Cohen I.M., Kundu P.K. Fluid Mechanics

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212 Irrotational Flow

15. Consider a two-dimensional, constant density potential ﬂow over a circular

cylinder of radius r = a with axis coincident with a right angle corner, as shown in

the ﬁgure below. Solve for the streamfunction and velocity components.

c 5 const

Literature Cited

Prandtl, L. (1952). Essentials of Fluid Dynamics, New York: Hafner Publishing.

Supplemental Reading

Batchelor, G. K. (1967). An Introduction to Fluid Dynamics, London: Cambridge University Press.

Milne-Thompson, L. M. (1962). Theoretical Hydrodynamics, London: Macmillan Press.

Shames, I. H. (1962). Mechanics of Fluids, New York: McGraw-Hill.

Vallentine, H. R. (1967). Applied Hydrodynamics, New York: Plenum Press.

Chapter 7

Gravity Waves

1. Introduction ........................ 214

2. The Wave Equation.................. 214

3. Wave Parameters .................... 216

4. Surface Gravity Waves............... 219

Formulation of the Problem ......... 219

Solution of the Problem ............. 221

5. Some Features of Surface Gravity

Waves .............................. 223

Pressure Change Due to Wave

Motion ........................... 224

Particle Path and Streamline ........ 224

Energy Considerations .............. 227

6. Approximations for Deep and

Shallow Water ...................... 229

Deep-Water Approximation.......... 230

Shallow-Water Approximation ....... 231

Wave Refraction in Shallow Water . . . 233

7. Inﬂuence of Surface Tension ......... 234

8. Standing Waves ..................... 237

9. Group Velocity and Energy Flux ..... 238

10. Group Velocity and Wave

Dispersion .......................... 242

Physical Motivation ................. 242

Layer of Constant Depth ............ 243

Layer of Variable Depth H(x) ....... 244

11. Nonlinear Steepening in a

Nondispersive Medium .............. 246

12. Hydraulic Jump ..................... 248

13. Finite Amplitude Waves of

Unchanging Form in a Dispersive

Medium............................. 250

Finite Amplitude Waves in Deep Water:

The Stokes Wave.................. 251

Finite Amplitude Waves in Fairly

Shallow Water: Solitons ........... 252

14. Stokes’ Drift ......................... 253

15. Waves at a Density Interface

between Inﬁnitely Deep Fluids ....... 255

16. Waves in a Finite Layer Overlying

an Inﬁnitely Deep Fluid ............. 259

Barotropic or Surface Mode .......... 261

Baroclinic or Internal Mode .......... 261

17. Shallow Layer Overlying an

Inﬁnitely Deep Fluid................. 262

18. Equations of Motion for a

Continuously Stratiﬁed Fluid ........ 263

19. Internal Waves in a Continuously

Stratiﬁed Fluid ...................... 267

The w = 0 Limit .................... 269

20. Dispersion of Internal Waves in a

Stratiﬁed Fluid ...................... 270

21. Energy Considerations of Internal

Waves in a Stratiﬁed Fluid........... 272

Exercises ............................ 276

Literature Cited ..................... 277

213

DOI: 10.1016/B978-0-12-381399-2.50007-1

214 Gravity Waves

1. Introduction

It is perhaps not an overstatement to say that wave motion is the most basic feature

of all physical phenomena. Waves are the means by which information is transmitted

between two points in space and time, without movement of the medium across the

two points. The energy and phase of some disturbance travels during a wave motion,

but motion of the matter is generally small. Waves are generated due to the existence of

some kind of “restoring force” that tends to bring the system back to its undisturbed

state, and of some kind of “inertia” that causes the system to overshoot after the

system has returned to the undisturbed state. One type of wave motion is generated

when the restoring forces are due to the compressibility or elasticity of the material

medium, which can be a solid, liquid, or gas. The resulting wave motion, in which the

particles move to and fro in the direction of wave propagation, is called a compression

wave, elastic wave, or pressure wave. The small-amplitude variety of these is called

a “sound wave.” Another common wave motion, and the one we are most familiar

with from everyday experience, is the one that occurs at the free surface of a liquid,

with gravity playing the role of the restoring force. These are called surface gravity

waves. Gravity waves, however, can also exist at the interface between two ﬂuids of

different density, in which case they are called internal gravity waves. The particle

motion in gravity waves can have components both along and perpendicular to the

direction of propagation, as we shall see.

In this chapter, we shall examine some basic features of wave motion and illustrate

them with gravity waves because these are the easiest to comprehend physically. The

wave frequency will be assumed much larger than the Coriolis frequency, in which

case the wave motion is unaffected by the earth’s rotation. Waves affected by planetary

rotation will be considered in Chapter 14. Wave motion due to compressibility effects

will be considered in Chapter 16. Unless speciﬁed otherwise, we shall assume that the

waves have small amplitude, in which case the governing equation becomes linear.

2. The Wave Equation

Many simple “nondispersive” (to be deﬁned later) wave motions of small amplitude

obey the wave equation

∂

∂t

= c

∇

η, (7.1)

which is a linear partial differential equation of the hyperbolic type. Here η is any

type of disturbance, for example the displacement of the free surface in a liquid,

variation of density in a compressible medium, or displacement of a stretched string

or membrane. The meaning of parameter c will become clear shortly. Waves traveling

only in the x direction are described by

∂

∂t

= c

∂

∂x

, (7.2)

which has a general solution of the form

η = f

(

x − ct

)

+ g

(

x + ct

)

, (7.3)

2. The Wave Equation 215

where f and g are arbitrary functions. Equation (7.3), called d’Alembert’s solution,

signiﬁes that any arbitrary function of the combination (x±ct) is a solution of the wave

equation; this can be veriﬁed by substitution of equation (7.3) into equation (7.2). It

is easy to see that f(x−ct) represents a wave propagating in the positive x direction

with speed c, whereas g(x + ct) propagates in the negative x direction at speed c.

Figure 7.1 shows a plot of f(x−ct) at t = 0. At a later time t, the distance x needs to

be larger for the same value of (x −ct). Consequently, f(x−ct) has the same shape

as f(x), except displaced by an amount ct along the x-axis. Therefore, the speed of

propagation of wave shape f(x − ct) along the positive x-axis is c.

As an example of solution of the wave equation, assume initial conditions in the

form

η(x, 0) = F(x) and

∂η

∂t

(x, 0) = G(x), (7.4)

Then equation (7.3) requires that

f(x)+ g(x) = F(x) and −

G(x),

which gives the solution

f(x) =



F(x)−



G(ξ) dξ



, g(x) =



F(x)+



G(ξ) dξ



(7.5)

The case of zero initial velocity [G(x) = 0] is interesting. It corresponds to an initial

displacement of the surface into an arbitrary proﬁle F(x), which is then left alone.

In this case equation (7.5) reduces to f(x) = g(x) = F(x)/2, so that solution (7.5)

becomes

η =

F(x − ct) +

F(x + ct), (7.6)

The nature of this solution is illustrated in Figure 7.2. It is apparent that half the initial

disturbance propagates to the right and the other half propagates to the left. Widths

of the two components are equal to the width of the initial disturbance. Note that

boundary conditions have not been considered in arriving at equation (7.6). Instead,

the boundaries have been assumed to be so far away that the reﬂected waves do not

return to the region of interest.

Figure 7.1 Proﬁles of f(x − ct) at two times.

216 Gravity Waves

Figure 7.2 Wave proﬁles at three times. The initial proﬁle is F(x) and the initial velocity is assumed to

be zero. Half the initial disturbance propagates to the right and the other half propagates to the left.

3. Wave Parameters

According to Fourier’s principle, any arbitrary disturbance can be decomposed into

sinusoidal wave components of different wavelengths and amplitudes. Consequently,

it is important to study sinusoidal waves of the form

η = a sin



2π

(x − ct)



. (7.7)

The argument 2π(x − ct )/λ is called the phase of the wave, and points of constant

phase are those where the waveform has the same value, say a crest or trough. Since η

varies between ±a, a is called the amplitude of the wave. The parameter λ is called the

wavelength because the value of η in equation (7.7) does not change if x is changed

by ±λ. Instead of using λ, it is more common to use the wavenumber deﬁned as

k ≡

2π

, (7.8)

which is the number of complete waves in a length 2π. It can be regarded as the

“spatial frequency” (rad/m). The waveform equation (7.7) can then be written as

η = a sin k(x − ct). (7.9)

3. Wave Parameters 217

The period T of a wave is the time required for the condition at a point to repeat itself,

and must equal the time required for the wave to travel one wavelength:

T =

. (7.10)

The number of oscillations at a point per unit time is the frequency, given by

ν =

. (7.11)

Clearly c = λν. The quantity

ω = 2πν = kc, (7.12)

is called the circular frequency; it is also called the “radian frequency” because it is

the rate of change of phase (in radians) per unit time. The speed of propagation of the

waveform is related to k and ω by

c =

(7.13)

which is called the phase speed, as it is the rate at which the “phase” of the wave

(crests and troughs) propagates. We shall see that the phase speed may not be the

speed at which the envelope of a group of waves propagates. In terms of ω and k, the

waveform equation (7.7) is written as

η = a sin(kx − ωt). (7.14)

So far we have been considering waves propagating in the x direction only. For

three-dimensional waves of sinusoidal shape, equation (7.14) is generalized to

η = a sin(kx + ly + mz − ωt) = a sin(K

•

x − ωt), (7.15)

where K = (k,l,m)is a vector, called the wavenumber vector, whose magnitude is

given by

= k

+ l

+ m

. (7.16)

It is easy to see that the wavelength of equation (7.15) is

λ =

2π

, (7.17)

which is illustrated in Figure 7.3 in two dimensions. The magnitude of phase velocity

is c = ω/K, and the direction of propagation is that of K. We can therefore write the

phase velocity as the vector

c =

, (7.18)

where K/K represents the unit vector in the direction of K.

218 Gravity Waves

Figure 7.3 Wave propagating in the xy-plane. The inset shows how the components c

and c

are added

to give the resultant c .

From Figure 7.3, it is also clear that the phase speeds (that is, the speeds of

propagation of lines of constant phase) in the three Cartesian directions are

. (7.19)

The preceding shows that the components c

, c

, and c

are each larger than the

resultant c = ω/K. It is clear that the components of the phase velocity vector c do

not obey the rule of vector addition. The method of obtaining c from the components

and c

is illustrated at the top of Figure 7.3. The peculiarity of such an addition

rule for the phase velocity vector merely reﬂects the fact that phase lines appear to

propagate faster along directions not coinciding with the direction of propagation,

say the x and y directions in Figure 7.3. In contrast, the components of the “group

velocity” vector c

do obey the usual vector addition rule, as we shall see later.

We have assumed that the waves exist without a mean ﬂow. If the waves are

superposed on a uniform mean ﬂow U, then the observed phase speed is

= c + U.

A dot product of the forementioned with the wavenumber vector K, and the use of

equation (7.18), gives

= ω + U

•

K, (7.20)

where ω

is the observed frequency at a ﬁxed point, and ω is the intrinsic frequency

measured by an observer moving with the mean ﬂow. It is apparent that the frequency

4. Surface Gravity Waves 219

ofawaveisDoppler shifted by an amount U

•

K due to the mean ﬂow. Equation (7.20)

is easy to understand by considering a situation in which the intrinsic frequency ω is

zero and the ﬂow pattern has a periodicity in the x direction of wavelength 2π/k.If

this sinusoidal pattern is translated in the x direction at speed U , then the observed

frequency at a ﬁxed point is ω

= Uk.

The effects of mean ﬂow on frequency will not be considered further in this

chapter. Consequently, the involved frequencies should be interpreted as the intrinsic

frequency.

4. Surface Gravity Waves

In this section we shall discuss gravity waves at the free surface of a sea of liquid of

uniform depth H , which may be large or small compared to the wavelength λ.We

shall assume that the amplitude a of oscillation of the free surface is small, in the sense

that both a/λ and a/H are much smaller than one. The condition a/λ  1 implies

that the slope of the sea surface is small, and the condition a/H  1 implies that the

instantaneous depth does not differ signiﬁcantly from the undisturbed depth. These

conditions allow us to linearize the problem. The frequency of the waves is assumed

large compared to the Coriolis frequency, so that the waves are unaffected by the

earth’s rotation. Here, we shall neglect surface tension; in water its effect is limited

to wavelengths <7 cm, as discussed in Section 7. The ﬂuid is assumed to have small

viscosity, so that viscous effects are conﬁned to boundary layers and do not affect the

wave propagation signiﬁcantly. The motion is assumed to be generated from rest, say,

by wind action or by dropping a stone. According to Kelvin’s circulation theorem,

the resulting motion is irrotational, ignoring viscous effects, Coriolis forces, and

stratiﬁcation (density variation).

Formulation of the Problem

Consider a case where the waves propagate in the x direction only, and that the

motion is two dimensional in the xz-plane (Figure 7.4). Let the vertical coordinate z

Figure 7.4 Wave nomenclature.

220 Gravity Waves

be measured upward from the undisturbed free surface. The free surface displacement

is η(x, t). Because the motion is irrotational, a velocity potential φ can be deﬁned

such that

u =

∂φ

∂x

w =

∂φ

∂z

. (7.21)

Substitution into the continuity equation

∂u

∂x

∂w

∂z

= 0, (7.22)

gives the Laplace equation

∂

∂x

∂

∂z

= 0. (7.23)

Boundary conditions are to be satisﬁed at the free surface and at the bottom. The

condition at the bottom is zero normal velocity, that is

w =

∂φ

∂z

= 0atz =−H. (7.24)

At the free surface, a kinematic boundary condition is that the ﬂuid particle never

leaves the surface, that is

Dη

= w

at z = η,

where D/Dt = ∂/∂t + u(∂/∂x), and w

is the vertical component of ﬂuid velocity

at the free surface. This boundary condition is the specialization of that discussed in

Chapter 4.19 to zero mass ﬂow across the wave surface. The forementioned condition

can be written as

∂η

∂t

+ u

∂η

∂x



z=η

∂φ

∂z



z=η

. (7.25)

For small-amplitude waves both u and ∂η/∂x are small, so that the quadratic term

u(∂η/∂x) is one order smaller than other terms in equation (7.25), which then sim-

pliﬁes to

∂η

∂t

∂φ

∂z



z=η

, (7.26)

We can simplify this condition still further by arguing that the right-hand side can be

evaluated at z = 0 rather than at the free surface. To justify this, expand ∂φ/∂z in a

Taylor series around z = 0:

∂φ

∂z



z=η

∂φ

∂z



z=0

+η

∂

∂z



z=0

···

∂φ

∂z



z=0

Therefore, to the ﬁrst order of accuracy desired here, ∂φ/∂z in equation (7.26) can

be evaluated at z = 0. We then have

∂η

∂t

∂φ

∂z

at z = 0. (7.27)

4. Surface Gravity Waves 221

The error involved in approximating equation (7.26) by (7.27) is explained again later

in this section.

In addition to the kinematic condition at the surface, there is a dynamic condition

that the pressure just below the free surface is always equal to the ambient pressure,

with surface tension neglected. Taking the ambient pressure to be zero, the condition is

p = 0atz = η. (7.28)

Equation (7.28) follows from the boundary condition on τ

•

n, which is continuous

across an interface as established in Chapter 4, Section 19. As before, we shall simplify

this condition for small-amplitude waves. Since the motion is irrotational, Bernoulli’s

equation (see equation (4.81))

∂φ

∂t

+ w

) +

+ gz = F(t), (7.29)

is applicable. Here, the function F(t) can be absorbed in ∂φ/∂t by redeﬁning φ.

Neglecting the nonlinear term (u

+ w

) for small-amplitude waves, the linearized

form of the unsteady Bernoulli equation is

∂φ

∂t

+ gz = 0. (7.30)

Substitution into the surface boundary condition (7.28) gives

∂φ

∂t

+ gη = 0atz = η. (7.31)

As before, for small-amplitude waves, the term ∂φ/∂t can be evaluated at z = 0

rather than at z = η to give

∂φ

∂t

=−gη at z = 0. (7.32)

Solution of the Problem

Recapitulating, we have to solve

∂

∂x

∂

∂z

= 0. (7.22)

subject to the conditions

∂φ

∂z

= 0atz =−H, (7.24)

∂φ

∂z

∂η

∂t

at z = 0, (7.27)

∂φ

∂t

=−gη at z = 0. (7.32)