Environmental Fluid Dynamics || Surface Waves

CHAPTER88
Surface Waves Contents
8.1. 2D Long Surface Waves 3338.2. Linear Surface Waves 3358.3. Reflection and Transmission of Surface Waves 341References 349

Surface gravity waves are a common feature in nature and we observe them

whenever a water body is unconfined, such as in our swimming pool, the

neighborhood river, estuaries, coastal seas and even the deep ocean. The

objective of this chapter is to provide some introductory material on surface

waves; long waves, linear waves and reflection and transmission of linear

surface waves as an introduction to internal waves discussed in Chapter 09.

8.1. 2D LONG SURFACE WAVES

Consider a free surface undulation, h(x1, t), with a horizontal length scale l

in a shallow water body of depth h, as shown in Fig. 8.1.1, and supposeh

l< 1. Under these conditions the pressure, p(x1, x3, t), in the water will be

hydrostatic:

pðx1; x3; tÞ ¼ r0gðhðx1; tÞ � x3Þ; (8.1.1)

The momentum equation reduces to a simple balance of the net pressure

force, increasing or decreasing the momentum in the control volume. Now

x3

x1h

Control Volume

Hydrostatic Pressure

Figure 8.1.1 Schematic of long wave analysis control volume.

Environmental Fluid Dynamics � 2013 Elsevier Inc.ISBN 978-0-12-088571-8, DOI: 10.1016/B978-0-12-088571-8.00008-5 All rights reserved.

333 j

334 Environmental Fluid Dynamics

the net pressure force Fp on the control volume, of width dx1, is approxi-

mately given by:

dFpz� rghvh

vxdx: (8.1.2)

The momentum conservation law applied to the control volume shown

in Fig. 8.1.1, neglecting the advective momentum flux:

� rghvh

vx1dx ¼ rh

vv1vt

dx1: (8.1.3)

The movement of the free surface must be such that the mass of water, in

the control volume, is conserved and, since water is incompressible, volume

is conserved:

hvv1vx1

¼ � vh

vt: (8.1.4)

Combining (8.1.3) and (8.1.4) leads to a single equation for h:

v2h

vt2� c2

v2h

vx21¼ 0; (8.1.5)

where

c ¼ffiffiffiffigh

p: (8.1.6)

Equation (8.1.5) is fundamental for much of physics and is called the

“wave equation”. The basic property, important to us here, is that for any

function f, the expression

h ¼ f ðx� ctÞ; (8.1.7)

is a solution of (8.1.5). To see this let us carry out the differentiation:

v2h

vt2¼ c2

v2f

vz2�; (8.1.8)

2 2
v h
vx2¼ v f

vz2�; (8.1.9)

where z� ¼ x � ct. Thus clearly (8.1.7) is a solution of (8.1.5) for any

function f, provided the function has a second order derivative.

Surface Waves 335

The simple interpretation of (8.1.7) is that along a trajectory

x ¼ z� � ct; (8.1.10)

in the (x,t) plane the solution remains constant for constant z�. In other

words the solution consists of a simple translation of our function to the right

(�ve) or to the left (þve) with a speed c. These displacements are thus called

waves and the interpretation offfiffiffiffigh

pis the wave phase speed of long waves.

In x5.8, we introduced the concept of the Froude number:

Fr ¼ Uffiffiffiffigh

p ; (8.1.13)

where U is the discharge velocity in the channel. Thus, the Froude number

has the kinematic interpretation that it is the ratio of the discharge velocity

to the wave speed. Provided that there is no interaction between the flow

and the waves, then Fr> 1 means waves can only propagate downstream,

but if Fr< 1 then waves can propagate both down and upstream. Hence we

assign the word supercritical for Fr> 1 flows and subcritical for flows with

Fr< 1. When Fr¼ 1 we say the flow is critical.In the derivation of (8.1.5), two simplifying assumptions were made.

First, the advective momentum flux was assumed to be small compared to

the unsteady inertia:

v1vv1vx1

þ v3vv1vx3

vv1vt

wv1T

l¼ v1h

lc� hhmax

l2; (8.1.14)

where hmax is the maximum value of the surface displacement and T is the

period of the motionh

c. Second, we assumed the pressure was hydrostatic,

requiring that:vv3vtr0g

whv1

lr0gT� hhmax

l2: (8.1.15)

8.2. LINEAR SURFACE WAVES

Consider once again the configuration shown in Fig. 8.1.1, but we now

inquire whether we can relax the constraint (8.1.14). From x8.1, we saw that

the velocity at the bottom is oscillatory under a surface gravity wave and


from x3.6 we note that an oscillatory outer flow gives rise to a confined

viscous boundary layer attached to the bottom. If this layer thickness is thin

and remains confined close to the bottom, then we may assume that the

outer flow, the region between the boundary layer and the free surface, will

be irrotational as there is no source of vorticity. For such flows, we showed

in x4.5, that there exists a velocity potential f(x1, x3) that has the property

that the velocity {vi} is given by:

vi ¼ f;i: (8.2.1)

Combining this with conservation of volume (incompressible flow)

leads to:

vi;i ¼ f;ii ¼ 0; (8.2.2)

or in cartesian coordinates

v2f

vx21þ v2f

vx23¼ 0; (8.2.3)

which is called the Laplace Equation.Consider surface waves in a fluid of depth hmoving over a flat horizontal

bottom as shown in Fig. 8.1.1. The boundary condition to be satisfied at the

bottom, at x3 ¼�h, is zero vertical velocity (the outer problem) or in terms

of the velocity potential:

vf

vx3¼ 0 x3 ¼ �h: (8.2.4)

Before we can solve (8.2.2), we must determine the boundary condi-

tions at the free surface. The first one of these is that the pressure on the free

surface is zero (atmospheric pressure). If the wave amplitude is small, we can

assume the pressure over a distance h, the height of the wave, is hydrostatic

so that the condition of zero pressure at x3 ¼ h may be applied at x3 ¼ 0

so that:

pðx1; 0; tÞ ¼ rgh: (8.2.5)

Now if we neglect the non-linear acceleration in the momentum

equation we may write:

vv1vt

¼ � 1

r

vp

vx1¼ �g

vh

vx1; (8.2.6)

Surface Waves 337

the same as (8.1.3). Substituting from (8.2.1) yields

v2f

vtvx1¼ �g

vh

vx1: (8.2.7)

Integrating once with respect to x1 yields the required pressure boundary

condition:

vf

vt¼ �gh; x3 ¼ 0: (8.2.8)

However, (8.2.8) contains two unknowns, f and h, so we need one further

equation at the boundary x3 ¼ 0. This is obtained by noting that the water

surface moves with the same velocity as the fluid immediately below. Hence

v3 ¼ vh

vt¼ vf

vx3; x3 ¼ 0: (8.2.9)

Combining (8.2.8) and (8.2.9) by eliminating h yields the equation:

gvf

vx3þ v2f

vt2¼ 0; x3 ¼ 0: (8.2.10)

Thus the water motion is the solution to (8.2.2) subject to the boundary

conditions (8.2.4) and (8.2.10). The difference between (8.2.2) and (8.1.8)

is that here we have not assumed that the pressure is hydrostatic only that the

advective non-linear terms are small in setting up the surface boundary

condition.

Progressive Waves: Let us inquire whether a periodic solution of the form:

fðx1; x3; tÞ ¼ jðx1; x3Þeiut; (8.2.11)

exists to this problem. Substituting (8.2.11) into (8.2.2) to (8.2.10) and

(8.2.4):

gvj

vx3� u2j ¼ 0; x3 ¼ 0; (8.2.12)

2 2
v j
vx21þ v j

vx23¼ 0; � h < x3 < 0; (8.2.13)

vj

vx3¼ 0; x3 ¼ �h: (8.2.14)


Let,

jðx1; x3Þ ¼ hðx1Þf ðx3Þ; (8.2.15)

Substituting (8.2.15) into (8.2.12)–(8.2.14) leads to a separation of

variables:

gdf

dx3� u2f ¼ 0; x3 ¼ 0; (8.2.16)

2 2
1
h

d h

vx21¼ � 1

f

d f

vx23; � h � x3 � 0; (8.2.17)

vf

vx3¼ 0; x3 ¼ �h: (8.2.18)

From (8.2.17) it follows immediately that:

d2h

vx21þ k2h ¼ 0; (8.2.19)

and

d2f

vx23� k2f ¼ 0; (8.2.20)

where k is a constant, yet to be determined. The solutions to (8.2.19) and

(8.2.20) are

h ¼ Aeikx1 þ Be�ikx1 ; (8.2.21)

and

f ¼ C coshðkx3Þ þD sinhðkx3Þ; (8.2.22)

where A, B, C and D are constants yet to be determined from compatibility

with the boundary conditions.Consider first the boundary condition at x3 ¼ 0. Substituting (8.2.22)

into (8.2.16) and setting x3 ¼ 0 yields:

gDk� u2C ¼ 0: (8.2.23)

Similarly substituting (8.2.22) into (8.2.18):

� Ck sinhðkhÞ þDk coshðkhÞ ¼ 0; (8.2.24)

Surface Waves 339

or

D ¼ CtanhðkhÞ: (8.2.25)

Substituting (8.2.25) into (8.2.23) leads to an equation for k in terms of

the frequency u:

gk tanhðkhÞ ¼ u2: (8.2.26)

Equation (8.2.26) is called the dispersion relationship because if we

introduce the wave speed

c ¼ u

k; (8.2.27)

then (8.2.26) becomes

c2 ¼ g tanhðkhÞk

; (8.2.28)

which implies that waves with different wave numbers k (different wave-

lengths) move with a different wave speed c. The relationship (8.2.28) is

shown in Fig. 8.2.1. Given that tanh(kh)/ kh as kh/ 0 we see that in the

limit of long waves (kh / 0)

c ¼ffiffiffiffigh

p; (8.2.29)

which is the result derived in x8.1. Conversely, for very deep water

(kh / N) we get the result:

c ¼ffiffiffiffig

k

r; (8.2.30)

implying that, the shorter the waves, the slower they travel; waves in deep

water are dispersive.Substituting (8.2.21) and (8.2.22) into (8.2.15) and using the dispersion

relation as well as (8.2.23) leads to an expression for the velocity potential for

a progressive wave moving in the positive x1 direction:

4 ¼ ga

u

coshkðx3 þ hÞcoshkh

sinðkx1 � utÞ; (8.2.31)

where a is the wave amplitude.

kx = 1

Figure 8.2.1 Example of a mode one standing wave.


The associated velocity fields are

v1 ¼ v4

vx1¼ gak

u


cosðkx1 � utÞ; (8.2.32)

v3 ¼ v4

vx3¼ gak

u

sinhkðx3 þ hÞcoshkh

sinðkx1 � utÞ; (8.2.33)

��
h ¼ � 1
g

v4

vt��x3¼0

¼ a cosðkx1 � utÞ: (8.2.34)

The system of equations that we have used to derive this solution are

all linear, so we may combine solutions at will and the combination will

again be a solution. An example of some importance is that of a standing

wave, obtained by adding a left and right moving wave, of the type

(8.2.33) and (8.2.34).

h ¼ a

2fcosðkx1 � utÞ þ cosðkx1 þ utÞg; (8.2.35)

where we have chosen half the amplitude in order to make the final wave of

amplitude a. Simple application of the cosine summation formula from

trigonometry leads to the solution set:

4 ¼ ga

u


sinðkx1ÞcosðutÞ; (8.2.36)

v1 ¼ v4

vx1¼ gak

u


cosðkx1ÞcosðutÞ; (8.2.37)

v3 ¼ v4

vx3¼ gak

u

sinhkðx3 þ hÞcoshkh

sinðkx1ÞsinðutÞ; (8.2.38)

a
h ¼
2fcosðkx1ÞcosðutÞg: (8.2.39)

This is called a standing wave, as the water surface oscillates vertically with

no translation of the phase (Fig. 8.2.1). Given the form of (8.2.35), we see

that the horizontal velocity is such that it is zero whenever kx1 ¼ np,

Surface Waves 341

so a standing wave, as the name implies, may be fitted into a rectangular

basin as shown in Fig. 8.2.1.

8.3. REFLECTION AND TRANSMISSION OF SURFACE WAVES

Consider a simple linear plane progressive wave coming from x1 / �Nand moving toward x1 / þN. Suppose near x1 ¼ 0 there is a sill on the

bottom, as shown in Fig. 8.3.1, causing both a reflection of the surface wave

and transmission modification of the wave as it passes over the mound

moving toward x1 / þN.

Let the bottom be described by the function:

xðbÞ3 ¼ �hþ Df ðx1Þ; (8.3.1)

where f(x1) is a dimensionless function describing the shape of the bottom,

D is the amplitude of the bottom changes and h is the mean depth of the

water domain. The solution to the problem when the bottom undulations

are small was derived by Hurley and Imberger (1969).Suppose we have a single wave coming from the left and impinging on

the bottom undulation. This incoming wave may be written as (see x8.2)4ðx1; x3; tÞ ¼ Ai


sinðkx1 � utÞ; (8.3.2)

where k is the wave number and u is the frequency of the incoming wave.

As seen in x8.2, k and u are connected through the dispersion relationship:

gk tanhðkhÞ ¼ u2: (8.3.3)

From x8.2 we see that, in order to obtain an expression for both the

reflection and transmission properties, we must solve for the velocity

potential f that satisfies:

gv4

vx3þ v24

vt2¼ 0; x3 ¼ 0; (8.3.4)

Incident WaveTransmitted Wave

Reflected Wave

)( 1)(

3 xfhx b Δ

Δ

+−=

3 hx

3x

−=

Figure 8.3.1 Schematic of wave reflection and transmission.

v24 v24

vx21þ

vx23¼ 0; 0 > x3 > x

ðbÞ3 ; (8.3.5)


v4

vx3¼ D

df

dx1

v4

vx1; x3 ¼ x

ðbÞ3 ; (8.3.6)

where equation (8.3.6) comes directly from requiring that the velocity at the

bottom is tangential to the bottom:

v3

v1¼ D

df

dx1: (8.3.7)

Now, before proceeding, it is convenient to introduce the following

non-dimensional variables:

x�1 ¼ x1

h; x�3 ¼ x3

h; t� ¼ ut; 4� ¼ 4

Ai

; (8.3.8)

so that (8.3.4)–(8.3.6) become

v2f�

v2t�þ� g

hu2

�vf�

vx�3¼ 0; x�3 ¼ 0; (8.3.9)

2 � 2 �
v 4
vx�21þ v 4

vx�23¼ 0; 0 > x�3 > x

�ðbÞ3 ; (8.3.10)

v4� df v4�

vx�3¼ ε

dx�1 vx�1; x�3 ¼ x

�ðbÞ3 ; (8.3.11)

where ε¼ D

h. Given that (8.3.2) is periodic with a frequency u, the solution

f must also be periodic with a frequency u and the non-dimensional

variable, 4� must thus be periodic with a period of unity. Hence we seek

a solution of the form:

4�ðx�1; x�3; t�Þ ¼ jðx�1; x�3Þe�it� : (8.3.12)

Substituting (8.3.12) into (8.3.9) to (8.3.11) yields

j� Kj;3 ¼ 0; K ¼ g

u2h; x�3 ¼ 0; (8.3.13)

2 � 2 �
v j
vx�21þ v j

vx�23¼ 0; 0 > x�3 > x

�ðbÞ3 ; (8.3.14)

vj df vj ðbÞ
vx�3
¼ ε

dx�1 vx�1; x3 ¼ x3 ; (8.3.15)

For clarity and ease of understanding the following perturbation analysis,

we shall from now on drop the star superscript and also change to the index

notation. The boundary condition (8.3.15) now needs to be reduced to

a condition at x3¼�1. This may be achieved by noting that, so j,3$j,1 may

be expanded in a Taylor series about x3 ¼ �1.

jðx1; xðbÞ3 Þ ¼ jðx1;� 1Þ þ εf ðxÞj;3ðx1;� 1Þ

þ ε2

2!f 2ðxÞj;33ðx1;� 1Þ þ.

(8.3.16)

and now seek a perturbation solution of the form:

jðx1; x3Þ ¼ jð0Þðx1; x3Þ þ εjð1Þðx1; x3Þ þ ε2

2jð2Þðx1; x3Þ þ.

(8.3.17)

Substituting both (8.3.16) and (8.3.17) into equations (8.3.13)–(8.3.15)

and equating equal powers of ε leads to:

jðiÞðx1; 0Þ � KjðiÞ;3 ðx1; 0Þ ¼ 0; (8.3.18)

ðiÞ ðiÞ

Surface Waves 343

j;11ðx1; x3Þ þ j;33ðx1; x3Þ ¼ 0; (8.3.19)

ð0Þ
j;3 ðx1;� 1Þ ¼ 0; (8.3.20)
ð1Þ ð0Þ
j;3 ðx1;� 1Þ ¼ ð f j;3 Þ;1; (8.3.21)
� � �2

�
jð2Þ;3 ðx1;� 1Þ ¼ f j
ð1Þ;1

;1þ f

2jð0Þ;13

;1

; i ¼ 1; 2; 3: (8.3.22)

This set of equations is most easily solved by taking a Fourier Transform

with respect to x1. However, given that the function j(i)(x1,x3) does not go

to zero at x1 ¼�N, some care must be taken with the integration. We shall

use the theory of generalized function described in Lighthill (1962).


Define the Fourier pair by

GðuÞ ¼ 1ffiffiffiffiffiffi2p

pZþN

�N

gðx1Þe�iux1dx1; (8.3.23)

þN

gðx1Þ ¼ 1ffiffiffiffiffiffi2p

pZ

�N

GðuÞeiux1du; (8.3.24)

Taking the Fourier Transform of equations (8.3.18)–(8.3.22) leads to:

JðiÞ � KJ;ðiÞ33 ¼ 0; x3 ¼ 0; (8.3.25)

ðiÞ 2 ðiÞ
J;33 � u J ¼ 0; (8.3.26)
ðiÞ
J;3 ¼ GðiÞðuÞ x3 ¼ �1; (8.3.27)
where G(i)(u) is the Fourier transform of the RHS of (8.3.22) for i¼ 1, 2,

3,.

gð0Þðx1Þ ¼ 0; (8.3.28)

� ð0Þ�
gð1Þðx1Þ ¼ f j; 1 ;1; (8.3.29)
� � �2

�
gð2Þðx1Þ ¼ f j;
ð1Þ1 ;1 þ f

2!j;

ð0Þ13 ;1: (8.3.30)

Now the solution to (8.3.26) is

JðiÞðu; x3Þ ¼ AðiÞðuÞcoshðux3Þ þ BðiÞðuÞsinhðux3Þ: (8.3.31)

Substituting this solution into (8.3.25) reveals:

AðiÞðuÞ ¼ KBðiÞðuÞ; (8.3.32)

so that (8.3.31) may be written as:

JðiÞðu; x3Þ ¼ BðiÞðuÞðKu coshðux3Þ þ sinhðux3ÞÞ: (8.3.33)

This may now be substituted in the lower boundary condition (8.3.27):

BðiÞðuÞðu coshðuÞ � Ku2sinhðuÞÞ ¼ HðiÞ; (8.3.34)

Surface Waves 345

thus

BðiÞðuÞ GðiÞðuÞuðcoshðuÞ � Ku sinhðuÞÞ þ C

ðiÞ1 dðu� n0Þ þ C

ðiÞ2 dðuþ n0Þ;

(8.3.35)

where �n0 are the symmetric roots to the equation

coshðuÞ � Ku sinhðuÞ ¼ 0; (8.3.36)

which is (8.3.3) the dispersion equation. Equation (8.3.35) may now be

substituted into (8.3.33) to yield the solution:

JðiÞðu; x3Þ ¼ ku coshðux3Þ þ sinhðux3ÞuðcoshðuÞ � Ku sinhðuÞÞ G

ðiÞðuÞ þ ðKn0 coshðn0x3Þ

þ sinhðn0x3ÞÞðCðiÞ1 dðu� n0Þ � C

ðiÞ2 dðuþ n0ÞÞ;

(8.3.37)

where d(u � n0) is the Dirac Delta function.
The solution j(i)(x1, x3) may be found by inverting (8.3.37). From
(8.3.24) it is seen that inversion may be achieved by taking the Fourier

Transform of j�(i)(�u, x3). Further, since we are only after the reflection

and transmission properties it is sufficient to carry out an asymptotic Fourier

Transform of j�(i)(�u, x3) for x1. This is achieved (see Lighthill, 1962) by

extracting the singularities (poles) from (8.3.37) and transforming only

these. The singularities in (8.3.37) are where the denominator is zero:

uðcoshðuÞ � Ku sinhðuÞÞ ¼ 0: (8.3.38)

Designating the roots of (8.3.38) by (0, n0, þn0), we see that n0 is the

solution to the dispersion equation (8.3.3). Since (8.3.38) is simple zeros at

u ¼ �n0, we may write:

ku coshðux3Þ þ sinhðux3ÞuðcoshðuÞ � ku sinhðuÞÞ GðiÞð�uÞwGðiÞðn0Þ

� kn0 coshðn0x3Þ þ sinhðn0x3Þðn0 þ coshðn0Þsinhðn0ÞÞ

sinhðn0Þuþ n0

as juþ n0j/0

GðiÞð�n0Þ kn0 coshðn0x3Þ þ sinhðn0x3Þðn0 þ coshðn0Þsinhðn0ÞÞ

sinhðn0Þu� n0

as ju� n0j/0

(8.3.39)


Hence,

JðiÞð�u; x3Þ ¼ kn0 coshðn0x3Þ þ sinhðn0x3Þðn0 þ coshðn0Þsinhðn0ÞÞ

� sinhðn0Þ"(

HðiÞðn0Þuþ n0

�HðiÞðn0Þu� n0

)

þ n0 þ coshðn0Þsinhðn0Þsinhðn0Þ

��C

ðiÞ2 dðuþ n0Þ � C

ðiÞ1 dðu� n0Þ

�#þ FðuÞ;

(8.3.40)

where F(u) is an analytic function, the inverse of which does not contribute

to the solution J(i)(x1, x3) at x1 / �N. The inverse of eqn (8.3.40) for

x1 / �N may be written (see Lighthill, 1966) as

JðiÞð�u; x3Þwi�p2

�1=2 coshðn0ðx3 þ 1ÞÞðn0 þ coshðn0Þsinhðn0ÞÞ ½e

in0x1GðiÞn0ðsgn x1 � 1Þ

� e�in0x1GðiÞð�n0Þðsgn x1 þ 1Þ�;(8.3.41)

where the radiation conditions of x1 / �N were used to evaluate CðiÞ1ðiÞ

and C2 .

CðiÞ1 ¼ �ipGðiÞð�n0Þ sinhðn0Þ

ðn0 þ coshðn0Þ sinhðn0ÞÞ ; (8.3.42)

ðiÞ
C
ðiÞ2 ¼ �ipG ðn0Þ sinhðn0Þ

ðn0 þ coshðn0Þ sinhðn0ÞÞ ; (8.3.43)

so that at x1 / �N we have only the incoming wave.The reflection coefficient a and the transmission coefficient b become

a ¼ iffiffiffiffiffiffi2p

p

n0 þ coshðn0Þsinhðn0ÞnεHð1Þðn0Þ þ ε

2Hð2Þðn0Þ þ.o; (8.3.44)

ffiffiffiffiffiffip n o
b ¼ 1þ i 2p
n0 þ coshðn0Þsinhðn0Þ εGð1Þð�n0Þ þ ε2Gð2Þð�n0Þ þ. ;

(8.3.45)

Surface Waves 347

where

gð0Þðx1Þ ¼ 0 so that Gð0ÞðuÞ ¼ 0; (8.3.46)

and

gð1Þðx1Þ ¼ f j;ð0Þ11 þ df

dx1j;

ð0Þ11 ; (8.3.47)

so that

Gð1ÞðuÞ ¼ n0uFðuþ n0Þ; (8.3.48)

and

a ¼ iffiffiffiffiffiffi2p

p

n0 þ coshðn0Þsinhðn0Þ�εn20Fð2n0Þ þOðε2Þ; (8.3.49)

ffiffiffiffiffiffip
b ¼ 1þ i 2p
n0 þ coshðn0Þsinhðn0Þ�� εn20Fð0Þ þOðε2Þ; (8.3.50)

where F(0) is the area under the mound. We shall not take the analysis to

higher orders; the reader is referred to Hurley and Imberger (1969).Equations (8.3.50) and (8.3.51) offer a convenient tool to understand the

reflection and transmission of surface waves from submerged obstacles.

Consider the obstacle given by:

f ðxÞ ¼ 1ffiffiffip

pZxþls

x�ls

e�z2dz; (8.3.51)

that is shown schematically in Fig. 8.3.2 and represents a simple mound of

length O(2l ) with transitions at each length O(s).

Incident WaveTransmitted Wave

Reflected Wave

3 hx

3x

−=

l

Figure 8.3.2 Reflection from a uniform, constant height sill.


The Fourier transform of (8.3.51) is given by:

FðuÞ ¼ffiffiffiffiffi2

p

rsinðluÞ

ue�

s2u2

4 ; (8.3.53)

so that

a ¼ iεn0 sinhð2n0lÞn0hþ coshðn0hÞsinhðn0hÞ e

�s2v20 þOðε2Þ; (8.3.54)

2il
b ¼ 1þ
n0hþ coshðn0hÞsinhðn0hÞ þOðε2Þ; (8.3.55)

the i indicating a 90� phase shift and the wave number n0 must satisfy the

dispersion relationship:

gn0 tanhðn0hÞ ¼ u2

the amplitude jaj is shown in Fig. 8.3.3 where it is seen that jaj depends onsin(2n0l ), indicating that the front and the rear of the mound cause an equal

reflection, but with a different phase shift so that the reflections add

constructively when 2n0l ¼ ð2kþ 1Þp2

, k¼ 1,2,., and destructively when

2n0l¼ 2kp, k¼ 1,2,.,. From Fig. 8.3.4 we see that, asslbecomes large, jaj

decreases exponentially and we approach what is called weak reflection; the

incident wave can negotiate the sill without appreciable reflection.

1.00 1.25 1.50 1.75 2.00 2.250.00

0.01

0.02

h= 0.07

h= 0.20

Figure 8.3.3 Reflection coefficient for a rectangular sill.

1/12 1/6 1/40.00

0.01

0.02

0.03

h= 0.07

h= 4.0

= 1/4= 1/2

= 1/12

h= 4.9

h= 6.5

h= 3.8

Figure 8.3.4 Effect of slope width on the reflection coefficient for a sill.

Surface Waves 349

REFERENCESHurley, D.G., Imberger, J., 1969. Surface and internal waves in a liquid of variable depth.

B. Aust. Math. Soc. 1, 29–46.Lighthill, M.J., 1962. Fourier Analysis and Generalised Functions. Cambridge Univ. Press,

pp. 79.

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Environmental Fluid Dynamics || Surface Waves