‚xi t ‚t x ‚xj xi ‚xi xj

The subscripts indicate what is being left ¬xed when we di¬erentiate. We

must be careful about this, because we want to use the dispersion equation

to express ω as a function of k and x, and the wave-vector k will itself be a

function of x and t.

Taking this dependence into account, we write

‚ω ‚ω ‚ω ‚kj

= + . (7.39)

‚xi ‚xi ‚kj ‚xi

t t

k x

We now use (7.38) to rewrite this as

‚ki ‚ω ‚ki ‚ω

=’

+ . (7.40)

‚t ‚kj ‚xj ‚xi

t

x x k

194 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

Interpreting the left hand side as a convective derivative

dki ‚ki

+ (vg ·

= )ki ,

dt ‚t x

we read o¬ that

dki ‚ω

=’ (7.41)

dt ‚xi k

provided we are moving at velocity

dxi ‚ω

= (vg )i = . (7.42)

dt ‚ki x

Since this is the group velocity, the packet of waves is actually travelling at

this speed. The last two equations therefore tell us how the orientation and

wavelength of the wave train evolve if we ride along with the packet as it is

refracted by the inhomogeneity.

The formul¦

‚ω

™

k=’ ,

‚x

‚ω

™

x= , (7.43)

‚k

are Hamilton™s ray equations. These Hamilton equations are identical in form

to Hamilton™s equations for classical mechanics

‚H

p=’

™ ,

‚x

‚H

™

x= , (7.44)

‚p

except that k is playing the role of the canonical momentum, p, and ω(k, x)

replaces the Hamiltonian, H(p, x). This formal equivalence of geometric

optics and classical mechanics was mystery in Hamilton™s time. Today we

understand that classical mechanics is nothing but the geometric optics limit

of wave mechanics.

7.2. MAKING WAVES 195

7.2 Making Waves

Many waves occuring in nature are generated by the energy of some steady

¬‚ow being stolen away to drive an oscillatory motion. Familiar examples

include the music of a ¬‚ute and the waves raised on the surface of water by

the wind. The latter process is quite subtle and was not understood until the

work of J. W. Miles in 1957. Miles showed that in order to excite waves the

wind speed has to vary with the height above the water, and that waves of

a given wavelength take energy only from the wind at that height where the

windspeed matches the phase velocity of the wave. The resulting resonant

energy transfer turns out to have analogues in many branches of science. In

this section we will exhibit this phenomenon in the simpler situation where

the varying ¬‚ow is that of the water itself.

7.2.1 Rayleigh™s Equation

Consider water ¬‚owing in a shallow channel where friction forces keep the

water in contact the stream-bed from moving. We will show that the resulting

shear ¬‚ow is unstable to the formation of waves on the water surface. The

consequences of this instability are most often seen in a thin sheet of water

running down the face of a dam. The sheet starts o¬ ¬‚owing smoothly, but,

as the water descends, waves form and break, and the water reaches the

bottom in irregular pulses called roll waves.

It is easiest to describe what is happening from the vantage of a reference

frame that rides along with the surface water. In this frame the velocity

pro¬le of the ¬‚ow will be as shown in the ¬gure.

y

y0

U(y)

h

x

The velocity pro¬le U (y) in a frame at which the surface is at rest.

Since the ¬‚ow is incompressible but not irrotational, we will describe the

196 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

motion by using a stream function Ψ, in terms of which the ¬‚uid velocity is

given by

vx = ’‚y Ψ,

vy = ‚x Ψ. (7.45)

· v = 0, while the (z compo-

This parameterization automatically satis¬es

nent of) the vorticity becomes

2

„¦ ≡ ‚x vy ’ ‚y vx = Ψ. (7.46)

We will consider a stream function of the form2

Ψ(x, y, t) = ψ0 (y) + ψ(y)eikx’iωt , (7.47)

where ψ0 obeys ’‚y ψ0 = vx = U (y), and describes the horizontal mean ¬‚ow.

The term containing ψ(y) represents a small-amplitude wave disturbance

superposed on the mean ¬‚ow. We will investigate whether this disturbance

grows or decreases with time.

Euler™s equation can be written as,

v2

v+v—„¦=’

™ P+ + gy = 0. (7.48)

2

Taking the curl of this, and taking into account the two dimensional character

of the problem, we ¬nd that

‚t „¦ + (v · )„¦ = 0. (7.49)

This, a general property of two-dimensional incompressible motion, says that

vorticity is convected with the ¬‚ow. We now express (7.49) in terms of Ψ,

when it becomes

2™