This result can be extended to three dimensions with

‚ω

i

vgroup = (7.30)

‚ki

Example: de Broglie Waves. The plane-wave solutions of the time-dependent

Schr¨dinger equation

o

‚ψ 12

=’

i ψ, (7.31)

‚t 2m

are

ψ = eik·r’iωt , (7.32)

with

12

ω(k) = k. (7.33)

2m

The group velocity is therefore

1

vgroup = (7.34)

k,

m

which is the classical velocity of the particle.

190 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

7.1.3 Wakes

There are many circumstances when waves are excited by object moving at

a constant velocity through a background medium, or by a stationary object

immersed in a ¬‚ow. The resulting wakes carry o¬ energy, and therefore

ˇ

create wave drag. Wakes are involved, for example, in sonic booms, Cerenkov

radiation, the Landau criterion for super¬‚uidity, and Landau damping of

plasma oscillations. Here, we will consider some simple water-wave analogues

of these e¬ects. The common principle for all wakes is that the resulting wave

pattern is time independent when observed from the object exciting it.

Example: Obstacle in a Stream. Consider a log lying submerged in a rapidly

¬‚owing stream.

v v

Log in a stream.

The obstacle disturbs the water and generates a train of waves. If the log lies

athwart the stream, the problem is essentially one-dimensional and easy to

analyse. The essential point is that the distance of the wavecrests from the log

does not change with time, and therefore the wavelength of the disturbance

the log creates is selected by the condition that the phase velocity of the wave,

coincide with the velocity of the mean ¬‚ow1 . The group velocity does come

into play, however. If the group velocity of the waves is less that the phase

velocity, the energy being deposited in the wave-train by the disturbance will

be swept downstream, and the wake will lie behind the obstacle. If the group

velocity is higher than the phase velocity, and this is the case with very short

wavelength ripples on water where surface tension is more important than

gravity, the energy will propagate against the ¬‚ow, and so the ripples appear

upstream of the obstacle.

1

In his book Waves in Fluids, M. J. Lighthill quotes Robert Frost on this phenomenon:

The black stream, catching on a sunken rock,

Flung backward on itself in one white wave,

And the white water rode the black forever,

Not gaining but not losing.

7.1. DISPERSIVE WAVES 191

Example: Kelvin Ship Waves. A more subtle problem is the pattern of waves

left behind by a ship on deep water. The shape of the pattern is determined

by the group velocity for deep-water waves being one-half that of the phase

velocity.

C

D

A θ B

O

Kelvin™s ship-wave construction.

In order that the wave pattern be time independent, the waves emitted in

the direction AC must have phase velocity such that their crests travel from

A to C while the ship goes from A to B. The crest of the wave emitted from

the bow of the ship in the direction AC will therefore lie along the line BC ”

or at least there would be a wave crest on this line if the emitted wave energy

travelled at the phase velocity. The angle at C must be a right angle because

the direction of propagation is perpendicular to the wave-crests. Euclid, by

virtue of his angle-in-a-semicircle theorem, now tells us that the locus of

all possible points C (for all directions of wave emission) is the larger circle.

Because, however, the wave energy only travels at one-half the phase velocity,

the waves going in the direction AC actually have signi¬cant amplitude only

on the smaller circle, which has half the radius of the larger. The wake

therefore lies on, and within, the Kelvin wedge, whose boundary lies at an

angle θ to the ship™s path. This angle is determined by the ratio OD/OB=1/3

to be

θ = sin’1 (1/3) = 19.5—¦. (7.35)

Remarkably, this angle, and hence the width of the wake, is independent of

the speed of the ship.

The waves actually on the edge of the wedge are usually the most promi-

nent, and they will have crests perpendicular to the line AD. This orientation

is indicated on the left hand ¬gure, and reproduced as the predicted pattern

192 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

of wavecrests on the right. The prediction should be compared with the wave

systems in the image below.

Large-scale Kelvin wakes. (Image source: US Navy)

Small-scale Kelvin wake.

7.1. DISPERSIVE WAVES 193

7.1.4 Hamilton™s Theory of Rays

We have seen that wave packets travel at a frequency-dependent group ve-

locity. We can extend this result to study the motion of waves in weakly

inhomogeneous media, and so derive an analogy between the “geometric op-

tics” limit of wave motion and classical dynamics.

Consider a packet composed of a roughly uniformly train of waves spread

out over a region that is substantially longer and wider than their mean wave-

length. The essential feature of such a wave train is that at any particular

point of space and time, x and t, it has a de¬nite phase ˜(x, t). Once we

know this phase, we can de¬ne the local frequency, ω, and wave-vector, k,

by

‚˜ ‚˜

ω=’ , ki = . (7.36)

‚t x ‚xi t

These de¬nitions are motivated by the idea that

˜(x, t) ∼ k · x ’ ωt, (7.37)

at least locally.

We wish to understand how k changes as the wave propagates through a

slowly varying medium. We introduce the inhomogeneity by assuming that

the dispersion equation is of the form ω = ω(k, x), where the x dependence

arises, for example, as a result of a slowly varying refractive index.

Applying the equality of mixed partials to the de¬nitions of k and ω gives

us

‚ω ‚ki ‚ki ‚kj