Coastal Engineering

Professor A G L Borthwick Hilary Term 2010

Paper C2A: STRUCTURES AND FLUIDS

4CE16: COASTAL ENGINEERING

Lecture 2 Coastal Hydrodynamics II

2.1 Wave Breaking Wave breaking dissipates wave energy at beaches, produces turbulence, throws sediment into suspension and can cause large impact forces on coastal structures. Excess momentum (i.e. radiation stress) from non-uniform wave breaking is responsible for longshore currents, wave set-up and set-down.

Waves become unstable when their steepness (height divided by wavelength) reaches a sufficient value. A useful criterion for estimating the limiting steepness of a wave passing over a flat bed was given by Miche as  2 d  H max  0.142 tanh     where H is the wave height,  is the wave length and d is the water depth.

Waves break when the water particles begin to move faster than the wave celerity. At beaches, wave breaking depends on the bed slope, local water depth, and offshore wave conditions. The Iribarren (surf similarity) number is tan  NI  Ho o

13 Coastal Engineering where  is the bed slope angle to the horizontal, H o and o are the height and wavelength, respectively of the offshore incident wave. Wave breaking may be classified according to the Iribarren number as follows: spilling for NI  0.4; plunging for 0.4  NI  2.4 ; collapsing for

2.4  NI  3.4; surging for NI  3.4 . Most damage is caused by plunging and collapsing waves.

For shoaling and non-refracting waves, the wave height is

H  H o cgo cg

where c is the individual wave celerity, cg is the group celerity and the subscript o refers to deep water conditions. In deep water, cgo  co 2, whereas in shallow water, cg  c  gd . As d  0, cg  0 and H   which is clearly unsustainable. In fact, waves become unstable and break.

McCowan (1894) proposed that the breaking wave height is

Hb   db

where   0.78 and db is the water depth.

Later, Weggel fitted  to laboratory data on breaking waves and found that depends on the beach slope, m  tan  . Weggel determined the following empirical formula (recommended by the US CERC Shore Protection Manual):

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H H b    b  a b 1 1 2 db gT

where a1  43.81 exp(19m) and b1 1.56 1 exp(19.5m). Note that Weggel’s formula matches McCowan’s criterion as m  0 (i.e. flat beds).

For shoaling and refracting breaking waves,

Hb  Ho cgo cgb coso cosb where  is the wave direction. In shallow water, b  0 and cosb 1. Also, cgb  gdb . Hence, after substitution,

1/ 2 H  c  H  o  o cos  . b 1/ 4 2 o gd b   

Equating this to the McCowan formula ( Hb   db ), and rearranging, we obtain

2/5  H 2c  d   o o cos  . b  2 1/ 2 o   2 g 

This equation gives an estimate of the depth of water at which breaking occurs, but does not take account of bed slope. If instead, Weggel’s formula is used, then, after rearrangement,

2 5/2  a1 2 2 a1 db  db  db   0 g 2T 4 gT 2

2 2 1/ 2 where   Ho co coso  2b1 g . This can be solved iteratively. Goda (1970) produced a graphical approach by which the breaking wave

15 Coastal Engineering height, H b , and depth, db , may be estimated from curves based on the refracted onshore wave steepness, Ho o , where H o is the refracted wave height ( Ho  KR Ho and KR is the refraction coefficient).

2.2 Wave Run-up Run-up is the height above still water level reached by the wave uprush at a beach. For small amplitude non-breaking waves, the run-up on a smooth slope is Rs  Hs  (2s ) where s is the angle of the beach

(swash) face slope to the horizontal and H s is the wave height at the beach toe. For finite amplitude waves,

Rs  Hs   (2s )  kHs coth(kd)/ 2. For broken waves, use Hunt’s formula, Rs  Hs tan s Hs o . Note that for a vertical wall, s   2 and Rs  Hs . Total run-up is R  Rr Rs where the roughness factor

Rr [0.9,1.0] for smooth concrete slopes and Rr [0.5, 0.7] for rip-rap.

2.3 Interaction of Waves and Currents Assume that seawater is irrotational, inviscid and incompressible and the waves are small amplitude. Consider wave propagation in the presence of a steady uniform horizontal current U  iU  jV of magnitude

2 2 Uc  U V in which U and V are the horizontal current components acting in the onshore x and longshore y directions. Let the current be at an angle  to the waves.

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The wave speed relative to the fixed bed is ca   T where T is the intrinsic wave period which would be measured by a stationary observer.

The wave speed relative to the moving water mass is cr   T r where Tr is the wave period measured by an observer moving with the water mass. The current component collinear with the wave is

Uc cos  U k/ k . The wave number vector is k  ik cos  jk sin where  is the wave angle measured anticlockwise from the onshore x- direction. Hence,

ca  cr Uc cos .

By definition c   k , and multiplying by k  2  , the Doppler equation is

a  r Uck cos

in which r  2 Tr is the wave angular frequency relative to the water mass and a  2 T is the (intrinsic) wave frequency relative to the sea bed. The linear dispersion equation is

2 2 r  a Uck cos  gk tanh(kd)

17 Coastal Engineering where the mean water depth d  h  where h is the still water depth and  is the set-up. The wavelength is   2 k .

Hence, waves are elongated when in the same direction as the current. Likewise, an opposing current shortens the wavelength. The individual wave celerities relative to the water mass and sea bed are, respectively,

cr  r k   Tr ; ca  a k   T .

The corresponding group celerities are

d d c  r ; c  a . gr dk ga dk

Using the linear dispersion and Doppler equations,

cgr  ncr ; cga  cgr Uc cos where n  ½1 2kd/sinh(2kd). The Doppler equation,

r  a Uck cos , may be written in non-dimensional form after dividing through by g / d1/ 2 as

r a   U c    1 (kd) cos . g / d g / d  a d  

Substituting the linear dispersion equation into the left hand side, we have

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a   Uc    kdtanh(kd)  1 (kd) cos . g / d  a d  

This equation may be solved graphically by plotting

  fL (kd)   kdtanh(kd) , fL (kd)   kdtanh(kd) and fR (kd)  a [1 (kd)(Uc (a d))cos] g / d .

There are four possible solutions:

 A: Uc cos  0

 B:  cgr Uc cos  0

 C:  cr  Uc cos  cgr

 D: Uc cos  cr

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When A = B, Uc cos  0 and there is no current acting in the wave direction.

When B = C, Uc cos  cgr and cga  0 . We have a stopping current that prevents wave energy being transferred upstream For the case when   0 , Uc  cgr  ncr . Letting Vo  g a  gT (2) and making use of the Doppler and linear dispersion equations, we can derive

2 U 1   2kd   c   tanh(kd)1    . V 4  sinh(2kd)  o    

In deep water, tanh(kd) 1 and sinh(2kd)   , and so Uc  Vo 4. In shallow water, ca and a  0 for a stopping current. Also tanh(kd)  kd and so Uc  gd as would be expected.

When C = D, Uc  cr and the waves are stationary.

For waves interacting with irrotational horizontal currents, the particle velocity components are

k Ag coshk(z  d) u  sin S cos U , r cosh(kd)

k Ag coshk(z  d) v  cosS sin V , r cosh(kd)

k Ag sinhk(z  d) w  cosS , r cosh(kd)

20 Coastal Engineering where A is the wave amplitude, z is measured vertically upwards from still water level, and the phase function is

S  at  k  x  rt  k  xr .

Note that x  i x  j y is measured from the origin of a stationary coordinate system, whereas xr  i xr  j yr moves with velocity U such that x  xr  Ut .

2.4 Long Waves Tides, storm waves, tsunamis, seiches and edge waves can be approximated as long waves (of period > 60 s) where the motions are predominantly horizontal. Consider a long wave in the Eulerian frame of reference. At a fixed location, the total depth is d  h  where h is the still water depth and  is the free surface elevation above still water level. Assume shallow water, such that kh  10 or h   20. Assume small-amplitude waves where A  ,h. Hence, vertical accelerations may be neglected, the pressure is hydrostatic (with p  g(  z) where z is upwards from still water level), and linearised theory applies. Ignore turbulence, Coriolis and bed friction effects. The Continuity and Euler momentum equations are then depth-averaged, linearised and approximated for small-amplitude to give the 2-D long wave equations

  q q   x y      , t  x y 

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q  q  x  gh and y  gh , t x t  y

noting that qx  Uh , qy  Vh, where the depth-averaged velocity components are

  1 1 U  u dz and V  v dz . d  d  h h

For a horizontal bed, the long wave equations are cross-differentiated to give

2 2 1   h  c2  t 2 which is the classical wave equation, where c  gh . Note that the above long wave equations may be extended to include convective acceleration, bed friction, turbulent mixing and Coriolis terms.

For a progressive long wave propagating in the x-direction, the 1-D version of the classical wave equation solves to give the free surface elevation as   Acos(kxt) where A = H/2. The corresponding depth-averaged velocity, obtained by substituting  into the 1-D long wave equations and integrating, is

A kAg kg c U  cos(kxt)  cos(kxt)    , kh   h

22 Coastal Engineering noting that  2  gk 2h in shallow water. The total energy per surface area of a progressive long wave is E  (gH 2 ) /8 and the energy flux is

1/ 4 1/ 2 F  Enc  E gh . Hence, H2 H1  h1 h2  b1 b2  , analogous to the wave shoaling formula. This is Green’s law when b1  b2 .

2.4.1 Seiching in a Rectangular Basin The long wave equations may be applied to free oscillations in a rectangular basin of length a, width b, and still water depth h. The free surface profile is   (x, y)exp(it) in which i  1. The wave equation becomes

2 2 2   h  k  0 t 2  where k  /c. For boundary conditions,  0 at x  0 and a, and x   0 at y  0 and b, the solution of the wave equation is  y

  m x n y    Amn cos cos . m0n0 a a

Hence, the free oscillation period in the rectangular basin is 2 Tmn   m2 n2  gh    2 2   a b  where m =0, 1, 2, … and n = 0, 1, 2, …, and c  gh .

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2.4.2 Co-oscillating 1-D Tide in a Rectangular Basin Consider a long wave that propagates from a deepwater inlet into a semi-enclosed basin of constant depth, which has a perfectly reflecting wall at one end.

The free surface elevation is the sum of incident and reflected components

  Acos(kxt)  Acos(kxt)  2Acos(t)cos(kx) .

The depth-averaged velocity obtained by substituting for  in the 1-D long wave equations is 2A 2kAg U  sin(kx)sin(t)  sin(kx)sin(t) . kh 

At x  0,   2Acos(t) and U  0. At x  L ,   2Acos(kL)cos(t). The tidal range is 2(L)  4Acos(kL) . Hence, (0) (L) 1 cos(kL) which gives resonant conditions when kL (2n 1) / 2 and n = 0, 1, 2, …

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