed shap e and so it was hop ed that this device would bring the reected

amplitude into line The at b ed depths for the Eckart problem in x and

x were chosen so that the Eckart and mildslop e wavenumb ers were identical

at each and over these at regions This guaranteed that at eachvalue of

and the mildslop e and Eckart problems had the same incidentwave A

varietyofchoices of depth prole for joining the two at b ed regions together

were used However none of the results over any of the articial b eds caused the

reected amplitude predicted byEckarts approximation to b e any closer to that

predicted by the mildslop e approximation than the reected amplitude obtained

using in Eckarts approximation instead of

Employing in Eckarts equation ensures that this approximation has

practically the correct wavenumber at eachvalue of and and thus the

mildslop e and Eckart equations have practically identical solutions over a at

b ed Currently this is the b est p ossible improvementtoEckarts equation but

it still has an inferior reected amplitude problem as compared with the MSE

The discovery of a trial function for the variational principle by Miles which

delivers the function and a new corresp onding U function should remove

this problem

A nal p ointofinterest in Eckarts approximation is the metho d by whichit

was derived Eckart obtained his equation by rstly transforming the linear

b oundaryvalue problem for the timeindep endentvelo citypotential into

an integral equation He then discarded without justication an assumed small

integral and converted the resulting approximate problem back to a dierential

equation It was hop ed that an investigation into this metho d would yield an

error b ound b etween the full linear integral equation and Eckarts approximate

integral equation However the complexity of the discarded term an integral

whose integrand contained a dierential op erator amongst its terms stopp ed this

line of approach So an error b ound whichshows how accurately the mildslop e

Eckart and shallowwater mo dels approximate still do es not exist

ve scattering problems Eckarts ap In this section wehave seen that for wa

proximation gives an inferior reected amplitude compared with that given by the

mildslop e approximation Currently the b est p ossible improvementtoEckarts

approximation is eected by replacing the function in Eckarts equation

by The new equation still gives a slightly inferior reected amplitude than

that given by the MSE The very accurate explicit approximation to the

ro ot of the disp ersion relation has never b een seen b efore and is an excel

lent rst term to use in an iteration metho d to givemachine accurate solutions

of See Newman for details of such iterative metho ds

Symmetry Prop erties

Here certain intrinsic prop erties of a particular typ e of secondorder dierential

system referred to as symmetry prop erties are derived When the secondorder

system is sp ecied to b e one of the mo del equations mildslop e Eckart or shallow

water the wellknown symmetry prop erties of the reection and transmission

co ecients are found These prop erties were rst derived by Newman for

the full linear problem More recentlyChamb erlain used integral equation

metho ds to derive these prop erties for the mildslop e approximation

Consider the general secondorder dierential equation in the form

Lxr xx mx

where the dierential op erator L is dened by

q xx Lx px x

and where p q r and m are continuous realvalued functions p is dierentiable

and is a given real parameter the prime denotes dierentiation with resp ect to

x Supp ose that satises the b oundary conditions

a b c

C where the constants a b and c i

i i i

Now supp ose that satises together with the b oundary conditions

g e f

e f g

where the constants e f and g i C

i i i

Then an application of Greens theorem on that is

L L dx p

yields

m dx p

a e g c

b f f b

e g c a

b f f b

If mx is set identically to zero then gives relations b etween the endp oint

values of solutions of the dierential equation

We are considering the scattering of plane harmonic waves normally incident

on a given scaled b ed prole H H x that varies only in some nite interval

of x so that

H x

where H and H are constants for a given problem and H x is assumed to

be continuous on In these circumstances all three mo del equations

mildslop e Eckart and shallowwater can b e written in the general form

U U

i t

where Re xe is an approximation to the free surface elevation The

functions U U x and x are as previously dened in each of the mo del

equation cases The dierential equation reduces to x

x where and Therefore as in and to

Chapter we take

i x i x

x A e B e

i x i x

A e B e x

denotes the prescrib ed amplitudes of the incident plane waves propa where A

gating from x resp ectively and B denotes the unknown amplitudes of the

scattered waves propagating towards x resp ectively As already noted in

Chapter the reection and transmission co ecients for an incidentwave from

the left A are dened by

B B

R and T

A A

and those for an incidentwave from the rightA are dened by

B B

and T R

A A

Using the equations and enforcing the continuityof and at x

and x gives the b oundary conditions

i A i

i A i e

Cho osing p q r and m of the dierential equation to b e the corre

sp onding terms in dierential equation that is p U q U r

and m reduces to the identity

e g c a

b f f b

e g c a

b f f b

Now the symmetry relations of the reection and transmission co ecients can

b e easily found Firstlycho ose to b e the solution of and for an

incidentwave of unit amplitude from the left so A A andcho ose

to b e the complex conjugate of Hence satises and the constants in

the b oundary conditions and satised by and resp ectively are

given by

a i e b f c i g

a i e b f c g

and from and we see that the functions and have endp oint

values

and T e R

Substituting and into the identity gives

U i T U i R T R

which is easily rearranged to give the relation

jT j jR j

For the same cho ose to b e the solution of and for an incident

wave of unit amplitude from the rightsoA A This results in the

constants in the b oundary condition satised by to b e redened as

e i f g

e i f g i e

now has endp ointvalues given by

i i

T and e R e

Substituting and into the identity gives the relation

U T U T

Nowcho ose to b e the complex conjugate of the solution of and for

an incidentwave of unit amplitude from the right This redenes the constants

of to b e the complex conjugate of those dened by Similarly the

endp ointvalues of are just the complex conjugate of those dened by

Substituting into the identity results in the relation

R T U T R U

The nal symmetry relation is derived bycho osing to b e the solution of

and for an incidentwave of unit amplitude from the right and

retaining the most recent The constants in the b oundary conditions

satised by are given by

a i b c

a i b c i e

now has endp ointvalues given by

i i

T and e R e

Substituting into the identity gives the nal symmetry relation

jT j jR j

Therefore the reection and transmission given by the three approximations

satisfy these symmetry relations with the and U functions dened according

to the approximation used

These symmetry relations can also b e derived within the framework of integral

equations using an integral equation form of the identity with the same

choices of input functions Chamb erlain has used this approach and it is briey

reiterated here

From Chapter we recall the integral equation namely

i x i x i jxtj

xa e b e e t t dt j

j j j j

where the subscripts have b een intro duced to denote incidentwave direction

from the left or right resp ectively Therefore from Chapter we see that the

constantsa and b j are dened by

j j

b e

U iU

b e e

U U

It follows that L satises

a f b f KP j

j j j j

where the op erators K and P are dened by

i jxtj

e t dt and Pxx x Kx

and all the other constants functions and op erators are as dened in Chapter

Noting that if satises the dierential equation and b oundary conditions that

give rise to the integral equation namely

and

i i a

b i i e

then also satises the same dierential equation and the complex conjugate of

the b oundary conditions Therefore satises the integral equation with

a and b replaced bya and b where

j j j j

e b

iU U

e b e

U U

Now supp ose that L j satises

g KP j

j j j

where g f f for some and j and consider the functional

j j j j j

S given by

g Pg i S P

We denote the adjoint of an op erator T by T Then as P P we see that

i S PI K P PI K P

P I KP P I KP

Substituting x and x into the integral equation satised by j

yields

f tt t dt

j j j j

i i

f tt t dt e e

j j j j

and hence it follows that

S e

This identity is equivalent to the identity whichwas derived within the

framework of dierential equations Before the symmetry relations are rederived

the denitions of the reection and transmission co ecients as given previously

ws in Chapter are restated here for convenience as follo

T e

i i

R e e

where the subscripts distinguish b etween waves incident from the left or the

right resp ectively

and Substituting and therefore a b and

therefore a b inS gives

S j j j j

and employing reduces this to the symmetry relation namely

jT j jR j

By the same pro cedure

S implies U T U T

S implies U R T U T R

S implies jR j jT j

In the nal part of this section we need to recall the rank two system of

equations dened in Chapter This relates and j the end

j j

p ointvalues of the solutions j of the integral equation with

the inner pro ducts Pf j k The j L are the

j k j

solutions of the real valued integral equations

I LP f j

j j

where I is the identity op erator and the op erators L and P are dened by

Lx sin jx tj t dt and Px xx

and where the free terms are dened by f x cos xand f x sin x

In Chapter and the rst part of Chapter wehave used variational techniques

to approximate the values of these inner pro ducts For convenience wegive the

rank two system of equations here namely

c b c b B B b b

CC B C C B B C BB

AA A A A

c b c b B B b b

B B b c b

B B C C B C

i j

A A A

b B B c b

where the values of known constants b and c i are chosen according

i i

to whether corresp onds to the solution of the integral equation for an

incidentwave from the left or right or the complex conjugate of the solution of

for an incidentwave from the left or right as seen earlier in this section

and

A A and B A iA A B

and

Pf j k A

j k jk

Now the reection and transmission co ecients are dened by and

in terms of and j and therefore through the rank two

j j

system of equations in terms of the inner pro ducts Pf j k

j k

When calculating the reection and transmission co ecients it was noticed that

even by using p o or approximations to the inner pro ducts Pf j k

j k

the resulting reection and transmission co ecients still satised the symmetry

relations This prop erty is explained by the following theorem

THEOREM The reection and transmission coecients dened by

and satisfy the symmetry relations for any

approximation to the values of the inner products Pf j k

j k

Pro of

The rank two system of equations is used to express and and

thus the reection and transmission co ecients in terms of the inner pro ducts

Pf j k

j k

Two approaches can now b e used The direct approachistoinvert the rank

two system and substitute the complicated resulting expressions for R and

T i into the symmetry relations After some long but simple algebraic

manipulation one nds that all the terms containing these inner pro ducts cancel

Therefore whatever values these inner pro ducts take the resulting reection and

transmission co ecients which are dened in terms of them will always satisfy

the symmetry relations

A second more illuminating approach follows from rewriting the rank two

system in three parts which are given by

d B B d

j j j

B C B B C C

i j

A A A

e e B B

j j j

where

d b b b

j j

B B C C B B C C

A A A A

e b b b

j j

and

c c c

j j

B C B C B C B C

A A A A

c c c

j j

A substitution of and into yields the original rank two system

For ease of notation we write as

j iW y x

where

B B d d

j j j

C B B C C B

x and W y

A A A

e e B B

j j j

Nowas B is real W W where denotes the conjugate transp ose

It follows that

iy iy y y y y x x iW y W y Wy x

and so the identity

y y x x

results Recapping the ab ove pro cedure the two forms of have b een

used to eliminate the W matrix to give the identity The inner pro d

ucts Pf j k only app ear in the W matrix which only o ccurs in

j k

equation of our breakdown of the rank two system So now an identity

very similar to the previous identity found in our integral equation

framework has b een derived that relates and j and is inde

j j

p endent of the inner pro ducts Therefore this identitywillalways b e satised no

matter what values the inner pro ducts take Finally the symmetry relations are

found by using similar choices of j in to those used earlier

One consequence of this theorem is that the symmetry relations can no longer

b e used as a metho d of checking the accuracy of the computed reection and

transmission co ecients

We cannot prove from the Greens identity approach that the reection and

transmission co ecients given by an approximation to the solution of the b ound

ary value problem and automatically satisfy the symmetry relations

This is b ecause this approach relies on the fact that the righ thand side of the

dierential equation is zero to derive the symmetry relations An approxi

mation to the solution of the BVP and will not satisfy the DE

exactly and so the righthand side of will no longer b e zero Chamb erlain

and Porter consider this problem using a dierent approach They prove

that for an approximation to the full linear wave scattering problem of the form

and the reection and transmission co ecients given byany ap

proximation to the solution of the BVP and automatically satisfy

the symmetry relations It follows that the symmetry relations are an intrinsic

part of the problem rather than of its exact solution in the sense that they are

always satised whatever the accuracy of the solution

In this chapter several extensions to the work app earing in Chamb erlain

have b een presented A new integral equation metho d has b een develop ed

whichsolves the mildslop e Eckart and linearised shallowwater equations over

a range of their parameters in less than one half of the CPU time required by

Chamb erlains integral equation pro cedure Eckarts approximation has b een

investigated and improved and as a bypro duct a new explicit and very accu

rate approximation to the solution of the disp ersion relation has also b een found

Finally after rederiving the symmetry relations of the reection and transmis

sion co ecients of these approximations wehave shown that these co ecients

satisfy the symmetry relations even when they are inaccurately calculated an

unexp ected prop erty

Chapter

A new approximation to wave

scattering

In this chapter a new approximation to the full linear wave scattering problem

is derived A Galerkin approach is used to derive an approximation to the time

indep endentvelo city p otential whichtakes account of decaying wave mo des as

well as progressivewave mo des The present approachusesannterm approxi

mation based on the propagating wave mo de and the rst n decaying wave

mo des over a at b ed If none of the decaying wave mo de terms are used and if

we discard terms that are secondorder on the basis of the mildslop e assumption

jrhj kh where h is the undisturb ed uid depth and k is the corresp onding

wavenumb er then this approach reduces to the mildslop e approximation The

extended approximation is then tested on a selection of b eds of varying steep

ness and the results are compared with the corresp onding results given by the

mildslop e approximation

A Galerkin approximation metho d

satises Recall from Chapter that the timeindep endentvelo city p otential

r hz

on z

rhr on z hx y

where r and r We also require additional conditions

x y z x y

on lateral b oundaries or a radiation condition if the uid extends to innityto

completely sp ecify For the moment we do not concern ourselves with these

additional conditions as our initial aim is to reduce the dimension of the b oundary

value problem for by approximating its dep endence on the z coordinate This

is achieved via a direct application of the classical Galerkin metho d

We seek a weak solution of in the sense that the residual

r is required to b e orthogonal to a given function Inotherwords we require

dxdy

where D can b e any domain in the plane z Integrating by parts gives

r dz dxdy

zz z z

z h

which b ecomes

dxdy r dz rhr

zz z z

z z h

when b oundary conditions and are imp osed on Equation is

aweak form of the b oundaryvalue problem and can b e used to

generate approximations to the solution of that problem

of the form We shall use a Galerkin approximation

x y z x y w x y z

j j

where w j n are given functions and j n are

j j

to b e determined from Wecho ose our given function as

x y z w x y z

for some k n After some simple manipulation which includes

use of the identity

rw r w r w w r r

j j j j j j j j

it is found that the functions k n must satisfy the coupled system

of dierential equations

w w dz

r f r g

j k

w j jk j dz jk j

for k n where

w rw dz f f w w rh w w

k j jk jk j k j k

z h

and

w w

k k

g g w w w w w rhrw w

jk jk j k j k k j j

z z

z z h

w r w dz

k j

Chamb erlain and Porter have used this Galerkin approach with a term

approximation that is n in to derivea newapproximation to that

contains the mildslop e approximation as a sp ecial case They also show that

this new approximation to can b e derived via a variational approach which

is similar to the recentwork of Miles Indeed Chamb erlain and Porter

use the same trial function in b oth the Galerkin and variational approaches The

variational principle used in is L where L is the functional given by

dxdy

r dz

By considering variations whichvanish on the lateral b oundary C h where

C is the b oundary of D itfollows that L is stationary at if and only if

satises So the variational principle L can b e used to generate

approximations In particular if we use the approximation given by

then after imp osing L forallvariations in k n which

vanish on C h weeventually nd after some straightforward manipulation

that k n must satisfy

The theory up to this p oint holds for an y w j n but now

a particular choice of these functions is made Todothiswenow consider wave

scattering problems that are indep endent of the y coordinate The separation

solution of for uniform depth given in section suggests wecho ose

the functions w j n as

ig cosB z h

w w w j n

j j j

cosB h

in which h hx is the undisturb ed uid depth and the functions

B B hB xj n are the rst n real p ositiveroots

j j j

of the relation

B tanB h

j j

arranged in ascending order of magnitude Equation has an imaginary ro ot

B ik and so we can writew in the form

coshk z h

cosh kh

where k k h is the real p ositive ro ot of the lo cal disp ersion relation

k tanhkh

For each xed value of the equations and implicitly dene

B B hj n and k k h resp ectively Notice that these

j j

w j n are an orthogonal set for z h and they satisfy the

same surface condition as namely

on z j n w

It follows that the function w is an approximation to the progressivewave

mo de part of and the functions w w w are approximations

n n

st nd th

to the n decaying wave mo de parts of resp ectively

With this choice for the functions w j n it follows from

equations and that at each x

j k

w w dz

j k

u h j k

where

B h

u h tan B h j n

j j

B sinB h

j j

with B ik The approximate solution x z w satises the

j j

same free surface condition as namely

on z

We recall from Chapter that over a at b ed the general solution of

X X

is given by w Hence over a at b ed w is the only solution

j j j j

j j

of corresp onding to the progressivewavemodeandtherstn

decaying wave mo des Also since the free surface elevation is dened by

i t

x tRe x e

o n

i t

then the approximate solution is such that Re e

Substituting and into reduces to the coupled system

o n

k n g f B u

jk j jk k k

j k k

where the prime denotes dierentiation with resp ect to x B hik h

f xf w w h w w w w dz

jk jk j k j k k

z h

and

i h

g xg w w w h w w w dz

jk jk j k k k

j j

z h

If the second step of the mildslop e approximation is used in which terms

O h jh j are assumed to b e negligibly small then the coupled system

reduces to

n o

B f k n u

k jk k

k k j

where the terms g j k n are omitted b ecause

g O h jh j j k n

See b elow for details

The metho d wehave employed to derive clearly shows that the two

approximations w and g are quite indep endent Therefore we

j j jk

may supp ose that the retention of g j k n gives the coupled

system a wider scop e than and so we fo cus our attention on the

coupled system

In the simple case when is just a term trial function then the coupled

system reduces to

u k u g

This is a relatively new approximation to the progressivewavemodepartof

and equation is known as the mo died mildslop e equation MMSE This

equation was rst derived by Chamb erlain and more recently byChamb erlain

and Porter via the Galerkin and variational pro cedures given earlier in this

section Investigations of the mo died mildslop e equation over a varietyofbed

proles are carried out in Chamb erlain and Porter and in a subsequent pap er

by Chamb erlain and Porter Similarly with the term trial approximation

the coupled system reduces to

k u u

the wellknown mildslop e equation MSE Miles adjusted a variational prin

ciple for nonlinear free surface ows due to Luke so that it applied to lin

earised free surface problems Miles then used this mo died variational principle

to derive the mildslop e equation by apparently discarding a term equivalentto

g in the pro cess The variational principle used by Miles diers from the

one used by Chamb erlain and Porter in that Miles variational principle is for

a realvalued p otential instead

Massel used the same Galerkin metho d as used here with the same choice

for the functions w but allowed the depth function to vary in b oth the x and y

wever allowing h to vary in this waymakes the ab ovechoice for directions Ho

the orthogonal functions w inappropriate in diraction problems for example

as the approximation do es not allow for cylindrical waves which arise in these

problems Massel should have reduced the dimension of the problem by one

by removing the y dep endence b efore making the ab ovechoice of the functions

w b ecause this is the situation that these w are appropriate for as wehave

j j

already demonstrated The system of dierential equations Massel derives do es

however reduce to the system when the y dep endence is removed from

the problem although this is not obvious on rst reading of the pap er b ecause of

Massels complicated notation In his pap er Massel only solves the system

in the simplied case of a term trial function In other words Massel solves the

mo died mildslop e equation Massel left the solution of in the case

of an n term trial function to another pap er whichisyet to app ear

Other approximations for wave scattering byabedofvarying top ography that

include decaying wave mo de terms have b een given by OHare and Davies

and Rey The approximation used by b oth sets of authors is very similar

and involves replacing the b ed prole by a series of horizontal shelves joining at

vertical steps Over each at shelf the velo city p otential has an innite series

representation see Chapter section which contains b oth progressive and

decaying wave mo de terms Continuity of the velo city p otential and its horizon

tal derivative is imp osed throughout the uid depth at the ends of each shelf

which gives a matrix system to solve As a large number of vertical steps is re

quired in order to obtain reliable results the resulting matrix system is large and

consequently b oth these metho ds are computationally exp ensive

The complicated notation used by Massel in his version of the system

is avoided in this approachbyevaluating f and g explicitly

jk jk

Dierentiating and with resp ect to h gives

w B

j j

secB h h B tan B hcosB z h

j j j j

h h

z hB sinB z h

j j

and

h B h sinB h

j j

for j n Now dierentiating and with resp ect to h gives

w cosB z h B B B

j j j j j

h h B tan B h tan B h

j j j

h cos B h h h h

B B B

j j j

z h B z

h h h

B sinB z h B B

j j j j

B tan B hh h

j j

cosB h h h h

B B B

j j j

z h B tanB h

j j

h h h

and

h i

B B

j j

B h sinB h cos B h

j j j

h B h sinB h

j j

for j n It follows that

h j n w

and

w w

j j

h h j n w

h h

Therefore the functions f and g j k n dened by and

jk jk

resp ectively can b e written as

f x w w w dz h

jk j k k

z h

and

w g x dz h

k jk

w w

j j

w dz h w

k k

h h

z h

A little algebra incorp orating the use of relations and shows that the

integrals app earing in and are given by

secB hsecB h k j

k j

B B

dz w

sec B h

sinD D cosD k j

k k k

D sinD

k k

and

B secB hsecB h B B B sin B h

k j j

j B k j k C

k j

dz w

D sinD

h k k

h B B B B

k j k j

and

B sec B h

k k w

D D sin D

dz w k k k

D sinD

k k

D D sinD sin D cosD

k k k k k

D sin D cos

k k

where D B h and B ik It is simple to see that the remaining terms

k k

in and are given by

secB hsecB h j k n w w

k j k j

z h

and

w secB hsecB hB sin B h

j k j j j

w j k n

h B h sinB h

j j

z h

Scaling

Wecho ose the same class of depth proles as in Chapter whicharevarying

only in some nite interval of xWe assume that

h x

h x l

where h h and l are given constants and where hxiscontinuous on

We allow h to have a slop e discontinuity at the ends of the varying b ed that is

at x and x l At the momentwe shall consider the scattering of plane

harmonic waves normally incidentona given depth prole The generalisation

to obliquely incidentwaves will b e dealt with later

The scaling pro cess now employed is the same as that used in Chapter

Therefore welet

hl x H x

u l x k n U x

k k

W x z w l x h z k n

k k

l x k n x

k k

F x f l x j k n

jk jk

G x g l x j k n

jk jk

Again weintro duce dimensionless parameters and k n by

h l and B l

k k

Rememb ering that B ik we also dene the real dimensionless parameter

by i As in Chapter we shall discard the accents from the scaled

indep endentvariable and from the k n in the pursuit of a simple

notation

In terms of these dimensionless quantities the coupled system of equations

o n

k n G F U

jk j k k jk k

j k

where the prime denotes dierentiation with resp ect to x The functions U are

given by

U tan H k n

k k

sin H

k k

and the functions are the p ositive real ro ots of

tan H k n

k k

arranged in ascending order of magnitude and i where is the p ositive

real ro ot of

tanh H

For j k n where j k the functions F and G are given by

jk jk

j k

H H F sec Hsec

j jk k

k j

and

H G sec Hsec H

jk k j

k j

sec Hsec H sin H

j k j j k j k j

C B

H sin H

j j

k j k j

For k n the functions F and G are given by

kk kk

sec H

sinD D cosD H F

k k k kk

D sinD

k k

and

sec H

G sinD D cos D H

kk k k k

D sinD

k k

sec H

k k

D D sinD

k k k

D sinD

k k

D sinD cos D cosD D

k k k k k

sin D sinD H

k k

where D H

k k

If we apply the second approximation step of the mildslop e approximation

to the coupled system then the reduced system is the same as with

just the G terms omitted

Once H and have b een assigned then all the other quantities in equations

n must can b e calculated The functions and k

b e calculated numerically This can b e done using the ecient iterative metho ds

given for example by Newman

The only other information we require is the b oundary or asymptotic condi

tions on k n

Boundary conditions

On the at b ed for x the nondimensional system of dierential equations

reduces to the decoupled set of equations given by

k n

k k

where the notation and k n is used Similarly

on the at b ed for x the system reduces to the decoupled set of

equations given by

k n

k k

where the notation and k n is used In

this section we shall only consider the case of a talud with the simplication

to a hump arising from putting h h and therefore and

k k

k n

The uid domain under consideration extends to innity Therefore we pre

scrib e radiation conditions for k n that are based on the radiation

condition for describ ed in Chapter Hence we assume that two plane waves

propagating from x with known co ecients A resp ectively are incident

on the talud The dimensional analogue of the radiation condition for implies

that the outgoing wave solutions must b e b ounded at x Hence there

heading towards will result outgoing plane waves with unknown co ecients B

ve x resp ectively There will also result n outgoing decaying wa

mo des with n of these heading towards x with unknown co ecients

B k n and with n heading towards x with unknown

co ecients B k n Therefore we assume

i x i x

A e B e x

i x i x

A e B e x

B e x k n

We shall now use this information to give b oundary conditions on by

returning to our original approximation given by and employing the

Galerkin pro cedure again It follows from and that our approxima

tion throughout the uid domain after scaling is given by

x z x

x z

x z x

where

i x i x

x z A e B e W B e W

j j

W x z

j j

i x i x

e B e W B W x z A e

j j

Here wehave used the notation W W z W W zj n

j j

W and the functions W j n are given by where W

j j j

cosh z H cos z H

W x z and W x z

cosh H cos H

n o

The set of functions W j IN is orthogonal for z H and in par

ticular

k j

W W dz

j k

U k j

where U U xisdenedby

j j

We wish our approximation to p ossess as many prop erties of as p ossible

are continuous at the ends of andsowe certainly need to require that and

the talud that is at x and x throughout the uid depth In other

words we require

x H z

x x

x H z

x x

Boundary conditions on j n are now derived from the ab ove

matching equations by employing the same Galerkin pro cedure used in section

to derive the dierential equation system satised by j n

Invoking the continuityof at x gives

n n

X X

C W W H z

j j j

j j

where

A B j

B j n

Multiplying by W for some k n and integrating with

resp ect to z from H to gives

C k n

In the same manner invoking continuityof at x gives

k n C

where

i i

A e B e k

k n e B

As we allow the depth function H xtohave a slop e discontinuityat x

and x thenitfollows that for j n

W W W W

j j j j

and

x x x x

since the depth function H x is constantforx and x Therefore

at x gives invoking continuityof

n n

X X

W W D

j j

j j j

j j

where

A B j i

B j n

Multiplying by W for some k n and integrating with

resp ect to z from H to gives

k n d H tan Hj D

j k k k

jk k k

where

sin z H cos z H

j k

z H dz

d x

cos H cos H

j k

H x

and d d Evaluating the integral in the expression for d gives

jk jk

sec H sec H H

j k j

j k

k k j k j

d x

cos H H

k k

H tan H j k

k k

sin H H

k k k

at x gives Similarlyinvoking the continuityof

k n d H tan Hj D

j k k k

jk k k

where

i i

k e i B A e

B e k n

k k

and d d Substituting into the expression for D and into

jk k

the expression for D gives the coupled b oundary conditions

i H tanh H j d i A

i H tanh H j d A i e

H tan Hj d

k k k k j

k jk

H tan Hj d

j k k k k

jk k

where k n and where the derivatives are evaluated inside the interval

The approximation to the free surface elevation is given by

n o

i t

x t Re e x x

These b oundary conditions also make the approximation to the free surface con

tinuous at x and x However the approximation to the slop e of the free

surface is continuous at x and x only when the slop e of the b ed is also

continuous at x and x

Massel uses the same approachtoderive the b oundary conditions for his

version of the z indep endent system However in his approach Massel

terms when he imp oses his version of the matching condition omits the

at x H z and omits the terms when he

x x x

imp oses his version of at x H z These omissions

x x

imply that Massels b oundary conditions are only correct when

W W

j j

x x

that is when the depth function has a continuous slop e at the ends of the talud

Massel only go es on to give solutions in the case of a term approximation

However as a means of testing the term approximation he considers the talud

problem considered by Bo oij for which Bo oij computed solutions of the full

linear problem Unfortunately this depth function has slop e discontinuities where

the talud joins the at b eds Therefore the results given by Massel for

the term approximation for Bo oijs test problem are wrong b ecause he uses

inappropriate b oundary conditions

In the case of a term approximation the dierential equation system

reduces to the mo died mildslop e equation

U U G

This is the dierential equation that Massel solved with his incorrect b ound

ary conditions Chamb erlain and Porter have also used this equation in

avariety of test problems The ab ove equation reduces to the wellknown mild

slop e equation if the G term is omitted Some of the authors that have used this

equation include Berkho Smith and Sprinks Bo oij Kirby

OHare and Davies Chamb erlain Rey Chamb erlain and Porter

and For b oth the mo died mildslop e and mildslop e equations all the

ab ove authors have used the b oundary conditions which arise from enforcing the

continuityof and at the junctions where the varying depth region meets

the at b eds For the scaling used in this chapter these b oundary conditions are

given by

i i A

i i e A

where A denotes the co ecient of the incidentwave from the right left

None of the ab ove authors returned to the approximation W and required

W and W to b e continuous throughout the uid depth at the junctions

where the varying depth region meets the at b eds which is the approachwe

have used in this section The b oundary conditions used by Massel with

the mo died mildslop e equation corresp ond to the ab ove b oundary conditions

and p ossibly explains why he omits the terms in his derivation of

the b oundary conditions

The b oundary conditions that wehave derived for the term approximation

are

H cosh H

i A i

sinh H H

H cosh H

i i e A

sinh H H

As far as is known these b oundary conditions are completely new and reduce to

only when the varying b ed has a continuous slop e at the junctions with

the at b eds

We shall refer to the sets and of b oundary conditions for the mild

slop e and mo died mildslop e equations as the old set and new set of b oundary

conditions resp ectively

In Section we compare results given by the mo died mildslop e and mild

slop e equations with b oth the new and old sets of b oundary conditions for Bo oijs

test problem We see that the results given by b oth equations with the new

set of b oundary conditions are much closer to the full linear results justifying the

approachwehave used here to derive the b oundary conditions

All that remains to b e done in this section is to dene the reection and

transmission co ecients of the outgoing plane waves whichwe shall denote by

R and T and the decay co ecients of the decaying wave mo des whichwe

k k

shall denote by R and T k n The denitions we shall use

here corresp ond to the denitions of the reection and transmission co ecien ts

used in Chapter for the mildslop e approximation In equation wehave

dened the co ecients of the incidentwaves from x as A resp ectively

Linearity allows us to sup erp ose through the scattering matrix given b elowin

solutions corresp onding to waves incident from x with solutions

corresp onding to waves incident from x Therefore we do not need to solve

the problem with two incidentwaves Instead by solving the problem for only

one incidentwave its amplitude may b e without loss of generality set equal to

unity The notation we use is summarised as follows

B B

k k k k

and T k n If A then R

A A

B B

k k

If A then R and T k n

A A

From equations and we see that A B

i i

A e B e B k n and B e

k k

k n We can use these expressions to write the reection transmis

sion and decaycoecients in terms of the endp ointvalues of k n

as follows

If A

R T

A A

k k

R T k n

A A

If A

R T e

A A

k k

T k n R

A A

We note that for an incidentwave from the left the transmitted decaying

k n and for an incidentwave from wavemodeat x is T e

the right the reected decaying wavemodeat x is R e k n

Therefore these quantities can b e calculated regardless of the size of the functions

k k k k

k n Once the co ecients R RT and T k n

e can deduce values for B k n the have b een determined then w

unknown co ecients of the outgoing plane wave and decaying wave mo des in

terms of A the given incidentwave co ecients The outgoing plane wave

co ecient B comprises of two parts the part of A transmitted b eyond the

talud that is a plane wave and the part of A reected back from the talud that

is a plane wave A similar argument applies to the other outgoing plane wave

co ecient B and to the decaying wave mode coecients B k n

and we can summarise these resulting relationships as follows

k k

B A T R

C B C B C B

k n

A A A

k k

B A T R

The matrix on the righthand side of equation is called the scattering

matrix

We only need to consider the problem of approximating the reection trans

mission and decay co ecients and then the B k n can b e deter

mined through for any A

AsystemofFredholm integral equations

We wish to solve the coupled dierential equation system together with

b oundary conditions Weknow from Chamb erlain that when

the system is a scalar equation that is when is generated using a

term approximation then an integral equation pro cedure can b e used to solve the

b oundaryvalue problem to a high degree of accuracy When the system

is a vector equation that is when is generated using an n n term

approximation an integral equation solution metho d is much more dicult to

implement A metho d of converting the system and b oundary conditions

into a system of Fredholm integral equations is nowgiven The

resulting system of integral equations presents serious problems for numerical

solution metho ds and no attempt is made to solve this integral equation system

here

Chamb erlain exp ends much eort in nding a straightforward metho d to

convert the mildslop e equation and its b oundary conditions into an integral equa

tion He uses a variation of parameters metho d to obtain the integral equation

We can also use this metho d for our system and b oundary conditions

As the idea here is only to indicate the form of the system of integral

equations that results we just give the ma jor steps that o ccur in the conversion

pro cess to the vector integral equation

Weintro duce the variable changes

x x k n

k k

U x

U k n Substituting into and re where U

arranging we nd that the functions k n satisfy the coupled

dierential equation system

N N M

j jk jk jk k k k

j k k

j k

Here wehave used the notation

U G U

k k

k n

k k

U U U

k k k

M U G k n U

jk jk

U U F k n N

jk jk

where G and F j k n are given by equations

jk jk

and the functions k n are given by equations and

As usual wehave used the notation and U U

k k k

k n

The b oundary conditions satised by the functions k n can

b e found in exactly the same metho d as that used to nd the b oundary conditions

for k n in section Omitting the details of this pro cess it

turns out that we can write these b oundary conditions in the form

r k n

k k

k k k

e s k n

k k

k k k

where

r i A H tanh H d

i U

U U e

i i A e H tanh H s

i U U

U U

r d H tan H

k j k k k

e U

s H tan H

k k k k

k k

X u

U U

for k n and where U U k n

The merit of writing the b oundary conditions for k n in the

form given by and is that it is now simple to use a variation of

parameters pro cedure to convert the b oundaryvalue problem into

the integral equation system given by

x x

k k

x r e s e

k k k

n o

jxtj

e t t P t tN t t dt

k k jk j jk

j k

for k n where for j k n and j k

t U

N t t P tM

jk jk jk

U t

To convert this into a Fredholm system of integral equations we need to

removethe terms in the integral whichwe can do on integrating by parts Let

jxtj

e N t tdt

j k

x x

k k

N e N e

jk j jk j

j k

jxtj

sgn x t N tN t tdt e

jk j

k jk

From section we recall that for j k

h i

F x W W H

jk j k

W dz

z H

k j

sec Hsec H H j k n

k j

where W z H cos z H sec Hk n and for j k

k k k

sec Hsec H j k n

k j

W dz

k j

Therefore it follows that we can rewrite F for j k as

W W

j k

F x H x

W W dz

k j

H H

Hence for j k we nd that F and G are related by

W W

k j

H x F x

W dz W

j k

H H

W W

k j

W H x W

W dz W

j k

H H

z H

G G j k n

jk kj

For j k wecannow see that N is given by

U x U

x U

j j

N x M x j k n N xM x

kj jk jk

U x U x U

j k

After substituting R into the system and rearranging we nd that the

functions k n satisfy the system of Fredholm integral equations

given by

x x

k k

x r e s e

k k k

jxtj

e t t Q t t dt

k k jk j

j k

for k n where

N k n r r

jk j k k

j k

s s N k n

k k jk j

j k

and for j k

t U

M t N t j k n sgn x t Q x t

kj jk jk

U U t

We can express this equation in op erator form as

x x x f x L

where

x xx x

i x i x x x x x

n n

x r e s e r e s e r e s e f

n n

and the op erator L is dened by

l x t l x t t

B C B C

L dt B x C B C

B C B C

A A

l l x t t

n n n n

where the terms in the kernel are dened by

jxtj

e t k j

l x t

jxtj

e Q x t k j

It is p ossible to use a variational approach analogous to that used by Cham

b erlain and summarised in Chapter to nd approximate solutions of

However this issue is not pursued here b ecause numerical evaluation of the ker

nel which is discontinuous is formidable and an ideal solution pro cedure has not

b een formulated yet It is for this reason that alternative solution metho ds were

sought

Solution pro cedure

Another approachtosolve the system of dierential equations could b e to

decouple the system by dierentiating with resp ect to xHowever this pro cess

do es not advance the cause b ecause the co ecients in the resulting dierential

equations are a lot more cumb ersome than those app earing in

The solution pro cedure we shall employ is to rewrite the secondorder b ound

ary value problem given by the dierential equation system and b oundary

conditions as a system of rstorder dierential equations satisfying

some initial conditions There is then a great wealth of numerical solution meth

ods available to solve the rstorder system see for example Lamb ert We

convert the secondorder system into a rstorder system byintro ducing

the functions

k n

Therefore the system can b e rewritten as

p x p q

where the n vectors p and q are given by

n n

B C

F G

j j j j

B C

F G

B C

j j j j

B C

F G

jn j jn j n

Let denote the solution of the b oundaryvalue problem

and If we knew the initial conditions that satisfy at

x then wewould only require a numerical metho d to solve one initialvalue

problem given by and these initial conditions to nd an approximation

at x However this is not the case b ecause we only know the b oundary to

value problem satised by Therefore weonlyknow n conditions on and

at x with the other n conditions on and b eing given at x

This means that wehave to resort to nding n indep endent solutions of

and then the solution of the b oundaryvalue problem is given by the particular

linear combination of these indep endent solutions which satises the b oundary

conditions

We can write the b oundary conditions in vector form as

D s

and

D s

Here the n n matrices D and D are given by

d d d

n n n n n

and

d d d

n n n n n

where

d d H tan Hj k n

k k

kk kk

and

d d H tan Hj k n

k k

kk kk

and s are given by The n vectors s

s A

and

s e A

Here wehaveemployed the usual notation j n

j j

and i with and j n the solutions of the relations

and resp ectively

denote linearly indep endent solutions of and Now let

therefore also linearly indep endent solutions of the coupled system Then

as wehave already noted the solution of the b oundaryvalue problem is given

c c c

for some constants c Cj n These constants are chosen so that

given by satises the b oundary conditions and There

fore substituting into and leaves a matrix equation for the

constants c j n whichisgiven by

Mc s

m m with the n where M is a n n matrix given by M m

j nands vectors m given by

B C

j j

B C

B C j n m

j j

and

B C

B C s

the solution of the Once has b een solved then we can nd the value of

b oundaryvalue problem at x and hence calculate the reection transmission

and decay co ecients

All that remains to b e done is to nd the n indep endent solutions of

We shall use a RungeKutta metho d of the form

p p h d q

j n n

in which h is the step size x x x nh p x and p

n n

q q h These RungeKutta x h p q to approximate p

n j js

j n s

numerical schemes are dened on cho osing h and j R

j js

and s j and their use is welldo cumented see Lamb ert for ex

ample The results we pro duce in this chapter will b e found using the stage

metho d R given byFehlb erg which is of order that is accurate

to O h This fthorder RungeKutta pro cedure uses a corresp onding fourth

order RungeKutta pro cedure for step size control Fehlb erg uses the fact that

the dierence b etween his order RungeKutta metho d and the corresp onding

order metho d provides an approximation of the leading term of the trunca

tion error in the order metho d He assumes that if the truncation error is

represented with sucient accuracyby its leading term then a step size control

can easily b e implemented into the order RungeKutta pro cedure A test is

made to see whether the truncation error as obtained from the dierence b e

th th

tween the and order metho ds exceeds a certain preset tolerable error

If it do es the step size is halved the step is recomputed and tested again On

the other hand if the truncation errors are much smaller than the tolerable error

then the step size is doubled Fehlb erg nds this typ e of step size control is quite

reliable b ecause unlike some other metho ds incorp orating step size control it is

based on a complete coverage of the leading term of the truncation error

We exp ect that the use of decaying wave mo de terms to approximate the

scattering problem will cause greater numerical problems than those which o ccur

when the approximation uses only progressivewave mo de terms This is b ecause

for wave scattering problems over any depth prole the decaying wavemode

functions k n satisfying increase with k and are much

larger than the corresp onding progressive mo de function which satises

for any given values of the parameters and We can see this by noting that

the k ro ot of the equation

H H tan H k IN

k k

can b e redened as

k IN H k

k k k

It follows that

k H k k IN

Therefore

k IN

k k

For example with a depth prole given by

cosx x H x

and parameter values and the functions and k

evaluated at x and x are given by

Therefore the inclusion of decaying wave mo des is certainly going to make

the terms in the coupled system more rapidly varying This means that

anynumerical solution metho d will require man y more steps in for a prob

lem including decaying wave mo de terms than a metho d whichsolves the problem

without decaying wave mo de terms to achieve the same solution accuracy Hence

a solution metho d involving step size control should b e more eectiveincontrol

ling the accuracy of the solution in a wave scattering problem involving decaying

wave mo de terms than a xed step metho d

We shall pro ceed bycho osing the simple initial conditions for the n vectors

j ngiven by

j n

and

j n

j n n

j nhave a in the j entry and zeros in the where the n vectors e

rest This choice of initial conditions clearly makes j n linearly

indep endent

The solution pro cedure is to use the RungeKutta metho d to nd approx

imations to j nwith j ngiven by

j j j

and Then wesolve the matrix system to nd the constants

c j n The general solution given by is then constructed

and the reection transmission and decay co ecients are found using

The solution pro cedure was implemented on the MATLAB software package

pro duced by the Math Works Inc This was used b ecause it is purp ose built for

handling vectors and for solving matrix equations The RungeKutta pro cedure

outlined in this section is the default numerical solution metho d for rstorder

systems of dierential equations on MATLAB Full details of how this Runge

Kutta pro cedure is implemented in MATLAB can b e found in Forsythe Malcolm

and Moler

Numerical results

Wenowpresent test examples which use the pro cedure outlined in section

to solve the b oundaryvalue problem given by the dierential system and

b oundary conditions We nd results using one two three and

four decaying wave mo de terms and compare them with the existing results given

by the mildslop e and mo died mildslop e approximations We examine whether

the results converge as the numb er of decaying mo de terms in the approximation

is increased We make no attempt to justify the choices used for the parameters

and asweonlywishtoseeifthenumerical metho d can solve the problem

accurately

In the results presented in this section and in section we refer to the

approximation given by

x z xw x z

j j

which generates the dierential system as the N term approximation

Therefore the term approximation is the mo died mildslop e approximation

The results given by the mo died mildslop e approximation will b e referred to as

the results given by the MMSE and the results given by the mildslop e approxima

tion will b e referred to as the results given by the MSE The b oundary conditions

used with the MMSE and MSE in this section are those given by that is

the new b oundary conditions that were derived in section

For the term approximation we illustrate the convergence of the numerical

metho d as we increase the tolerance in the step size control and go on to give

the solutions of interest Wetheninvestigate the b ehaviour of the initial value

j naswe increase the numb er of terms in the problem solutions

trial approximation and for dierentvalues of the parameters and

Example

Supp ose that a wave of unit amplitude is incident from x on a talud

whose scaled depth prole is given by

H x x x

This depth prole represents a concave talud Wecho ose parameter values

This could represent the physical situation where h m l m gs

We shall seek approximations to the reection co ecient of the progressivewave

and the co ecients of the reected decaying wave mo des at x

Concentrating on the term approximation wenow use the computer pro

j for a series of tolerances gram to generate approximations to

in the RungeKutta metho d The results are presented in Table

Tolerance

C C C C B B B B

A A A A

B B B B C C C C

A A A A

C C C C B B B B

A A A A

Table Approximations to j

We observe that the step size control on the RungeKutta metho d works

well with the approximations to j converging to signicant

gures sf for a tolerance of For all these tolerances the M matrix

in equation has a condition number MATLAB can solve

very accurately to give the constants c j correct to dp that is

s O Table presents the approximation to the amplitude of Mc

the reected plane wave jR j and to the amplitude of the reected decaying wave

mo de at x jR j calculated using the ab ove tolerances in the RungeKutta

metho d We can see that b oth amplitudes of reection have converged to dp

Tolerance jR j dp jR j dp

Table Approximations to jR j and jR j for the term approximation

when the tolerance in the RungeKutta metho d is

Now remember that we wish to compare our results with those given by the

mildslop e equation MSE and the mo died mildslop e equation MMSE We

can either use Chamb erlains integral equation pro cedure to do this or use the

RungeKutta metho d given in this chapter In keeping with the spirit of this

chapter we use the RungeKutta metho d with a tolerance of to nd the

following approximations to the co ecient of the reected plane wave

j i jR MSE R

MMSE R i jR j

These results are accurate to dp in the sense that they agree to dp with

results given by the RungeKutta metho d with an increased tolerance of

Now with a term trial approximation the co ecient of the reected plane wave

using a tolerance of in the RungeKutta metho d is given by

term R i

Weshallnowinvestigate the solutions of the initialvalue problem

and as we increase the numb er of terms in the trial approximation

For the MMSE that is the term approximation the RungeKutta metho d

gives approximations to and as

and

Comparing these solutions of the initialvalue problem and

with those given in Table for the term approximation illustrates that in the

term approximation case the solutions of the initialvalue problem are growing

From the at b ed solutions of the dierential system weknow that the sys

tem has decayinggrowing wave mo de solutions that b ehaveasdecayinggrowing

exp onentials Applying b oundary conditions to then re

moves these growing wave mo de solutions In our solution metho d as we are

solving with some initialvalues that do not corresp ond to the initial val

ues satised by the solution of the BVP then these grow

ing wave mo des are still present and cause the growth of the initialvalue solution

from x to x The degree to which these growing wave mo de terms aect

the solution is certainly dep endent on the magnitude of k n

Wehave already shown that k IN This implies that if we increase

k k

the numb er of terms in the approximation then the magnitude of the solutions

of the initialvalue problem and at x will also increase

If wenow use the term approximation then with a tolerance of in the

RungeKutta metho d we nd that

B B B C C C

A A A

B B B C C C

A A A

clearly illustrating the large growth in the solutions of the initial value problem

from x to x The condition numb er of the matrix M in equation

is now

condM

and MATLAB can calculate the constants c j correct to dp The

co ecient of the reected progressivewaveisgiven by

term R i jR j

The ab ove estimate for R agrees to dp with that given when the tolerance

in the metho d is increased to Notice that this reection co ecient agrees

with the corresp onding reection co ecientgiven by the term approximation

in the rst decimal places

If wenow use the term approximation then with a tolerance of in the

RungeKutta metho d we nd that

O O O O

again illustrating the large growth in the solutions of the initial value problem

from x to x The condition numb er of the matrix M in equation

is now

condM

MATLAB can still solve but the constants c j are only given

correct to dp The co ecient of the reected progressivewaveisgiven by

term R i jR j

agrees to dp with that given when the tolerance in The ab ove estimate for R

the metho d is increased to With a tolerance of in the RungeKutta

metho d the estimate for R is given by

R i jR j term

This estimate for R agrees to dp with that given when the tolerance in the

metho d is increased to So as the size of the solutions of the initialvalue

problem grow we need to increase the tolerance in the RungeKutta metho d in

order to maintain the solution accuracy

Finallyifwe use the term approximation then with a tolerance of in

the RungeKutta metho d we nd that

O O O

O O O O

illustrating huge growth in the solutions of the initial value problem from x

to x The condition numb er of the matrix M in equation is now

condM

MATLAB can still solve but the constants c j are only given

correct to dp The co ecient of the reected progressivewaveisgiven by

term R i jR j

The ab ove estimate for R agrees to dp with that given when the tolerance in

the metho d is increased to

With or more terms in the approximation we nd that the solutions of

the initialvalue problems at x are so large that MATLAB can no longer

solve and so solutions of the BVP cannot b e found It follows from the

relation that once and have b een prescrib ed the maxim um value of

the functions k n o ccurs at the minimum value of H that is

at the minimum depth In this example the minimum depth is at x and the

maximum values are

Obviouslywe wish to stop the solutions of the initialvalue problem

and growing Todothiswe need to know the initial conditions that

the solution of the BVP satises as these initial conditions would remove the

growing exp onential terms from the initialvalue problem However we do not

know these initial conditions and so the question then b ecomes whether we can

improve the situation In other words can wecho ose initial conditions so that the

solutions of the rstorder system do not grow as rapidly as the solutions

of that satisfy the initial conditions and

Wehave already noted that the dierential equation system has decay

er a at b ed with undisturb ed uid depth inggrowing wave mo de solutions ov

H given by

x x

k k

xS e T e x k n

k k k

for some constants S and T k n Initial conditions of the form

k k

k n

k k

clearly remove the growing exp onential term in these at b ed solutions Wehave

found that if we use these n initial conditions with the rstorder system

the solutions of the revised IVP at x are smaller by up to orders in

magnitude than the solutions obtained with the original initial conditions

j n

Wehave not yet found an improved choice for the remaining n initial conditions

and so we retain the present ones Therefore the initial conditions we are now

going to employ with the rstorder system are

j n

and

j n

j n n

j n and d is the zero vector With this choice of e where d

j j

initial conditions the functions j n are clearly linearly indep en

dent

Wenow return to the example at hand For the term approximation using

a tolerance of in the numerical metho d we nd that the solution of

together with initial condition is

B C

which is a reduction of orders in magnitude on the previous solution Now

condM which is a reduction of order of magnitude on the previous

value

For the term trial approximation using a tolerance of in the numerical

metho d the solutions and of with initial conditions given by

are

B C B C

A A

and have b een reduced by orders of magnitude from the previous So

solutions Now condM which is again a reduction of one order in

magnitude from the previous value

For the term approximation using a tolerance of in the numerical

metho d the solutions and of with initial conditions

given by are nowsuchthat

O O O

has b een reduced by order of magnitude has b een reduced by So

orders of magnitude and has b een reduced by orders of magnitude from

the previous solutions NowcondM which is again a reduction of

order in magnitude from the previous solution

For the term approximation using a tolerance of in the numerical

and of with initial conditions metho d the solutions

given by are nowsuchthat

O O O O

is the same order of magnitude as b efore has b een reduced by So

order of magnitude has b een reduced by orders of magnitude and

has b een reduced by orders of magnitude from the previous solutions Now

condM which is again a reduction of order in magnitude from the

previous solution Now MATLAB can determine the constants c j

correct to dp The co ecient of the reected progressivewaveisgiven by

j i jR term R

The ab ove estimate for R agrees to dp with that given when the tolerance in

the metho d is increased to

However the solutions of the initialvalue problem and

for a or higher term approximation are still to o large and so MATLAB cannot

solve the system

Letusnow compare these nterm approximation estimates for R where

n From ab ovewe recall that

MMSE R i jR j

term R i jR j

From these results we can see that as we increase the numb er of terms in the

approximation the estimate of R converges We also notice that jR j given by

the term and term approximations agree to dp

The co ecient of the rst decaying wavemodeevaluated at x given by the

term term term and term approximations calculated using a tolerance

resp ectively are of and

term R i jR j

j i jR term R

term R i jR j

j i jR term R

Again we see convergence in the estimate of R as the numb er of terms in the trial

function is increased with jR j given by the term and term approximations

the same to dp

The co ecient of the second decaying wavemodeevaluated at x given

by the term term and term approximations calculated using a tolerance of

and resp ectively are

term R i jR j

term R j i jR

The convergence is again evident with jR j given by the term and term

approximations the same to dp

The co ecient of the second decaying wavemodeevaluated at x given by

the term and term approximations calculated using a tolerance of are

term R i jR j

which agree to the rst decimal places The co ecient of the fourth decaying

wavemodeevaluated at x given by the term approximation calculated

using a tolerance of is

term R i jR j

The co ecient of the rst decaying wavemodeevaluated at x is one order

of magnitude larger than the corresp onding co ecient of the second decaying

wave mo de As k IN the k decaying wavemodedecays

k k

away more rapidly than the k as x This clearly illustrates that the

rst decaying wavemodeismuch more signicant than the second decaying wave

mo de which will b e more signicant than the third etc

Weshallnowinvestigate the solutions of the initialvalue problem

and for a dierentchoice of the parameters and Ifwenow

assign the following values to and given by

then the maximum values of the functions k n in the interval

are now

So for these values of and the functions k n are much

larger than at the previous values of and Consequently the solutions of

the initialvalue problem and evaluated at x will also b e

larger in magnitude at these values of and Indeed using a tolerance of

in the RungeKutta metho d we nd for a term approximation that

C C B B

A A

C C B B

A A

Now condM where M is the matrix in equation and

MATLAB can determine the constants c j accurate to dp

For a term approximation

O O O

Now condM where M is the matrix in equation and we

cannot now determine the constants c j

In this section wehaveshown that as the numb er of decaying wavemode

terms in the approximation is increased the results given can visibly b e seen to

converge Wehave demonstrated that the solution metho d we are employing to

solve the BVP is restricted by the magnitude of the func

tions k n which limits the numb er of terms we can use in the

approximation to Wehave also seen that the tolerance in the RungeKutta

metho d needs to b e increased as the size of the initialvalue problem solution

s increase in order to maintain solution accuracy The nterm approximation

should give dierent and more accurate approximations to the co ecients of

the scattered waves over steep b ed proles than the MSE and the MMSE b e

cause the decaying mo des are more signicant for steep er b ed proles As the

slop e of the b ed prole reduces the results given by these appro ximations should

b ecome very similar as the eect of the decaying mo des diminishes In section

we examine the results given by these approximations as the steepness of

a depth prole is varied Weshowhow the maximum values of the functions

k n in the interval vary as the steepness of the b ed is

varied We nd that the steep er the b ed prole the smaller the magnitude of

the functions k n As the slop e of the b ed prole decreases so

k n increase in magnitude So the solution metho d given here

can b e used to obtain the results given by the and term approximations

for steep b ed proles which are the results of prime interest We nd that the

results given by the and term approximations converge to those given by

the MMSE b efore the b ed prole b ecomes to o mild for results to b e found These

results are presented graphically in section

Graphical results

In this section we consider three examples where the results are b est presented

graphically In the rst example we plot the approximation to the free surface

at two dierent time intervals for normal incidence In the other two examples

weshowhow the amplitude of the reected progressivewavevaries with the

steepness of the b ed prole We consider two shap es of b ed prole a talud

and a hump and compare results given by the MSE MMSE term term and

term approximations We also compare results given by the MSE and MMSE

with the two sets of b oundary conditions discussed in section

Example

Here the depth prole is given by

cos x x H x

which represents an asymmetric hump whose height is half the still water depth

The parameters and are chosen to b e

The nterm approximation to the free surface elevation is given by

i t

Re e

For an incidentwave of unit amplitude from x the approximation to the

free surface at time t j j INisgiven by

i x i x

e R e R e x

x t Re

x x

i x

T e T e x

The results displayed in Fig were obtained by running the computer program

in the RungeKutta metho d for the MSE MMSE term with a tolerance of 4−term 3−term 2−term MMSE

MSE

Figure Free surface elevation

and term approximations and a tolerance of for the term approximation

With these tolerances the solutions for each approximation had converged to

dp The depth prole is also displayed in Fig and wehave exaggerated the

amplitude of the waves to make the gure clearer The dashed lines represent the

amplitude of the incidentwave and the dotted line represents the undisturb ed free

surface At this instant we observe that destructiveinterference pro duces a wave

which has an amplitude smaller than that of the incidentwavein x All the

decaying wave mo des decayawayvery rapidly as they moveaway from the hump

and the contribution they all make to the free surface shap e at the left and right

hand ends in Fig is negligible We note that the nterm approximation to the

free surface has converged as far as one can tell by the eye when n with

only a small dierence in results given by the MMSE and term approximations

So for these values of and the hump is steep enough for the decaying wave

mo des to make a small but noticeable contribution

The ys the same free surface after t has increased by Fig displa

diagram indicates the presence of constructiveinterference in x Again

there is a small dierence in the results given by the MMSE and the term

approximation 4−term 3−term 2−term MMSE

MSE

Figure Free surface elevation after t has increased by

Example

Wenow return to the talud problem considered by Bo oij in which the

accuracy of the mildslop e approximation was tested as the gradient of the talud

varied This was done by comparing the amplitude of the reection co ecient

jR j given by the MSE with the corresp onding value computed using the full

linearised theory The b oundary conditions used by Bo oij with the MSE

corresp ond to the set given in section The varying depth prole we

are considering here is given by

x x H x

so at x and x where the talud meets the at b eds the slop e of the

depth prole is discontinuous In the unscaled problem Bo oij considered had

at b ed depths h m and h m Bo oij computed jR j using the full

linearised theory at values of a parameter W which denotes the length of the

talud Our parameters and are dened in terms of W by

so that

Returning to the denitions of and we recall that

h l

So with and given in terms of W as ab ove weseethat

It follows that varying W is equivalenttovarying the length l of the talud and

the deep water wave therefore the steepness of the talud at each xed value of

numb er Following Bo oij we seek results for our new approximation for values

of W from to at intervals of As W varies from to

s s

and decreases monotonically Therefore the functions k n

the ro ots of the relations

H H tan H k IN

k k

monotonically increase as W increases Note that the minimum depth in this

example o ccurs at x and so the maximum value of k n in

the interval is at x When W we nd that

When W we nd that

Wesaw in section that when the maximum value of k n was

greater than solutions of the b oundaryvalue problem could not b e found

Therefore as W increases there will b e a value of W at which the function

s s

is large enough to make the solutions of the initialvalue problem for the

term approximation to o large for the system to b e solved to a minimum

accuracy The minimum accuracy we require is three decimal places For the

term approximation the value of W at which the system cannot b e solved

to this minimum accuracy will b e smaller than that for the term as

and so on for the term approximations

Using the solution pro cedure outlined in sections and it turns out that

for the term approximation when W

condM and equation cannot b e solved to the minimum

accuracy wehave sp ecied When W and the results giv

en by the term approximation cannot b e found to the minimum accuracy for

W When W and the results given by the term

s s

approximation cannot b e found to the minimum accuracy for W For

all the nterm approximations we initially use a tolerance in the RungeKutta

metho d of which rises to as W increases in accordance with the size

of the solutions of the initial value problem The initialvalue problem solutions

obtained therefore have all converged to at least signicant gures Remember

that for the MSE and MMSE we do not have a problem with the initialvalue

problem solutions growing b ecause these approximations do not contain anyex

p onentially growingdecaying wave mo des Therefore for the MSE and MMSE

we use a tolerance of in the RungeKutta metho d over the whole W range

whichgives solutions accurate to dp

The rst set of results wegive are those given by the MSE and MMSE with

the old set and new set of b oundary conditions These are presented

in Fig It is clear from the graphs that the results given by b oth the MSE

and MMSE with the new set of b oundary conditions are much closer to

the amplitude of the reection co ecient Bo oij computed using full linearised

theory The results given by the MMSE closely agree with all the computed results

for the full linearised theory The results given by the MSE closely agree with the

computed results for the full linearised theory for W which corresp onds

to taluds with gradientupto This implies that the MSE can give reliable

results for taluds with a maximum gradient of up to rather than a maximum

which Bo oij suggested The results given by the MSE with gradientofupto

b oundary conditions b egin to vary from the full linear ones for small W

b ecause the terms O r h jrhj that were neglected in this approximation are

now clearly not negligible The results given by the MMSE are much closer to

the full linear results for these small values of W as one would exp ect as no

terms of O r h jrhj have b een neglected As W tends to zero tends to

zero which is equivalenttothewavelength b ecoming much larger than the water

depth In other words the limit as W tends to zero is the shallowwater limit

s x H er the depth prole v Figure Reected amplitude o 10 10 |R1 | − − 10 2 1 j − 1 R j yof Full linear MSE witholdBCs MMSE witholdBCs MSE withnewBCs MMSE withnewBCs the MMSE s e conclude that the h closer than an W uc p p p h h W es s h h 10 e the b oundary conditions is the correct 0 j R j t rom the evidence in Fig w bs result and certainly m x ed namely x ximations F t case It can b e seen from Fig that when b deriv h used in section h to Notice also deriv from Fig that the amplitude of the oscillations of h is quite close to Lam hLam alue of the reection co ecien approac approac whic with the new b oundary conditions giv the other appro in the presen ater whic ww p j R j t should tend to the exact v tends to zero the talud is tending to a step and so the amplitude of s W e incidence on a step in shallo v a Also as the reection co ecien for w

given by the MSE and MMSE with the new b oundary conditions is much

larger than that given with the old b oundary conditions Bo oij claimed

that the results given by the full linear theory and the MSE with the old b oundary

conditions were in go o d agreement for W He do es not publish any

values of jR j for full linearised theory for W to conrm this claim but

from the evidence in Fig it is reasonable to supp ose that these results would

be much closer to those given with the new b oundary conditions rather

than those given by the old ones as claimed

In Fig we present the results obtained from the b oundaryvalue problem

−1 10 | 1 |R

2−term 3−term 4−term MMSE Full linear

−2 10 −1 0 10 10

Figure Reected amplitude over the depth prole H x x x

for the MMSE term term and term approximations

It is clear from the graph that as the numb er of terms in the trial function

is increased the closer the results b ecome to the full linear ones The results

given by the term approximation haveconverged to those given by the MMSE

when W which corresp onds to taluds with gradient less than In other

words taluds with gradients of up to are mild enough to make the contribution

from the rst and largest decaying mo de negligible Therefore it is not of

great imp ortance that we cannot nd results for the term approximation for

W as the results will b e the same as those given by the MMSE whichwe

can nd for W

For W that is for gradients less than the results given by the term

approximation have converged to those given by the term approximation So

the contribution from the second decaying mo de b ecomes negligible for taluds

with gradientupto

Finally the results given by the term approximation are practically the

same as those given by the term approximation with no observable dierence

From the similaritybe for W that is for taluds with gradients less than

tween the term and term approximation results we conclude that the nterm

approximation has essentially converged when n and so we do not compute

solutions for the term or higher term approximations

Rey also considers this problem by using a metho d whichapproximates

the depth prole as a series of horizontal shelves separated byabruptvertical

steps The results presented in Fig are in go o d agreement with those found

by Rey

Let us now consider another problem where the depth prole is nowgiven by

ahump

Example

We consider a depth prole given by

H x x x x

This corresp onds to a hump whose heightishalfthestillwater depth which has

slop e discontinuities at x andatx where it meets the at b eds We

wish to solve the b oundary value problem at values of a

parameter starting at ending at with intervals of The parameters

and are dened in terms of by

With these denitions for and varying corresp onds to varying the length

l of the hump or equivalentlytovarying the steepness of the depth prole as in

Example As increases and decreases so as in Example the

functions k n monotonically increase The minimum depth in

this example o ccurs at x and so the maximum value of k n

When we nd that in the interval is at x

and when we nd that

As in Example as increases there is a value of at which the function

is large enough to make the solutions of the initialvalue problem for the term

approximation to o large for the system to b e to b e solved to the minimum

accuracy dp Similarly at smaller values of solutions for the term and

term approximations will not b e available It turns out that for the term

cond M and equa approximation when

tion cannot b e solved to the minimum accuracy wehave sp ecied When

and the results given by the term approximation cannot

b e found to the minimum accuracy for When

and the results given by the term approximation cannot b e found to the mini

mum accuracy for As in Example for all the nterm approximations

we initially use a tolerance in the RungeKutta metho d of which rises to

as increases in accordance with the size of the solutions of the initial

value problem Again we use a tolerance of in the RungeKutta metho d for

the MSE and MMSE whichgives solutions accurate to dp over the whole

range

As in the previous example the rst set of results wegive are for the MSE

and MMSE with the old set and new set of b oundary conditions

These are presented in Fig It is clear from the graphs that the results given

by b oth the MSE and MMSE with the new set of b oundary conditions

are very dierent to those given with the old b oundary conditions The 1

0.9 MMSE with new BCs 0.8 MSE with new BCs MMSE with old BCs 0.7 MSE with old BCs

0.6 |

1 0.5 |R

0.4

0.3

0.2

0.1

0 0 1 2 3 4 5 6 7 8 9 10

Figure Reected amplitude over the depth prole H xx x x

results given by the MMSE with the old b oundary conditions just decayaway

as increases Those given by the MSE with the old b oundary conditions make

small oscillations For the MMSE results with the new b oundary conditions

have practically converged to those given by the MSE There is a clear overall

decreasing trend in these results which is b ecause as increases the physical

water depth b ecomes equal to a large number of wavelengths and consequently

less energy is reected The b ehaviour in the results as tends to zero is a

result of the mo delling b ecause for small values of the hump corresp onds to a

submerged thin step Decaying wave mo de terms are signicant in this case and

as the mildslop e or mo died mildslop e approximations do not contain decaying

wave mo de terms we cannot exp ect reliable results In Chapter we nd second

order accurate approximations to the reection and transmission co ecients for

the full linear wave scattering problem over a hump We nd that the results

given by the MSE and MMSE with the new b oundary conditions are very

similar to these full linear results with those given by the MMSE the closest

In Fig we present the results obtained from the b oundaryvalue problem

0.35

2−term 0.3 3−term 4−term 0.25 MMSE

0.2 | 1 |R 0.15

0.1

0.05

0 0 1 2 3 4 5 6 7 8 9 10

Figure Reected amplitude over the depth prole H xx x x

for the MMSE term term and term approximations

It is clear from the graph that as the numb er of terms in the trial function is

increased the results are converging The results given by the term approxima

tion have nearly converged to those given by the MMSE when In other

words for the slop e of the hump is mild enough to make the contribution

from the rst and largest decaying mo de almost negligible When the

results given by the term approximation havevery nearly converged to those

given by the term approximation So the contribution from the second decaying

mo de b ecomes essentially negligible for Finally the results given by the

term have nearly converged to those given by the term approximation when

Therefore the contribution from the fourth decaying mo de b ecomes

essentially negligible for From the similaritybetween the term and

term approximation results we conclude that the nterm approximation has

essentially converged when n and so we do not compute solutions for the

term or higher term approximations In this example wehave not seen the same

convergence in the results given bythe nterm n approximations as wedid

in the previous example This is b ecause the steepness of a hump of the same

maximum height as a talud and the same length is eectively twice that of the

talud So unfortunately the slop e of the hump do es not b ecome mild enough

for us to see the same convergence in the results given bythe nterm n

approximations as seen in Example b efore the results given by these nterm

n approximations b ecome inaccurate

For small values of the hump we are considering is similar to a thin rect

angular blo ck whose height is half the uid depth Mei and Black considered

this typ e of problem using full linear theory for blo cks of various width and the

limiting case of a thin barrierFor small wavenumb ers their results show that

there is a large dierence in the amplitude of the reected wavegiven by a thin

barrier and the blo cks they considered The results given in Fig by the

and term approximations when that is when the at b ed depth is

times the length of the hump are consistent with those of Mei and Black in the

sense that they lie b etween the corresp onding estimates Mei and Black give for

the thin barrier and the narrowest blo ck they consider where the at b ed depth

is half the length of the step

In Chapter we nd that our decay mo de approximation results for this

hump problem are in go o d agreement with the results we compute using full

linearised theory

We round o this chapter bydeveloping the theory given so far to encompass

obliquely incidentwaves

Obliquely incidentwaves

In this chapter the theory employed so far is only for dimensional problems

In other words it applies to waves that are normally incident on depth proles

which are indep endentofy We shall now generalise this to allowwaves incident

other than normally on such proles Consider a plane wave train arriving from

x whose direction of propagation makes an angle to the normal to the y

axis Let the angle which the transmitted wavemakes to the normal b e denoted

by where in general

If the uid depth in x is less than that in xthen Mei

for example We note that there exists a critical angle whichforany

crit

i h

total reection incidentwave with angle of incidence in the interval

crit

o ccurs a wellknown result in optics We do not consider these total reection

problems here as this would require us to derive alternative b oundary conditions

with whichwe are not concerned We shall consider wave scattering problems

in which part of the incidentwave is reected and part of it is transmitted If

the uid depth in x is greater than than that in x that is if h h

then this is the case depicted in Fig In this case the incidentwave

incident θ0

θ1

transmitted θ0

reflected

x=0 x=1 x

Figure Top view of an obliquely incidentwave approaching a talud for h h

is always transmitted whatever the angle of incidence For an oblique incidence

problem as depicted in Fig we can follow the same pro cedure used in section

to see that the approximation to the free surface elevation is given by

i t

Re e

Here the functions x y k n after scaling satisfy

k k

r U F G k n

k k k k jk jk j

with U U x x F F xand G G xj k n

k k k k jk jk jk jk

dened by and resp ectivelysince H H x

The wavenumb ers and of the incident and the scattered progressive

waves can b e calculated from the disp ersion relation once the parameters

and have b een prescrib ed Similarly the corresp onding terms and

k k

k n of the scattered decaying wave mo des can b e calculated from

The incident and scattered waves havean x and y comp onent with

the incidentwave comp onents in the x and y directions having wavenumb ers

cos and sin resp ectively As the depth prole H is indep endentofy

the y comp onent of the waves cannot change across the talud In other words

x y IR

i y sin

k x y xe n

k k

for some xk n Substituting into and dividing

i y sin

through by e gives

n o

U sin F G k n

k k k jk jk j

k j

with the prime denoting dierentiation with resp ect to x This equation reduces

to our original coupled equation in the event of an incidentwave of normal

incidence The asymptotic b ehaviour of k n can b e summarised

i x cos i x cos

e R e x

i x

T e x

x cos

R e x k n

T e x k n

Here is the xcomp onentofthewavenumb er of the transmitted progressive

wave and k n are the xcomp onents of the corresp onding terms

of the transmitted decaying wave mo des The pro cedure outlined in section

n from can b e used to derive the b oundary conditions for k

equations and The transmitted progressivewavehaswavenumber

and we knowinadvance that its y comp onentis sin Therefore

sin

Similarly the y comp onent of the functions is sin and therefore

sin k n

We can also deduce from the ab ovework that the angle b etween the normal

to the y axis and the direction of the transmitted wave is given by

sin sin

This is analogous to a wellknown result in optics known as Snells law and this

determines whether or

We can nowusethenumerical metho ds outlined in section to solvee

quation together with its appropriate b oundary conditions The required

solutions k n are then found using equation We return to

the problem considered in section where wegave plots of the approximation

to the free surface for a plane wave incident normally on the depth prole

H x cos x x

with parameter values

Now supp ose wehave an incidentwave of unit amplitude from x obliquely

to the normal The graphical repre incident on the b ed at an angle

sentations of the free surface given by the MMSE term term and term

approximations are very similar so we just present the results for the term

approximation here The term approximation to the free surface elevation at

time t j j INisgiven by

i x cos i x cos x cos

e R e R e x

i y sin

Re e

x x x

x i x

e x T e T

Fig displays the approximation to the free surface elevation given by the

term approximation Togive the gure more meaning wehave included the

depth prole used and the amplitude of the waves has b een exaggerated to

improve clarity Notice the presence of constructiveinterference in x

Figure Free surface for oblique incidence given by the term approximation

In the oblique incidence problem there is a p ossibility that at certain values

of the wavenumber and the angle of incidence waves may propagate in

the y direction with amplitude that tends to zero as jxj These trapp ed

waves therefore evade the radiation conditions used and it follows that at these

parameter values the solution is nonunique

It is clear that the numerical metho d that wehave used to solve the b oundary

value problem is restricted by the size of the functions

k n A ma jor ob jectiveofany future work will b e to devel

op a solution routine that overcomes this problem The size of the functions

k n will cause problems in most numerical solution pro cedures

of the b oundary value problem These problems mightbeavoided bydeveloping

an approximate solution metho d to solvetheintegral equation equivalenttothis

b oundaryvalue problem whichwe presented in section

In this chapter wehaveshown that by incorp orating decaying wavemode

terms into the formulation of the original mildslop e approximation to the ve

lo citypotential for the full linearised wave scattering problem we can nd go o d

approximations to the velo citypotential This new approximation has b een com

pared with two older approximations that only contain progressivewavemode

terms namely the mildslop e and mo died mildslop e approximations Wehave

sho wn that for steep depth proles where the decaying wave mo des are signi

cant the results given by the new approximation agreed much more closely with

results Bo oij obtained using full linear theoryFrom the results given in sec

tions and the new decaying mo de approximation of the co ecients of

the scattered waves is seen to essentially converge when the number of decaying

wave mo des included has reached three even for the steep est b ed proles The

milder the depth prole the fewer the numb er of decaying mo des needed for con

vergence until eventually the gradient of the depth prole b ecomes mild enough

to make all the decaying mo des negligible In the course of developing this ap

proximation it has b een found that the b oundary conditions that have b een used

in the past by all authors with the mildslop e approximation are incorrect By

this we mean that an alternative set of b oundary conditions can b e found which

make the mildslop e approximation results agree far more closely with computed

results of the full linear problem than the original ones

Chapter

The full linear problem for a

hump

In this chapter the depth proles that are considered are lo cal elevations in an

otherwise at uniform b ed whichwe refer to as humps A secondkind integral

equation is derived from the full linear b oundaryvalue problem BVP satised

by the velo city p otential whichwas given in Chapter The BVP was dened

there in terms of two orthogonal horizontal coordinates x and y and a vertical

coordinate z We shall b e considering the twodimensional problem whichis

indep endentof y in this chapter

Initially as the kernel of the integral equation presents serious numerical prob

lems to evaluate accuratelywemakeanapproximation The approximation

which is equivalent to one that is used in the mildslop e and mo died mildslop e

approximations is to remove the decaying wave mo de terms to leave just the

progressivewave mo de terms This pro cess makes the kernel of the secondkind

integral equation simple to compute and Chamb erlains integral equation pro ce

dure whichwas reviewed in Chapter can b e used to solveitvery accurately

Unfortunately this approximation is found to b e quite inaccurate when the results

ximations are compared with those from reliable appro

It is found however that the secondkind integral equation for the p otential

can b e converted into a rstkind integral equation for the tangential uid velo c

ity The kernel of the rstkind equation is given in terms of a Greens function

whereas the kernel of the secondkind equation is given by the normal derivative

of another Greens function evaluated on the hump It follows that the kernel in

the rstkind equation is much easier to compute numerically than the kernel of

the secondkind equation It is shown that the rstkind integral equation can b e

derived directly from the corresp onding b oundaryvalue problem for the asso ci

ated stream function A variational principle is used to deliver approximations to

the co ecients of the scattered waves which are secondorder accurate compared

to the approximation of the solution of the rstkind equation The subsequent

results are used to further test the accuracy of the decaymodeapproximation

derived in Chapter and also to test the accuracy of the mo died mildslop e and

mildslop e approximations

A secondkind integral equation

The problem under consideration is that of nding the velo city p otential

x z forwaves incident from x on an arbitrary that is sym

metric or asymmetric hump of nite length l This depth prole is such that

hxh x l and hx h x l The problem domain

is as depicted in Fig

- A A+

- B B + hh0 (x)

x=0 x=l

Figure The problem domain

From Chapter we recall that in this situation satises

r hz

on z

on z hx

and together with radiation conditions in the form where r

x z g

h i

cosh k z h

ik x ik x

x z A e B e as x

cosh k h

and

h i

cosh k z h

ik x ik x

A e B e as x x z

cosh k h

As discussed in Chapters linearity removes the need to solve the problem

for two incidentwaves Therefore for an incidentwave from x A

the co ecients of the reected and transmitted waves are dened as

B B

and T R

A A

Similarly for an incidentwave from x A the co ecients of the

reected and transmitted waves are dened as

B B

and T R

A A

The co ecients of the outgoing waves B for two incidentwaves are given in

terms of these reection and transmission co ecients Following the steps used

in Chapter this relationship is given by

B T R A

B C B C B C

A A A

B R T A

where the matrix on the righthand side of the ab ove equation is called the scat

tering matrix

The secondkind integral equation is derived via Greens identity

r G Gr dxdz dc G

n n

where the domain D and b oundary C will b e dened as required and where

is the outward normal derivative which is dened as

h x

z x

h x

where the prime denotes dierentiation with resp ect to x The Greens function

G is chosen to satisfy

r G x x z z h z

G on z

on z h

together with a radiation condition ensuring that G behaves like an outgoing

waveas jxj In other words

ik x

G e as x

This Greens function is wellknown and one can nd derivations of its integral

or series forms in Wehausen and Laitone and Thorne The innite series

representation of G is given by

ik jxx j

G x z jx z c coshk z h cosh k z h e

jxx j B

z h e z h cos B cos B c

n n n

where

k h sinh k h

c n IN

B h sin B h

n n

Here k is the ro ot of the at b ed disp ersion relation

k tanhk h

are the ro ots of the equation and B

B tanB h n IN

n n

arranged in ascending order of magnitude C

D (x0 ,z0 )

Figure First domain used in Greens identity

If Greens identity is now applied to the domain depicted in Fig

where x z is inside D and the vertical b oundaries are assumed to b e a great

distance from the hump so that the radiation conditions and

apply there then it is simple to show that

h i

cosh k z h

ik x ik x

A e A e x z

cosh k h

z hx

where C is the curved part of C Noticing that dc h x dx then

this equation can b e written as

i h

cosh k z h

ik x ik x

x z A e A e

cosh k h

G G

h x dx

x z

z hx

Equation is an integral representation of x z in terms of x hx

for x l that is in terms of evaluated on the hump Applying Greens

identity to a slightly dierent domain depicted in Fig where x z

now lies on the hump gives

h i

cosh k z h

ik x ik x

A e A e x z

cosh k h

z hx

Here the lefthand side is now zero b ecause r G r inD and the

extra term on the righthand side arises from the line integral around the small

indentation at x z The ab ove equation is a secondkind integral equation for

(x0 ,z0 )

Figure Second domain used in Greens identity

on the hump On the hump z hx and so we can rewrite this equation

i h

cosh k h hx

ik x ik x

A e A e x hx

cosh k h

G G

h x x hx dx

z hx

x z

z hx

Therefore in order to nd the velo city p otential in the uid domain the inte

gral equation must b e solved and its solution then substituted into equation

The kernel of the ab oveintegral equation is not easy to compute numerical

ly b ecause the series which denes it has p o or convergence prop erties Hence we

wish to avoid solving this secondkind integral equation Initiallywe try making

an approximation in which simplies the secondkind integral equation and

this pro cess is outlined in the next section

We nish this section by nding expressions for the co ecients of the reected

and transmitted waves in terms of the velo city p otential evaluated on the hump

Supp ose wehave an incidentwave from x so that A then if in

equation we take the limit x and compare the result with the

e nd that the reection co ecient appropriate radiation condition for w

is given by

B c cosh k h

R x cosh k h hx h

A A

ik x

i sinh k h hx e x hx dx

Similarlyby taking the limit x we nd that the transmission co ecient

is given by

B c cosh k h

T h x cosh k h hx

A A

ik x

i sinh k h hx e xhx dx

For an incidentwave from x so that A by the same pro cess as ab ove

we nd that the co ecients of the reected and transmitted waves are given by

c cosh k h B

h x cosh k h hx R

A A

ik x

i sinh k h hx e x hx dx

and

B c cosh k h

T h x cosh k h hx

A A

ik x

i sinh k h hx e xhx dx

An approximation

The mildslop e and mo died mildslop e approximations to the velo city p otential

are derived on the basis that decaying mo des are negligible It is our intention to

make an approximation in the same spirit as these approximations in the integral

equation Therefore the approximation we make in is to omit all the

decaying wave mo de terms in the Greens function The resulting approximation

x hx to x h x on the hump is the solution of the integral equation

h i

cosh k h hx

ik x ik x

A e x A e

cosh k h

sgnx xh xcoshk h hx cosh k h hx

ik jxx j

i sinhk h hx e xdx

where xx hx and the sgn function is dened by

x x

sgnx x

x x

The approximation to the velo city p otential in the uid is given by

i h

cosh k z h

ik x ik x

A e A e x z

cosh k h

cosh k z h sgnx xh xcoshk h hx

ik jxx j

i sinhk h hx e xdx

We shall refer to this new approximation to the velo city p otential as the integral

approximation

Therefore in order to nd the approximation x z tothevelo citypo

tential in the uid wemust solve the secondkind integral equation and

substitute its solution into equation The kernel of the secondkind in

tegral equation is discon tinuous at x x which will cause problems in

numerical integration routines if we attempt to solvethisintegral equation as

it stands These problems are avoided byusinga variable change and some s

traightforward manipulation to give a new secondkind integral equation whose

kernel is continuous

We dene a new function xby

xsechk h hx cosh k h x

Then equation can b e rewritten as

h i

ik x ik x

A e x A e

c sgn x xh xcosh k h hx

ik jxx j

e xdx sinhk h hx

Wenow dierentiate the ab oveintegral equation twice with resp ect to x to nd

the b oundaryvalue problem satised by Dierentiating once gives

h i

ik x ik x

x ik A e A e h x cosh k h hx x c

h i

ik x x

ik c h xcosh k h hx sinhk h hx e xdx

i h

ik xx

ik c sinhk h hx e xdx xcosh k h hx h

where the prime denotes dierentiation with resp ect to x Dierentiating this

equation for gives

h i

x ux x u x k v x k x x l

where

h x cosh k h hx ux c

and

v x c sinh k h hx

From equation and the equation for we deduce that satises the b ound

ary conditions

ik ik A u

ik l

l ik l ik A e ul l

where the functions are evaluated at x and x l

By writing the dierential equation satised by in selfadjoint form

is of the same form as we see that the b oundaryvalue problem satised by

the b oundaryvalue problem which arises from b oth the mildslop e and mo di

ed mildslop e approximations to the velo city p otential We can therefore use

Chamb erlains integral equation pro cedure whichwas reviewed in Chapter

to nd highly accurate estimates of and thus the reection and transmission

co ecients given byourintegral approximation

Using the same scaling pro cedure as used in Chapter whichisbrieysum

marised as

x Uxlul xVxlv l xHx hl x x l x

l h l

and discarding the accents the dierential equation in terms of these di

mensionless quantities is

i h

x x x U x x U x V x

where

U x C H x cosh H x

V x sinh H x C

sinh

and k l is the p ositive real ro ot of

tanh

and where and are two dimensionless parameters given by

h l

We need to convert into selfadjoint form whichisdonebyintro ducing an

integrating factor J given by

J x exp U xdx

H x sinh H x exp

The selfadjoint form of equation is then given by

x J x U x V x J x x

This equation is of the same form as the mildslop e equation whichwas given in

ChaptersThereforewecansolve this equation using the integral equation

pro cedure Chamb erlain used to solve the mildslop e equation Following this

pro cedure which is outlined in Chapter weintro duce a new variable dened

x x J x x

and nd that satises the secondkind integral equation

i x

x c c c e

i x

c c c e

i jx tj

e t t dt

where

J t J t

U t V t t

J t J t

and where the constants c j are given by

c A c C c H

C H c A c c

From the denition of the integrating factor J it is simple to see that the ab ove

equation for can b e rewritten as

tU tU t V t

Notice that is continuous for t and so the kernel in the secondkind

integral equation for is also continuous

The co ecients of the reected and transmitted waves for our integral ap

proximation are given by replacing in the denitions of the

reection and transmission co ecients for the full linear problem byourintegral

approximation to namely It is then simple to see that the reection and

transmission co ecients for our integral approximation are given in terms of

the solution of the integral equation as follows

For an incidentwave from x so that A the co ecients of the

reected and transmitted w aves are given by

e T

For an incidentwave from x so that A the co ecients of the reected

and transmitted waves are given by

R e

Wenow use Chamb erlains pro cedure to nd approximations to these re

ection and transmission co ecients given by the integral approximation which

are secondorder accurate compared to the estimate of the solution of the inte

gral equation We compare the results given by the integral approximation

with the corresp onding results given by the mildslop e and mo died mildslop e

approximations We shall use the new b oundary conditions derived in Chapter

with b oth the mo died mildslop e and mildslop e equations

Example

Supp ose that a wave of unit amplitude is incident from x on a hump

whose scaled depth prole is given by

H xx x x

This corresp onds to a hump whose height is one quarter of the stillwater depth

and whose slop e is discontinuous at x andat x where it meets the at

b eds Wecho ose parameter values

Using a dimensional trial space we nd that for the integral approximation

maximum error in jR j

R i

jR j

Similarly using a dimensional trial space we nd that for the mildslop e ap

proximation gives

maximum error in jR j

R i

jR j

Finally using a dimensional trial space we nd that for the mo died mildslop e

approximation gives

maxim um error in jR j

R i

jR j

It is clear that the integral approximation predicts a much larger reected ampli

tude than either the mildslop e or mo died mildslop e approximations

Wenow consider an example where we nd solutions given by the mo died

mildslop e mildslop e and integral approximations as wevary the steepness of

the hump depth prole

Example

We consider a depth prole given by

H x x x x

This corresp onds to a hump whose heightishalfthestillwater depth which has

slop e discontinuities at x andatx where it meets the at b eds We

seek solutions given by the mo died mildslop e mildslop e and integral approxi

mations at values of a parameter starting at nishing at with intervals

of The parameters and are dened in terms of by

Wehave already considered this problem for the new decaying mo de approxi

mation in Chapter and recall that with these denitions for and varying

corresp onds to varying the length l of the hump which translates to varying

the steepness of the depth prole

We use the cheap solution metho d given in Chapter to nd results for

each approximation imp osing the usual two signicant gure tolerance in the

error The results are depicted in Fig The reected amplitude given by

0.9 Integral approximation 0.8 Mild−slope approximation Modified mild−slope approximation 0.7

0.6 |

1 0.5 |R

0.4

0.3

0.2

0.1

0 0 1 2 3 4 5 6 7 8 9 10

Figure Reected amplitude over the depth prole H xx x x

the integral approximation has much higher p eaks than the corresp onding p eaks

given by the mo died mildslop e and mildslop e approximations The number of

p eaks in the reected amplitude given by all three approximations is the same

and for large values of that is as the slop e of the hump b ecomes milder

the reected amplitude given by the new integral approximation is closer to that

given bytheothertwo approximations

The evidence given by these two examples clearly suggests that the integral

approximation to whichwas derived by omitting all the decaying wavemode

terms in the kernel of the secondkind integral equation is not equivalent

to either the mo died mildslop e or the mildslop e approximations From the

evidence given in Chapter for Bo oijs talud problem wesaw that b oth the

mo died mildslop e and the mildslop e approximations gave good approximations

to the full linear results When decaying mo des were added to the approximation

the results b ecame even closer to the full linear ones that Bo oij had computed

Similarlywe exp ect that the results given in Chapter by the new decaying

mo de approximation for this currenthump problem are tending to those giv

en by full linear theory as the numb er of decaying mo des is increased From

Fig in Chapter we see that the height of the p eaks in the amplitude of

the reected wave for this scattering problem given by the new decaying mo de

approximations and the mo died mildslop e and mildslop e approximations are

very similar As the p eaks in the amplitude of the reected wavegiven by the

in tegral approximation are nearly twice the heightofthepeaksgiven by b oth the

mo died mildslop e and mildslop e approximations then we conclude that the

integral approximation is a p o or approximation to the full linear problem

If we recall howwe made the integral approximation it is really quite sur

prising that the integral approximation pro duces recognisable results at all We

approximated the kernel of the integral equation which is an innite series

with just the rst term of this series From the results wehave obtained it is

clear that this estimate of the innite series is quite p o or a not surprising fact

Instead of trying to improve the approximation in the full linear integral e

quation we nd that accurate estimates of the solutions of the full linear

problem itself can b e obtained This is achieved by rstly converting the second

kind integral equation for the velo city p otential on the hump into a rstkind

integral equation for the tangential uid velo city on the hump

A rstkind integral equation

Wenow return to the full linearised problem whichwas stated in section The

Greens identity domain under consideration is as depicted in Fig In this pic

z C x

(x0 ,z0 ) D h(x)

s Ch s

x=0n x=l

Figure The Greens identity domain

ture denotes the outward normal derivative the tangential derivativeand

n s

s the arc length along C the curved part of C measured from x z h

The outward unit normal was dened in Chapter in terms of two orthogonal

horizontal coordinates x and y and a vertical coordinate z For the problem

domain we are considering which is indep endentof y the outward unit normal

n at x z onC is given by

h x n

h x

where the prime denotes dierentiation with resp ect to x The unit tangent s at

x z onC is given by

s h x

h x

It follows that the normal and tangential derivatives at x z on C are given by

h x

n z x

h x

and

h x

s x z

h x

Taking the normal derivative of equation the integral representation of the

velo city p otential in the uid gives

i h

cosh k z h

ik x ik x

x z A e A e

n n cosh k h

x z inD

z hx

The exp onential term in the innite series for G guarantees that G is an

innitely dierentiable function of x z andx z for x z x z As

under the integral in the x z isinD and x z ison C thenwe can take

ab ove equation to give

h i

cosh k z h

ik x ik x

x z A e A e

n n cosh k h

n n

z hx

where x z isinD

Notice that the Greens function G dened by can b e written as

Gx z jx z f x x z z h f x x z z

where

jxj ik jxj B

c cosh k z e c cos B z e f x z

n n

k B

As G is a Greens function singular at x z x z it follows that the function

f x x z z is a Greens function singular at x z x z satisfying

r f x z x z h z

The function f x x z z h is regular in D that is it satises Laplaces

equation at all p oints in D Wenow dene a new function L L x z jx z by

Lx z jx z f x x z z h f x x z z

It follows that L is also a Greens function in the sense that

r L x x z z h z

Notice that G behaves like a line source near x z and L behaves like a line

sink near x z From the denition of f it is simple to see that L is given by

the innite series

ik jxx j

L x z jx z c sinh k z h sinh k z h e

B jxx j

c sin B z h sin B z h e

n n n

Lemma Suppose G Gx z jx z and L Lx z jx z aredenedby

G L

and respectively Then and satisfy

n s

G L G L

and

n n s s s n n s

G L

for al l x z and x z provided x z x z In this sense and can

n s

be thought of as conjugate functions

Pro of

This follows directly from nding the ab ove derivatives of the innite series

and for G and L

It follows from Lemma that equation can b e rewritten as

i h

cosh k z h

ik x ik x

A e A e x z

n n cosh k h

z hx

where x z isinD The right hand side of the ab ove equation can b e rewritten

as the tangential derivative of a function by noticing that

h i

cosh k z h

ik x ik x

A e A e

n cosh k h

i h

i sinh k z h

ik x ik x

A e A e

s cosh k h

Therefore equation can b e written as

h i

i sinhk z h

ik x ik x

x z A e A e

n s cosh k h

z hx

where x z isinD Integration by parts gives

L ds ds

s s

z hx z hx

C C

h h

The velo city p otential is certainly b ounded on the b ed and from the innite series

for L it is clear that

L h jx z Ll h jx z x z inD

Therefore for x z inD

ds ds L

s s

z hx z hx

C C

h h

Equation can now b e written as

i h

i sinhk z h

ik x ik x

L ds

A e A e x z

n s cosh k h

z hx

where x z isin D If wenow let x z tend to a p ointonC then this

equation b ecomes

h i

i sinhk z h

ik x ik x

ds L

A e A e

s cosh k h

z hx

where x z ison C Integrating this equation with resp ect to s gives

i h

i sinhk h hx

ik x ik x

c A e L A e ds x l

cosh k h s

where c is a constanttobefound

In the neighb ourho o d of a corner of angle of a b oundary on which takes

prescrib ed continuous values the velo city p otential is given as the sum of the

b ounded separation solutions near r by

a r cos

where r are p olar coordinates relative to the corner a j IN are constants

and has prescrib ed values on and We consider the case when

on the b oundary as depicted in Fig It follows that the tangential

derivativeof on the b oundary is r

αφ=0 θ n

φ=0

Figure Polar coordinate system at a corner

asr since a jr

s r

The same result clearly holds on the b oundary Therefore as the depth

proles C that we are considering are all humps so that it follows that

evaluated on C is zero and therefore b ounded at the ends of the hump that

is at x and x l Itfollows that as Lx z j h forx z onC then

L ds

Substituting x in then gives c and so it follows that the tangential

uid velo cityevaluated on the hump satises the rstkind integral equation

i h

i sinhk h hx

ik x ik x

x l A e ds A e L

s cosh k h

Notice that in the uid the stream function asso ciated with the velo city

p otential is suchthat

and

x z z x

provided the derivatives exist and on the b ed is related to by

and

n s s n

as long as the derivatives exist In other words and are related by the

CauchyRiemann equations in any orthogonal coordinate system We can now

rewrite in terms of the normal derivativeof evaluated on the hump as

h i

i sinhk h hx

ik x ik x

ds A e A e x l L

n cosh k h

This rstkind integral equation for evaluated on C lo oks like the result

of applying Greens identity to and L Indeed in the next section we

show that this is precisely the case and therefore we can givea much more direct

derivation of equation than we did in this section

A direct derivation of the rstkind inte

gral equation

Let us recall that the problem domain we are considering is as depicted in Fig

and that the time indep endentvelo city p otential satises

r hz

on z

on z hx

where r together with the radiation conditions and

x z g

i h

cosh k z h

ik x ik x

as x A e B e x z

cosh k h

and

h i

cosh k z h

ik x ik x

x z A e B e as x

cosh k h

From equations and it follows that satises

r hz

on z

z z

on z hx

together with the radiation conditions

h i

i sinh k z h

ik x ik x

A e B e as x x z

cosh k h

and

h i

i sinh k z h

ik x ik x

x z A e B e as x

cosh k h

Note that the stream function is arbitrary to within a constantandwehave

chosen on z hx that is wehavechosen the streamline to

b e the b ed Wehave also expressed the radiation conditions on at x

in terms of the co ecients of the incident and scattered waves It follows that

the co ecients of the reected and transmitted waves due to an incidentwave

from either x or x are as given in section by equations

We shall show later that the reection and transmission co ecients can

b e expressed in terms of evaluated on C

We shall now derive the rstkind integral equation via Greens identity

r L Lr dxdz L dc

n n

where D is the domain with b oundary C depicted in Fig Following the work

in the previous section the Greens function L is chosen to satisfy

r L x x z z h z

L L

on z

z z

L on z h

together with a radiation condition that L behaves like an outgoing waveas

jxj In other words

ik x

L e as x

Wehave already shown that the Greens function L dened by the innite se

ries satises From equation it is clear that L also satises

conditions

The structure of the Greens function L given by is unusual in that L

is not dened in terms of an orthogonal sequence on h z b ecause

cos B h sin B h i j sin B z h sin B z h dz

i j i j

This raises the question of whether the Greens function L satisfying

can b e determined without rst nding the Greens function G given by

We can show that L is indeed given by without rst nding G by

given by considering a fourier series expansion for

x z jx z d xjx z v z

n n

Here v z c coshk z h v z c cosB z h n IN and the

n n

functions d n IN are to b e found All the other functions are dened

earlier in this chapter

with resp ect to z gives Integrating the ab ove expression for

L x z jx z d xjx z c sinh k z h

d xjx z c sin B z h

n n

where there is no constantofintegration b ecause werequireL on z h

It is clear that L as dened ab ove also satises the free surface condition

From Chapter weknow that the sequence fv n INg is orthonormal

for z h Therefore the unknown functions d are given by

n IN d xjx z

x z jx z v z dz

Integration by parts gives

c L

d xjx z sinh k h

w z dz

k z k

h z

L c

L n

d xjx z n IN sin B h

n w z dz

B z B

n n

z h

z h n IN where w z c sinhk z h and w z c sinB

n n

Dierentiating twice with resp ect to x and then integrating the result by

parts gives

k d xjx z c cosh k h

w z dz

d xjx z c cos B h n IN B

n w z dz

n n n

where the prime denotes dierentiation with resp ect to x

From it follows that

r L w z dz x x w z n IN

n n

Substituting equations and into and employing the equations

k tanh k h and B tan B h n IN we nd that

n n

L L

d c cosh k h d k x x k w z

x z

L L

x x B w z n IN d B d c cos B h

n n n

n n n n

x z

As x z isinD then r L onz and so as L satises the surface

condition then it also satises the surface condition

L L

on z

x z

Therefore the unknown functions d n IN satisfy the ordinary dierential

equations

d d x x k w z k

d B d x x B w z n IN

n n

n n n

These equations can b e solved using variation of parameters to give

ik x ik x

L e M e xx

d xjx z

w z

ik xx ik xx ik x ik x

xx e e L e M e

and for n IN

x x B B

n n

x x M e L e

n n

d xjx z

w z

B xx B xx B x B x

n n n n

e e x x L e M e

n n

Imp osing the radiation condition gives the unknown constants in the ab ove

expressions It follows that

ik jxx j

c sinh k z h e d xjx z

and

B jxx j

d xjx z c sin B z h e n IN

n n

Therefore the Greens function L satisfying is given by

ik jxx j

c L x z jx z sinh k z h sinh k z h e

jxx j B

z h e z h sin B sin B c

n n n

The metho d that wehave used to construct L here can b e used to con

struct the Greens function G given by Notice that L and G are only

dened for h z Hence it is clear that these two Greens function

s are not suitable to b e used with other typ es of depth proles that satisfy

hx h x l such as ripples in the sea b ed or a trench

as h is no longer the greatest uid depth The issue of nding the Greens

functions L and G for depth proles corresp onding to ripples and trenches is a

separate problem and is not pursued here This is the reason whywe just consider

hump depth proles in this chapter

If Greens identity is now applied to the domain depicted in Fig

where x z is inside D and the vertical b oundaries are assumed to b e a great

distance from the hump so that the radiation conditions and

apply there then

ds L

dx L

x z

n n

z z

z hx

z h

dx L

L dz

z z

x x

z h

L dz

L dx

x x

z z

As on z hx x and L on z h then the rst and

third integrals in are zero The fth integral in is nonzero b ecause

and L satisfy a secondorder b oundary condition at the surface z This

integral is given by

ik x ik x

dx L

A e Lj A e Lj

z z

h i

ic sinh k z h

ik x ik x

A e A e

k cosechk h

The sum of the fourth and sixth integrals in the right hand side of is given

dz L

L dz

x x

h i

ik x ik x

A e Lj A e Lj sinh k z h dz

x x

cosh k h

h i

ik x ik x

A e A e

k ic k h sinh k h

cosh k h k cosechk z h k

Substituting these expressions for the integrals into gives

h i

ik x ik x

A e A e

ic k h sinh k h

x z sinhk h

k cosechk z h cosh k h

ds L

z hx

which simplies to

i h

i sinh k z h

ik x ik x

ds L

A e A e x z

coshk h

z hx

If wenow let x z tend to a p ointonC then this equation reduces to the rst

evaluated on C that was derived in the previous kind integral equation for

section namely

i h

i sinhk h hx

ik x ik x

L x l A e ds A e

n cosh k h

From equation which relates and on the b ed we can rewrite this

rstkind equation as

i h

i sinhk h hx

ik x ik x

x l L A e ds A e

s cosh k h

or equivalently as

x hx dx Lj

z hx

i h

i sinh k h hx

ik x ik x

A e x l A e

cosh k h

If the b oundaryvalue problem for has a unique solution then

it is given by solving the ab ove rstkind integral equation

Evans whilst considering water wave scattering by a shelf showed that

as the p ointx z approaches x z

o o n n

G log x x z z log x x z z h

f g

where the notation f g means that is b ounded for all p oints on the

shelf This result holds for any b ed contours including humps The ab ove result

can also b e deduced from equation where we expressed G asasumoftwo

innite series It is simple to showthatonahump as x z x z then

o n

log x x z z

n Rx

where

h x

is the radius of curvature of C at x hx Therefore the ab ove normal

derivative is b ounded for all x l The other logarithm given ab ove is only

singular on C at x z h and x z l h It is simple to show

is that at these p oints the value of log fx x z z h g

n R

and resp ectively

It follows that except for discrete values of the parameters of the problem for

which the homogeneous form of the secondkind integral equation satised

by on C has nontrivial solutions the secondkind integral equation has

a unique solution and therefore so do es the b oundaryvalue problem

for

It is p ossible that there are discrete values of the parameters of the problem for

which the homogeneous form of the secondkind integral equation has non

trivial solutions which corresp onds to trapp ed mo des over the hump However

in all the problems wehave considered there has b een no evidence of suchmodes

nor do we exp ect them for these b ed geometries

on Wehave derived a rstkind integral equation for the tangential velo city

C We actually want to nd approximations to the co ecients of the reected

and transmitted waves due to a waveincident on the hump These co ecients

were dened earlier by equations in terms of evaluated on the

hump We can rewrite these equations in terms of evaluated on the hump as

follows

For an incidentwave from x the co ecient of the reected wavewas

dened as

c cosh k h

R h x cosh k h hx

ik x

i sinh k h hx e x hx dx

Integration by parts gives

h i

c cosh k h

ik x

R x hxe sinh k h hx

A k

ik x

x hxdx e sinh k h hx

As is b ounded on C that is on the hump then R is given in terms of the

tangential velo cityonthehump by

c cosh k h

ik x

e sinh k h hx x hxdx

A k

Similarly the transmission co ecient is dened by

c cosh k h

ik x

e sinh k h hx x hxdx

A k

For an incidentwave from x the co ecients of the resulting reected and

transmitted waves are given by

c cosh k h

ik x

x hxdx e sinh k h hx

A k

and

c cosh k h

ik x

x hxdx e sinh k h hx

A k

We are now ready to solve the rstkind integral equation for evaluated on

C given by for a waveincident from either x and hence calculate

the co ecients of the resulting reected and transmitted waves We do this using

avariational approach after wehave nondimensionalised the problem

Nondimensionalisation

We use the same scaling pro cedure as used in previous chapters which is briey

summarised as follows Let

H x hl x

x z l xh z

Then after discarding the accents from these denitions to simplify the notation

the rstkind integral equation may b e written in nondimensional form as

Lj x dx

z H x

i h

i sinh H x

i x i x

A e x A e

cosh

where x x H x and A A is the amplitude of the incidentwave

from x x The Greens function L is now redened to b e

i jxx j

sinh z sinh z e C L x z jx z

jxx j

z e z sin sin C

n n n

where

C C n IN

sinh sin

n n

k l is the p ositive real ro ot of

tanh

B l n IN are the p ositive real ro ots of

n n

n IN tan

n n

arranged in ascending order of magnitude and where and are two dimen

sionless parameters given by

h l

The co ecients of the reected and transmitted waves due to an incidentwave

from x are given by

C cosh

i x

e sinh H x xdx

C cosh

i x

e sinh H x xdx

and for an incidentwave from x are given by

C cosh

i x

e sinh H x xdx

C cosh

i x

e sinh H x xdx

Avariational principle

We start this section by recasting the rstkind integral equation on the

Hilb ert space L which is the set of all equivalence classes of complexvalued

j j This space Leb esgue measurable functions satisfying the condition

f has p ointwise op erations with inner pro duct dened byf g g and norm

f f Young This space includes functions f dened by kf k

where kf k so equivalence classes of functions are considered to reestablish

the norm Two functions f and g are said to b e equivalent if they are unequal

only on a set of measure zero such a set b eing one which can b e covered bya

sequence of arbitrarily small op en intervals Two equivalent functions are written

f g ae that is f and g are equal almosteverywhere All function equalities

in the Hilb ert space are meant in this almosteverywhere sense

Wenowintro duce an op erator K dened by

Kx k x x xdx x

where k x x the kernel of K is dened by

i jxx j

k x x C sinh H x sinh H x e

jxx j

C sin H x sin H x e

n n n

It is quite simple to show that if

jxx j

C sin H x sin H x e n IN f x x

n n n

then

Z Z

kf k jf x xj dxdx

n n

C e n IN

Hence f L L n IN

O asn see Now n INC is b ounded and for a xed

n n

Wehausen and Laitone for example Therefore it follows that kf k

converges and hence f converges in L L Hence

k x x f x x

converges in L L where

i jxx j

f x x C sinh H x sinh H x e

Wealsohave that for all n IN the functions f are continuous and are therefore

measurable and

Z Z

kk x xk jk x xj dxdx

Z Z

jf x xj dxdx

and so k is called an L kernel on From Porter and Stirling

it follows that the op erator K dened by is a b ounded linear map from

L to itself and that K is a compact op erator and so is not invertible The

adjoint op erator K of K is a unique b ounded linear map from L to itself

with the prop erty that

L K K

As k is an L kernel the adjoint op erator K is given by

K x k x x xdx x

The pro of of this result can b e found in Porter and Stirling From the

denition of the kernel k we can see that k x x k x xand k x x

k x x and so k is called complex symmetric

The rstkind integral equation can now b e written as an op erator

equation in L by dening g as

h i

i sinh H x

i x i x

g x A e A e

cosh

Then is the solution of

K g

in L

Notice that if wenow dene to b e the solution of for a waveofunit

amplitude incidentfromx requiring A A in then the

reection co ecient R is given by

iC cosh

g R

where

i sinh H x

i x

e g x

cosh

Similarlyifwenow dene to b e the solution of for a waveofunit

amplitude incidentfromx requiring A A in then the

reection co ecient R is given by

iC cosh

R g

where

i sinh H x

i x

e g x

cosh

Now g istheintegral of the pro duct of the solution of the op erator equation

and its free term g We can nd a variational principle that will givea

g compared to the estimate of the solution of of secondorder estimate to

C given by Consider the functional J L

J pp g Kp p

It is simple to establish that J g so that given an approximation p to

J p delivers an approximation to g

Lemma The functional J p dened by is stationary at p the

solution of with stationary value g

Pro of

Supp ose that p where represents the error in this approximation

to Then

J pJ

g g K K K

g g K K as k is complex symmetric

g O kk

Hence J is stationary at p with stationary value equal to g Therefore

using a given rstorder accurate approximation p to in the functional J delivers

a secondorder accurate approximation J pto g

Wenow need to consider the choice of our approximation to In other words

given the integral equation K g howdowecho ose p suchthatp

Wecho ose the trial function p in the form

a x px

j j

Cj N are unknown and j N where the constants a

j j

are orthogonal functions that satisfy the same b oundary conditions at x and

x as that is they satisfy

and j N

j j

Wehave already shown in section that evaluated at the ends of C is

zero In other words and Consequentlywecho ose

xsinjx j N

The approximation p a to the solution of the rstkind integral

j j

equation is therefore an element of the subspace E of L whichis

spanned by the orthogonal functions j Ngiven by

The constants a Cj N are determined by using the fact that

the variational principle J is stationary at p Therefore wecho ose p so

that

j N

Hence the unknown constants a Cj N satisfy the N simultaneous

linear equations

a K g n N

j j n n

We can actually evaluate the functional J without needing to nd explicitly

the approximation to This pro cedure is given in Porter and Stirling p

where the stationary value is expressed as the quotient of the determinants

of two N N matrices The entries in the j column of each matrix corresp ond

to an inner pro duct involving Ifwe did not require to know the values of the

constants a j N then the advantage of this metho d for evaluating the

stationary value of J is that it is not necessary to solve the set of simultaneous

equations However in our problem wewant to know the approximation to

so that we can calculate the norm of the residual error of the rstkind integral

equation that is calculate

g a K

j j

This norm gives an indication of the accuracy of the approximation Kp to K

a K Unfortunatelyeven if we show that g decreases as the di

j j

mension of the trial space N increases it do es not follow that the approximation

p a converges to as N increases b ecause K is not invertible How

j j

ever if weshowthatg a K decreases as N increases then this gives

j j

an indication that p maybeconverging to The unavailability of a pro of that

N is the price wehavetopaybycho osing a converges to as p

j j

to solve the computationally attractive rstkind integral equation for

rather than the computationally dicult secondkind integral equation for

evaluated on C

We can obtain secondorder accurate approximations to the transmission co

ecients directly from the secondorder accurate approximations to the reec

tion co ecients using the symmetry relations these co ecients satisfyItmaybe

shown by a simple application of Greens identityto and its complex conjugate

for an incidentwave from x that

jR j jT j

Similarlyby applying Greens identityto and its complex conjugate for an

incidentwave from x itmay b e shown that

jR j jT j

Applying Greens identityto for an incidentwave from x and to for

an incidentwave from x gives

T T

Finally applying Greens identityto for an incidentwave from x and

the complex conjugate of for an incidentwave from x gives

R R T T

These relationships are wellknown with rst derived by Kreisel and

the remainder derived by Newman

From equations it is simple to deduce that

jR j jR j

and

R R T n n ZZ

where z is the argument of the complex number z

Once wehave calculated the secondorder accurate approximations and

to R and R resp ectively the secondorder accurate approximation to T

is given by substituting these values into equations and The second

order approximation to T is then given by equation

Wearenow in a p osition to showhowwe implement this solution pro cess

Numerical solution metho d

In order to determine the approximation to the reection co ecients and to

we see from equation that we need to calculate the inner pro ducts

g j N

and

K j n N

j n

In the previous section weproved that the series for the kernel k of the

op erator K converged in L L It follows that wecanswap the

order of integration and summation in K In other words K can b e written

j j

i jxx j

w x w xe xdx K x

j j

j N

jxx j

C w x w xe xdx

n n j

where the functions w xn IN are dened by

w x sinh H x

w x sin H x n IN

ve form is that byintegrating b efore The advantage of writing K in the ab o

taking the sum weavoid the numerical problem of calculating the kernel of K

at x x where the kernel is logarithmically singular We nd that we only

need to take terms in the innite series for K for the series to converge to

dp for anyvalues of the parameters and This feature is illustrated in

the results section

For ease of computation we rewrite the functions K and inner pro ducts

K j n N in terms of realvalued integrals Since

j n

ijxx j

ie i cos x x sinjx x j x x IR

and IRj N then w e can write K in real and imaginary parts

j j

K x K x i K x j N

j r j i j

where

K x w x w xsin jx x j xdx

jxx j

xdx w x w xe

n n

and

K x w x cos x w x cos x xdx

sin x w xsin x xdx

Similarlywe write K j n N in terms of real and imaginary

j n

parts as

K K i K j n N

j n r j n i j n

g j n N in terms of real and imaginary parts Finally we express

For a wave of unit amplitude incident from x A we write

g g f i f j N

j j j j

and for a wave of unit amplitude incident from x A we write

g g f i f j N

j j j j

where the functions f and f are given by

sinh H x sinh H x

sin x and f x cos x f x

cosh cosh

It is clear that once the realvalued inner pro ducts on the righthand side of

have b een calculated to give g j N then as these are the same

inner pro ducts that app ear in the righthand side of we can nd g

j N without calculating any more inner pro ducts In other words we

only havetondasolutionforawave incident from x andwe can then

nd the solution for a wave incident from x without having to calculate any

more inner pro ducts

The functions K and K can only b e approximated at a nite number of

r i

P P

N N

p oints x Therefore the approximations p a and Kp a K

j j j j

j j

are only known at a nite number of points We use p oint comp osite Gauss

Legendre quadrature to calculate the integrals b ecause it delivers highly accurate

answers very economically for wellb ehaved integrands Details of this quadrature

rule can b e found in Johnson and Riess for example

Chamb erlain used GaussLegendre quadrature to calculate

f x sin jx x j dx I x

where f and its rst derivative are continuous realvalued functions He found

that the quadrature rule was hamp ered by the presence of the slop e discontinuity

which exists in the integrand at x x Chamb erlain showed that an ecient

remedy is given by writing

Z Z

I x f x f x sin jx x j dx f x sin jx x j dx

The integrand in the rst term of the righthand side has a continuous rst

derivative and the integral in the second term can b e found explicitly and is

given by

cos cos x sin jx x j dx

Details of how this pro cess improves the p erformance of the quadrature rule are

given in Chamb erlain Similarlywe write

Z Z Z

jxx j jxx j jxx j

e dx f x f x e dx f x f xe dx

so that the integrand in the rst term on the righthand side has a continuous

rst derivative and the integral in the second term is given by

h i

jxx j x x

e e e dx

The integration routine in our computer program therefore calculates the integrals

in K by using and

The solution pro cess is now clear Once the depth prole H and the pa

rameters and have b een chosen we calculate the functions K and K

r j i j

j N The inner pro ducts K j n N are then calcu

j n

lated byevaluating their comp onent realvalued inner pro ducts Similarly g

j N are calculated for an incidentwave from x and from

x We then substitute these inner pro ducts into equation and solve

it to give the constants a j N We then nd the approximation

s to the reection and transmission co ecients formulate p a and

j j

Kp a K and evaluate the residual of the rstkind integral equation

j j

that is evaluate kKp g k

Results

In this section we nd approximations to the reection and transmission co e

cients of the scattered wavesduetoawave of unit amplitude incident from either

x or x on a varietyofhump depth proles In the rst example we

vergence of K where j N givenumerical results and illustrate the con

and the convergence of the approximation to the reection co ecientasthenum

b er of terms in the series for K is increased After concluding that terms in

the series gives suciently accurate results we then illustrate the convergence of

the approximation to the reection co ecient and its mo dulus as the number of

terms in the trial function is increased Finallywe present some graphical results

compare them to results from previous mo del equations such as the mildslop e

equation and give some conclusions

Example

Supp ose that C is given by the depth prole

H xx x x

This corresp onds to a hump whose height is one quarter of the stillwater depth

with slop e discontinuities at x and x Wecho ose parameter values

and

Firstlywe shall examine the convergence of K K where

r j j

n n

w x K x w x sin jx x j xdx

j j

and

r jxx j

K x w x w xe xdx n M

j n n j

as M is increased

Instead of showing the convergence of K x forchosen values of x in

r j

weshow the convergence of kK k as M increases to giveanoverall

r j

P P P P

M M M M

r r r r

M K K K K

n n n n n n n n

Table Illustrating the convergence of K j

n n

impression of the convergence of K x in as M increases Table

r j

P P

M M

r r

gives K and K for j and j asM takes values from

j j

n n n n

to We can see from Table that for this depth prole and parameter

values kK k j converges quite slowly but has converged to dp

r j

when M

Now let us examine the convergence of the approximation to the reection

co ecientas M increases Table gives the approximation to the reection

co ecient R delivered by a term and an term trial approximation as M

takes values from to It is clear from Table that when M the

approximation to the reection co ecientdelivered by the term trial function

has converged to dp and that given by the term trial function has practically

converged to dp as well Similarlywe nd that for all choices of the parameters

and of interest and all depth proles tested using M in the series for

K gives an approximation to the reection co ecient which has converged to

M R ter m R ter m

i i

Table Illustrating the convergence of the approximation to R as M increases

dp for anynumb er of terms in the trial function p Therefore for the results

presented in the rest of this section we use M in the series for K

Wenow examine the b ehaviour of the approximation to the reection co e

cient R as we increase the numb er of terms in the trial function Table gives

the approximations to R jR j and kKp g k as the numb er of terms N in the

trial function is increased From Table it is clear that a term trial function

M R jR j kKp g k

Table Illustrating the convergence of the approximation to R as N increases

delivers an approximation to the reection co ecient that has converged to dp

We also note from Table that as the numb er of terms in the trial function

is increased so the norm of the residual error in the rstkind integral equation

kKp g k decreases This gives an indication that p maybeconverging to

For all other hump depth proles and values of parameters and wehave

found that a term trial function gives an approximation to the reection and

transmission co ecients that is correct to dp in the sense that the estimates of

K j Nhave converged to dp This accuracy is quite satisfactory

r j

and so the results given in the rest of this section are obtained using a term

trialfunction

The approximation to the reection co ecient R for this problem is

R i

Using the metho d describ ed in Section we nd that the transmission co e

cients are given by

T T i

In Table we compare the reection and transmission co ecients of the

ves given by the mildslop e MSE mo died mildslop e MMSE scattered wa

with b oth sets of b oundary conditions whichwere given in Chapter and the

nterm n trial approximations whichwere derived in Chapter with

the estimates wehave obtained for full linear theory It is clear from Table

that the results given by the MSE and MMSE with the new b oundary conditions

are in much b etter agreement with the estimates wehave obtained for full linear

theory than the results given by the MSE and MMSE with the old b oundary

conditions We can also see that as we increase the number of decaying mo des in

the approximation to the full linear velo city p otential the results b ecome closer

to those obtained for full linear theory

Wenow compare the estimates to the solutions of the full linear problem

with those given by the ab ovementioned approximate mo dels over a range of

parameter values In the rst example we return to the wave scattering problem

for a hump that was considered in Chapter

Approximation R T

Full linear i i

term i i

MMSE new i i

MSE new i i

MMSE old i i

MSE old i i

Table A comparison of estimates of R and T

Example

The depth prole is given by

H x x x x

This corresp onds to a hump whose heightishalfthestillwater depth which has

slop e discontinuities at x andat x where it meets the at b eds We seek

solutions given by the mildslop e mo died mildslop e and nterm n

approximations and estimates to the solutions of full linear problem at values

of a parameter starting at nishing at with intervals of The

parameters and are dened in terms of by

With these denitions for and varying corresp onds to varying the length

l of the hump which corresp onds to varying the steepness of the depth prole

Using a term trial function to give estimates of the reection and trans

mission co ecients for the full linear problem we nd that the maximum value

of kKp g k over the whole range is

The graphs of jR j against given using full linear theory MSE and MMSE

with the new and the old b oundary conditions are presented in Fig It is

clear from the graphs that the results given by b oth the MSE and MMSE with

the new set of b oundary conditions are very much closer to those estimated for 0.3 Full linear MMSE with new BCs 0.25 MSE with new BCs MMSE with old BCs 0.2 MSE with old BCs | 1 |R 0.15

0.1

0.05

0 0 1 2 3 4 5 6 7 8 9 10

Figure Reected amplitude over the depth prole H xx x x

the full linear problem than the results given by the MSE and MMSE with the

old b oundary conditions Indeed the results given by the MSE and MMSE with

the old b oundary conditions b ear little resemblance to the full linear results

From this evidence and that given in Chapter for Bo oijs talud problem

we conclude that the new b oundary conditions are the appropriate b oundary

conditions to use with the MSE and MMSE

The results given by the MMSE with the new b oundary conditions are the

closest to the full linear results as exp ected and give go o d agreement with the

full linear results for However the results given by the MSE with the

new b oundary conditions only agree with the full linear results as well as the

results given by the MMSE when This illustrates that the MMSE gives

results in go o d agreement with the full linear results for humps with up to nearly

twice the maximum gradient that can b e used with the MSE to give results that

are in go o d agreement with full linear theory

In Fig we present the graphs of jR j against given using full linear theory

and the MMSE term term and term approximations It is clear from the

0.35

0.3 Full linear 2−term

0.25 3−term 4−term MMSE 0.2 | 1 |R 0.15

0.1

0.05

0 0 1 2 3 4 5 6 7 8 9 10

Figure Reected amplitude over the depth prole H xx x x

graph that as the numb er of terms in the nterm approximation is increased

the closer the results b ecome to the full linear ones For the slop e of

the hump is mild enough for the term approximation to give go o d estimates to

the full linear solutions Similarly the and term approximations gives good

estimates of the full linear solutions when and resp ectively

As a nal example we consider a hump which is smo othly joined to the at

b eds at x andx Rememb er from Chapter that in this case the old

and new b oundary conditions for the MSE and MMSE are the same

Example

We consider a depth prole given by

H x cos x x

which corresp onds to a hump whose height is half the stillwater depth We seek

solutions given by the mildslop e mo died mildslop e and nterm n

approximations and estimates of the solutions of full linear problem at values

of a parameter starting at nishing at with intervals of The

parameters and are dened in terms of by

Again with these denitions for and varying corresp onds to varying the

steepness of the depth prole

Using a term trial function to give estimates of the reection and trans

mission co ecients for the full linear problem we nd that the maximum value

of kKp g k over the whole range is

In Fig we present the graphs of jR j against estimated using full linear

theory and the MSE MMSE term term and term approximations Again

0.25

Full linear 0.2 2−term 3−term 4−term MMSE 0.15 MSE | 1 |R

0.1

0.05

0 0 1 2 3 4 5 6 7 8 9 10

cosx x Figure Reected amplitude for depth prole H x

it is clear from the graph that as the numb er of terms in the nterm approxima

tion is increased the closer the results b ecome to the full linear ones The results

given by the MSE over the whole range are quite p o or The results given by

the MMSE are much closer to the full linear results than those given by the MSE

and give go o d agreement with the full linear results for However for

it is clear that the slop e of the hump is large enough to warrant the

inclusion of decaying mo des in the approximation to the velo city p otential As

usual we see that the greater the slop e of the hump b ecomes the more decaying

mo des we need to use in the approximation to the velo citypotential As there

is little dierence in the results given by the term and term approximations

we do not calculate results for the term approximation Again we notice that

the results given bythenterm approximation converge to those given by the

n term approximation as decreases that is as the slop e of the hump

b ecomes milder

In this chapter wehaveshown how to obtain estimates of the co ecients of

the scattered waves due to waves incident from x on hump depth proles

for full linear theory A new rstkind integral equation for the tangential uid

elo city v on the hump has b een derived and a variational approach has b een

used to generate approximations to the co ecients of the scattered waves which

are secondorder accurate compared to the approximation of the solution of the

rstkind equation Wehaveshown that estimates of the reection co ecient

can b e determined to dp using terms in the innite series of the kernel in

the rstkind equation and a term trial function Results have b een presented

of how the estimate of the mo dulus of the reection co ecient for the full linear

mo del varies as the slop e of the hump is varied These results were compared with

the corresp onding results given by earlier approximations to the full linear mo del

namely the mildslop e mo died mildslop e and the nterm approximations which

were derived in Chapter Further evidence was found to supp ort the evidence

given in Chapter that the new b oundary conditions derived in Chapter were

the appropriate ones to use with the MSE and MMSE We also showed that the

mildslop e and mo died mildslop e approximations give go o d agreement with the

full linear results for humps with mild slop es and that as the slop e of the hump

b ecomes large decaying wave mo de terms are required in the approximation to

gives results that are in go o d agreement with the full linear results

These estimates of the solutions of the full linear wave scattering problem over

humps will provide an invaluable new test to the accuracy of any new approxi

mation to wave scattering by an arbitrary sea b ed

Summary and Further Work

In this thesis the scattering of a train of small amplitude harmonic surface waves

on water by undulating one and quasionedimensional b ed top ography has b een

investigated

After reestablishing the full linear b oundaryvalue problem satised by the

velo city p otential for the scattering of waves byvarying top ography in Chapter

three approximations to this problem were given namely the mildslop e Eckart

and linearised shallowwater approximations A highly accurate integral equation

metho d given by Chamb erlain which can b e used to solve these approximate

problems was then reviewed

In Chapter several extensions to the work app earing in Chamb erlain

were presented A new integral equation metho d was develop ed which solves

the mildslop e Eckart and linearised shallowwater equations over a range of

their parameters in less than one half of the CPU time required by previously

implementedintegral equation pro cedures Eckarts approximation was also in

vestigated and improved and as a bypro duct a new explicit and very accurate

approximation to the solution of the disp ersion relation was also found Finally

after rederiving the symmetry relations satised by the reection and transmis

ed that these co ecients sion co ecients given by these approximations we show

satisfy the symmetry relations even when they are inaccurately calculated an

unexp ected prop erty

In Chapter we derived a new approximation to wave scattering that includ

ed b oth decaying and progressivewave mo de terms This new approximation

was compared with two older approximations that only contain progressivewave

mo de terms namely the mildslop e and mo died mildslop e approximations We

showed that for steep depth proles where the decaying wave mo des are signi

cant the results given by the new approximation agreed much more closely with

the results Bo oij obtained using full linear theory The maximum number of

decaying mo des used in the new approximation was three This was b ecause the

dierence in the results obtained using two and three decaying mo des in the new

approximation was only very small for even the steep est depth proles In oth

er words the results given by the new approximation had essentially converged

for the steep est depth proles when the numb er of decaying mo des included had

reached three The less steep the depth prole the fewer the number of decaying

mo des required for convergence until eventually the gradient of the depth prole

b ecomes mild enough to make all the decaying mo des negligible The solution

metho d in whichthegoverning system of secondorder dierential equations was

converted into a rstorder system and then solved using a RungeKutta pro ce

dure could not b e used for some values of the parameters of the problem which

corresp ond to mild b ed proles Further work is required here to nd a more ro

bust solution metho d This could p ossibly b e achieved by using a nite dierence

metho d to solve the secondorder system of dierential equations and asso ciated

b oundary conditions Alternativelydeveloping an approximate solution metho d

to solve the integral equation system equivalent to this b oundaryvalue problem

is another p ossibility

More work is still required to determine the prop erties of the reection and

transmission co ecients given by the new decaying mo de approximation The

ehave calculated of these co ecients satisfy the usual symmetry rela estimates w

tions However a pro of that the exact co ecients or any estimates of them satisfy

these symmetry relations has not yet b een found Chamb erlain develop ed a de

comp osition metho d for the mo died mildslop e and mildslop e equations which

allows wave scattering by complex b ed proles to b e deduced from wave scat

tering over simple b ed proles which together make up the complex b ed prole

This pro cess is built around the symmetry relations satised by the reection and

transmission co ecients It seems likely that such a metho d could b e develop ed

for the decaying mo de approximation to o once the prop erties of the reection

and transmission co ecients have b een determined

In the course of developing the decaying mo de approximation a new set

of b oundary conditions was derived for the mildslop e and mo died mildslop e

equations Previously the b oundary conditions for these equations had b een

obtained by enforcing the continuity of the approximation to the free surface and

its slop e at the ends of the varying depth region The new b oundary conditions

arise from enforcing the continuity of the approximation to the velo city p otential

and its horizontal velo city throughout the uid at the ends of the varying depth

region The results given by the mildslop e and mo died mildslop e equations

with these new b oundary conditions are in much b etter agreement with results

that have b een computed using full linear theory and results that have b een

found by using a typ e of approximation in the full linear wave scattering problem

dierent to the ones that have b een used in this thesis The mo died mildslop e

equation together with these new b oundary conditions can b e derived from a

variational principle The details of this pro cess are the sub ject of a pap er by

Porter and Staziker in preparation

Finallyweshowed how to obtain estimates of the co ecients of the scattered

waves due to plane wave incidence on hump depth proles for full linear theoryA

new rstkind integral equation for the tangential uid velo city on the hump was

derived and a variational approach used to generate approximations to the reec

tion co ecients which are secondorder accurate compared to the approximation

of the solution of the rstkind equation The symmetry relations satised by the

reection and transmission co ecients were then used to calculate the transmis

es sion co ecients An alternativevariational principle can b e found which giv

approximations to b oth the reection and transmission co ecients However

this alternative pro cess is computationally more exp ensive than the metho d used

b ecause a larger trial space is required The reection co ecientwas determined

to three decimal places using the rst terms of the innite series which denes

the kernel in the rstkind equation and a term trial function Further work to

remove the logarithmic singularity in the innite series will improve the conver

gence of this series and thus improve computational eciency These estimates

of the solutions of the full linear wave scattering problem over humps provide an

invaluable new test of the accuracy of any new approximation to wave scattering

by an arbitrary sea b ed

The mo died mildslop e approximation to the velo city p otential is derived by

using a term plane wave trial function in a variational principle The results ob

tained give go o d agreement with full linear theory for all but the steep est of depth

proles An apparently similar approximation in the variational principle which

was used for the rstkind integral equation is to omit all the decaying wavemode

terms in the series for the kernel so that only the rst term in the series is used

However the results obtained by this approximation are p o or and we nd that the

amplitude of the reected wave increases as the water depth increases whichis

exactly opp osite to the b ehaviour of the exact solution So the following question

arises what approximation in the variational principle for the rstkind integral

equation corresp onds to the mo died mildslop e approximation Further work

is clearly required here to fully understand this approximation pro cess Once

it is understo o d it could b e used in other more complicated integral equations

arising from wave scattering such as that derived byEvans for scattering

by a shelf of arbitrary prole So far this secondkind integral equation has not

b een solved due to the extremely complicated form of the kernel If an approxi

mation equivalent to the mo died mildslop e approximation could b e made in a

variational principle which is equivalent to solving this equation then not only

could p owerful solution techniques such as reiteration b e used to solve the ap

proximate equation but it could also b e p ossible to nd explicit b ounds on the

error incurred by making the approximation

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