第十七屆海洋玉程研討會 暨 1995 兩岸港口及海岸開發研討會治文集

1995 年11 月 9-10 日 'Edge ﹒The Differences Between the Full . and ShallowW ater- Theories

Kai Meng Mok 1

Abstract

Edge waves are known to have significant impact on the coasta1 hydrodynamics and sediment010gy. The two linearso1utions for edge waves are derived from the full and shallow water-wave theories ，respective1y. Since the shallow water-wave s01ution is on1y‘an approximation ，it shou1d work well on1y when the water is shallow. As 伽 water depth becomes deeper ，the shallow water-wave s01ution is bound to be different from the exact full water-wave s01ution. In the present study ，a genera1 comparison of the ~ull water-wave s01ution and the shallow water-wave approximation is made by p10tting re1ative errors for

different edge-wave modes showing 伽 effects of 伽 be 叫 slope and the offshore 4!stance The re1ative error charts of the water surface elevation for the mode-O ，mode-1 and mode-2 edge waves arederived ，respective1y. It is readi1y shown that the error of the shallow-water approximation is 1arger as the . s10pe gets steeper and the distance is farth 巴r from the shoreline.

1 Introduction

Over the years ， waves in the coasta1 zone have provided peop1e with both a p1ace of 1eisure activities and disasters. The subsequent 10ve-hate re1ationship motivated mariy researchers to irivestigate the characteristics of these wave actions. Since it is more apparent to the eye ，most of the coasta1 studies concentrate on normally incident waves which appear to have a more direct impact on the coasta1 region. In 1846 ，Sir George Stokes discovered theoretically that waves with crests perpendicu1ar to the shoreline and amplitudes which exponentially decay offshore can propagate a10ngshore on a uniformly s10ping beach. It is this exponentially diminished offshore charact 巴ristic which eamed this wave the name of "edge wave" from Sir Horace Lamb (1932) ，and his comment was that "it does not appear that th巳 type of motion h巴re referred to is very important". Lamb's comment reflects the genera1 fee1ings the research community had about the edge wave during the early twentieth century. Ironically ，concentrating energy nearshore ，the reason that the edge wave was ignored by the research community for more than one hundred years since its discovery ，is a1so why the edge wave is so important in coasta1 hydrodynamics. Since edge waves are trapped by refraction near the shore and their energy is concentrated near the

1AssistantProfessor，Facultyof ScienceandTechnology，Universityof Macau -393- shoreline ，the wave phenomenon has a major impact on the coastal sedimentology and nearshore flows such as the formation of crescentic bars ，beach cusps ，rip currents ，longshore currents and wave runup. Since the discovery of edge waves by Stokes in 1846，the problem was also solved by Eckart (1951) and Ursell (1952) ，respectively. Nevertheless ，Eckart's solution is based on the shallow water wave theory while Ursell's is based on the full water wave theory. Due to the fundamental differences between the assumptions of the full and shallow water wave theories ， the two solutions are bound to be different. The differences between these two solutions are discussed and demonstrated in the present study. Before commencing the theoretical comparison ，two terms are defined to facilitate the discussion: (i) FuU Water-Wave Theory: The formulation of the problem is based on the assumptions of an inviscid and homogeneous fluid ，irrotational f1uid motion ，constant pressure on the free surface ，negligible surface tension effect ，and only gravity as the restoring force. (ii) Shallow-Water (Long) Wave Theory: The assumptions of the problem formulation are the same as those of the full-water wave theory with further simplification. The vertical f1uid acceleration is neglected which means that the pressure variation with depth within the f1uid domain is hydrostatic ，and the horizontal component of the velocity field is uniform over the whole depth. With these assumptions ，the problem is reduced to two-dimensional by the depth-integration.

2 Theoretical DÎscussÎon and ComparÎson

As mentioned ，the edge wave on a uniformly sloping beach was discovered analytically by Stokes in 1846 through the linear water-wave theory. The solutions of Stokes' progressive edge wave can be written as

<þ= 去si咐叩(一份叫+的州州 U 一ωt) (1)

η= asinß exp( 一句 cosß)cos(kx 一ωt) ， (2)

with the relation

ω2 = gksinß ， (3)

where 伏，y，z) are the Cartesian coordinates of the alongshore ，offshore and vertically upward directions (the shorelineis at y = 0 and z= 0) ，ηis the displacement of the water surface from the equilibrium position ，a is the wave run-up amplitude on the beach ，ß is the beach sloping angle from the horizontal ，ω(= 2π/wave period) is the angular frequency and k (= 2n/wavelength) is the wavenumber in the longshore direction. The solutions show thatboth

-394- the velocity field and the wave amplitude are maximum at the shoreline and decrease exponentially offshore while the waves propagate in thealongshore direction with their crests perpendicular to the shoreline. A three dimensional plot of a Stokes-mode edge wave's surface profile on a uniformly sloping beach .of 150 is shown in figure 1. Since the wave profile is sinusoidal in the lòngshore (x-) direction ，only one wave cycle is plotted. Using the shallow-water (long) wave theory ，Eckart (1951) obtained solutions for other progressive edge waves;

。n- 于αp(-kny)Ln( 泣ny)sin(knx 一ω，t)， (4)

ηn = anßexp( 一kny )Ln (2kny) cos( knx 一ωt) ， . (5)

with the dispersion relation ，

ω2 = gknß(2n+ 1) ， (6)

where n is a non-negative integer (n = 0，1，2，3，...) representing the offshore mode number ，

an is the wave run-up amplitude ，and Ln(﹒)is the Lague 叮e polynomial of order n，which can be defined as

n Ln(s) =三寺 (e-ss ) (7)

The solutions show that other edge-wave modes besides the Stokes-mode edge wave can exist; in fact the shallow-water wave approximation gives an infinite number of mode solutions to the problem (i.e. no limitation to the value of n); the zero-th mode solution correspQnds toStokes' solution with ß → o and this solution is often called the Stokes mode. However ，theshallow- water wave approximation leads to two major questions. Are all the modes given by the solutions the ones which have their energy trapped in theregion near the shoreline? How well does the shallow water-wave approximation work for the higher-mode waves? As one can expect ，the long wave appro 主imation should work well when the water is shallow. For a uniforrnly sloping beach ，the water depth increases linearly and the long wave theory must break down as y →∞. In 1952 ，Ursell solved the same próblem with the full water-wave theory. His solutions

are in agreement with Ecka 哎's (1951) solutions proving the existence of modes other than

Stokes' mode in a sloping beach environment. Contrary to Ecka 哎 's (1951) shallow-water wave solutions ，the number of "trapped" modes is not infinite; it depends upon the beach slope with the maximum mode number satisfying

(2n+ 圳三? (8)

-395- There is also çontinuous spectrum mode whichradiates energy to the infinity in the offshore direction ，with the dispersion relation ω2 > gk. ，Note that with the ，shallow-water wave approximation ，one çannot distinguish betweenthe trapped modes and those that radiate energy offshöre. On the other hand; with the full~water wave theory ，the trapped mùdes are represented by this discrete spectrum while the radiating mode is by the continuous spectrum. (The proof of this distinction was given byUrsell ，1951.) Whitham (1979) explicitly showed that ，as r1 →∞， the shallow-water wave solution corresponds to the perfectly reflected cross- shore wave which is obviously not a (trapped) edge-wave mode. According to Ursell's solutions ，there can only be a finite number of modes existing for a given beach slope; the number of possible modes increases as the beach slope decreases. Ursell's progressive edge wave 'solutions can be expressed as

。n- 莘[位卅yc 吋向位 nß)

+ I，Amn{exp[-knycos(2m 一I)ß - knzsin(2m 一1)戶] m=l

+exp[ -knycos(2m + 1)戶 +kρin(2m + 1)戶口]sin( 卅一ωt) ， (9)

ηn- 的 Iexp( -knycosß) +主 Amn {exp[ - knycos(2m -1)ß] m=l

+exp[-knycos(2m + I)ßl}]cos(knx 一ωIt)， (10) where

mmtan(n 一j + I)ß Amn =(一1)m n (11) ;:1 tan(n + j) 戶， and αn is a constant. The corresponding dispersion relation is given by

ω2 = gkn sin(2n + I)ß. (12)

Note that when n = 0，Amnand αo are taken to be 0 and a sin戶，respectively. Ursell's solution is then reduced to Stokes' solution. However ，as the mode number ，n，becomes larger ，the edge-wave surface profile and flow field become more complex. Figures 2 and 3 give three- dimensional surface-profile plots of a mode-l and a mode-2 edge wave on a 150 slope ， respectively. The mode-l edge wave's offshore profile has one node (zero-crossing) while the mode-2 edge wave has two offshore nodes. Both of the twoedge waves' surface profiles approach the still water level asymptotically in the offshore direction showing that the energy of edge waves is confined to a region near the shoreline.

-396- As Ihentioned earlier ，the shallow-water wave theory does not provide good approximations in deep water. The discrepancies between the full water-wavesolution and the shallow-water wave approximation on a beach of 150 (戶 =π112) slope were demonstrated gra:phically by Yeh (1986) and the graph is reproduced here in figureA. It is shown that the full water~wave and the shallow-water wave solutions have good agreement for the mode-O edge wave. But they deviate from each other as either n or y increases; these deviations reflect the failure of the shallow-water wave approximation as y →∞ (in deep water).

A more generalcomparison of the full water-wave solution and the shallow water-wave approximation can be made by plotting relative errors for different edge-wave modes showing the effects ofthe beach slope and the offshore distance. To be consistent with Yeh's (1986) comparison ，only the relative errors of the water surface elevation ηfor the mode-O ，mode-l and mode-2 edge waves are shown in figures 5，6 and 7，respectively. The relative errors for the velocity potential are not shown due to similar characteristics. Each contour line in figures 5 to 7 represents a constant error value between the two solutions (full and shallow) ，and it is computed asηslηS ，y=o-ηflηf ，y=o，where 11s and ηf are from Eqn. (5) and Eqn. (10) ， respectively. It is noted that the plot range of ß is determined by Eqn. (8) for each edge-wave mode. It isreadily shown that the error of the shallow-water approximation is larger as the beach slope gets s悅 per and the distance is farther from the shoreline (y = 0). Similar to Yeh's (1986) demonstration ，the shallow water approximation gives less error for the mode-O edge wave and the relative error increases as the mode number increases.

3 Conclusions

Due to its simplicity ，many edge-wave transport models (e.g. Bowen & Inman (1971) and Holman & Bowen (1982)) are based on the shallow-~ater approximation. The use of the shallow-water approximation seems reasonable if the beach slope is mild ，only location relatively close to shore ，or only relatively low-mode edge waves are considered. The significant differences between the full and shallow water-wave solutions as the beach slope gets steeper ，the distance is farther from the shoreline ，or the mode number gets larger issue a warning to the modelers of the coastal hydrodynamics and the formationsof nearshore topography. In order to get a better and more exact prediction ，the full water-wave solution should be utilized at all time.

References

1. Bowen ，A. J. and D. L. Inman (1971) ，"Edge waves and crescentic bars ，" J. Geophys. Res. ，Vol. 76 ，pp. 8662-8671.

2. Eckart ，C. (1951) ，"Surface waves on water of variable depth ，"Wave Report No. 100， Univ. of Calif. ，Scripps Institution of ，Calif.

-397- 3. Holrnan ，R. A. and A. J. Bowen (1982) ，"Bars ，burnps ，and holes: rnodels for the generation of cornplex beach topography ，"J. Geophys. Res. ，Vol. 87，pp. 457-468.

4. Larnb ，H. (1932) ，Hydrodynarnics ，6th edn. Carnbridge: Cambridge University Press.

5. Stokes ，G. G. (1846) ，"Report on recent researches in hydrodynarnics ，"Rep. 16th JEeetingBrit. i\ssoc. Adv. Sci. ，pp. 1-20 = Math. Physics Papers ，Vol. 1，pp. 157-187 ， Carnbridge ，1880.

6. Ursel1，F. (1951) ，"Trapping rnodes in the theory ofsurface waves ，"Proc. Carn. Phil. Soc. ，Vol. 47 ，pp. 347-358.

7. Ursel1，F. (1952) ，"Edge waves on a sloping beach ，"Proc. R. Soc. Lond. A，Vol. 214 ， pp. 79-97.

8. Whitharn ，G. B. (1979) ，Lectures on Wave Propagation. Berlin: Springer-Verlag.

9. Yeh ，H. H. (1986) ，"Experimentalstudy of standing edge waves ，"J. Fluid Mech. ，Vol. 168，pp. 291-304.

-398- l

0.5 η llmax

1 ，/"ío":

，，/這/

kOx ，///4 koY

2 pi

(a)

、x y

η ηmax

2 4 8 10 koY

(b)

Figure 1 Three-dimensiona1 surface profile of a mode-O edge wave on a slope of rrl12; a) viewing from inshore to offshore ，b) viewing along shore.

-399- 1

0.5 η

l1max -0.5

8 6

k1y

2 pi

(a)

P 工 。 2

η

l1max

2 4 6 8 10 klY

(b)

Figure 2 Three-dimensional surface profile of a mode-l edge wave on a slope of 7tl12; a) viewing from inshore to offshore ，b) viewing along shore.

-400- 1

0.5 η llmax -0.5

kZX

2 Pi

(吋

O ~! ./' 2

η llmax

5 10 15 20

k2Y

(b)

Figure 3 Three-dimensional surface profileof a mode-2 edge wave on a slope of 1rI12; a) viewing from inshoreto offshore ，b) viewing along shore.

-401- η

ηat y = 0

0.6 (a)

koY 。 2 是 6 工

(b)

k1y

0.6 1

(c)

"6 k2y 0.2

0.6

Figure 4 Offshore profiles of various edge-wave modes with ß ==π112; a) n :::0，b) n = 1，c) n= 2. 一一一，full theory; 一一-， shallow-water approximation. (From Yeh ，1986 ， J. Fluid Mech.)

-402- 1.5 z-----一 ←一一一一一一一一-一一一-一一一←一一←一一一一一一

1.25

1

ß 0.75

0.5 -0.018

0.25

O

《 O 4 己 4 6 8 10

koY

Figure 5 Contour plot showing the differences on the water-surface elevation between the full water-wave .theory~~~dth~ shal.low-water wave approximation for the mode-O edge wave at various offshore locations and on varioûs beach slopes; the contour lines

represent the values of 115/11 5 ，y=O -ηflηf ，y=O ，where the subscripts "s" and "f" represent the values from the shallow and th己 full water-wave theories ，respectively.

-403- 0.5 一

0.4

0.3

戶

0.2

0.1

0.000

O O 2 4 6 8 10 klY

Figure 6 Contour plot showing the differences on the water-surface elevation between the full water-wave theory and the shallow-water wave approximation for the mode-l edge wave at various offshore locations and on various beach slopes; the contour lines represent the values of lls!lls ，y=O-llf/llf ，y=o，where the subscripts "s" and "f" represent the values from the shallow and the full water-wave theories ，respectively.

-404- 0.3 一

0.25

0.2

戶 0.15

吶仆.~ 0.1 -0.010

0.05

O 5 10 15 20

k2Y

Figure 7 Contour plot showing thedifferences on the water-surface elevation between the full water-wave theory and the shallow-water wave approximation for the mode-2 edge wave at various offshore locations and on various beach slopes; the contour lines represent the values of ll/r\s ，y=ü -ηr/llr ，y=o ，where the subscripts "s" and "f" represent the values from the shallow and the full water-wave theories ，respectively.

-405-