amplitude motion of particles at the surface we may, to a ¬rst approximation,

set x = x0 , y = h0 on the right-hand side. The orbits of the surface particles

are therefore approximately

ak

x(t) = x0 ’ cos(kx0 ’ ωt),

ω

ak

y(t) = y0 ’ sin(kx0 ’ ωt). (7.15)

ω

y

x

Surface waves on deep water.

For right-moving waves, the particle orbits are clockwise circles. At the

wave-crest the particles move in the direction of the wave propagation; in

the troughs they move in the opposite direction. The ¬gure shows that this

results in a characteristic up-down asymmetry in the wave pro¬le.

When the e¬ect of the bottom becomes signi¬cant, the circular orbits

deform into ellipses. For shallow water waves, the motion is principally back

and forth with motion in the y direction almost negligeable.

7.1. DISPERSIVE WAVES 187

7.1.2 Group Velocity

The most important e¬ect of dispersion is that the group velocity of the waves

” the speed at which a wave-packet travels ” di¬ers from the phase velocity

” the speed at which individual wave-crests move. The group velocity is

also the speed at which the energy associated with the waves travels.

Suppose that we have waves with dispersion equation ω = ω(k). A right-

going wave-packet of ¬nite extent, and with initial pro¬le •(x), can be Fourier

analyzed to give

∞ dk

A(k)eikx .

•(x) = (7.16)

’∞ 2π

x

A right-going wavepacket.

At later times this will evolve to

dk

∞

A(k)eikx’iω(k)t .

•(x, t) = (7.17)

2π

’∞

Let us suppose for the moment that A(k) is non-zero only for a narrow band

of wavenumbers around k0 , and that, restricted to this narrow band, we can

approximate the full ω(k) dispersion equation by

ω(k) ≈ ω0 + U (k ’ k0 ). (7.18)

Thus

dk

∞

A(k)eik(x’U t)’i(ω0 ’U k0 )t .

•(x, t) = (7.19)

2π

’∞

Comparing this with the Fourier expression for the initial pro¬le, we ¬nd

that

•(x, t) = e’i(ω0 ’U k0 )t •(x ’ U t). (7.20)

188 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

The pulse envelope therefore travels at speed U . This velocity

‚ω

U≡ (7.21)

‚k

is the group velocity. The individual wave crests, on the other hand, move

at the phase velocity ω(k)/k.

When the intial pulse contains a broad range of frequencies we can still

explore its evolution. We make use of a powerful tool for estimating the be-

havior of integrals that contain a large parameter. In this case the parameter

is the time t. We begin by writing the Fourier representation of the wave as

dk

∞

A(k)eitψ(k)

•(x, t) = (7.22)

2π

’∞

where

x

’ ω(k).

ψ(k) = k (7.23)

t

Now look at the behaviour of this integral as t becomes large, but while we

keep the ratio x/t ¬xed. Since t is very large, any variation of ψ with k

will make the integrand a very rapidly oscillating function of k. Cancellation

between adjacent intervals with opposite phase will cause the net contribution

from such a region of the k integration to be very small. The principal

contribution will come from the neighbourhood of stationary phase points,

i.e. points where

dψ x ‚ω

=’

0= . (7.24)

dk t ‚k

This means that, at points in space where x/t = U , we will only get contri-

butions from the Fourier components with wave-number satisfying

‚ω

U= . (7.25)

‚k

The initial packet will therefore spread out, with those components of the

wave having wave-number k travelling at speed

‚ω

vgroup = . (7.26)

‚k

This is the same expression for the group velocity that we obtained in the

narrow-band case. Again this speed of propagation should be contrasted

with that of the wave-crests, which travel at

ω

vphase = . (7.27)

k

7.1. DISPERSIVE WAVES 189

The “stationary phase” argument may seem a little hand-waving, but it can

be developed into a systematic approximation scheme. We will do this in

later chapters.

Example: Water Waves. The dispersion equation for waves on deep water is

√

ω = gk. The phase velocity is therefore

g

vphase = , (7.28)

k

whilst the group velocity is

1 g 1

vgroup = = vphase . (7.29)

2 k 2

This di¬erence is easily demonstrated by tossing a stone into a pool and

observing how individual wave-crests overtake the circular wave packet and

die out at the leading edge, while new crests and troughs come into being at