yc yc

We see that either sign of γ is allowed by our analysis. Thus the resonant

interaction between the shear ¬‚ow and wave appears to lead to either ex-

ponential growth or damping of the wave. This is inevitable because our

inviscid ¬‚uid contains no mechanism for dissipation, and its motion is neces-

sarily time-reversal invariant. Nonetheless, as in our discussion of “friction

without friction” in section 5.2.2, only one sign of γ is actually observed.

This sign is determined by the initial conditions, but a rigorous explanation

of how this works mathematically is not easy, and is the subject of many

papers. These show that the correct sign is given by

’1

3

‚2U |ψ(yc )|2

ωR ‚U

γ = ’π . (7.63)

2gk 2 ‚y 2 |ψ(y0 )|2

‚y

yc yc

Since our velocity pro¬le has ‚2 U /‚y 2 < 0, this means that the waves grow

in amplitude.

We can also establish the correct sign for γ by a computing the change of

momentum in the background ¬‚ow due to the wave. Details may be found in

G. E. Vekstein Landau resonance mechanism for plasma and wind-generated

water waves. American Journal of Physics, vol.66 (1998) pages 886-92. The

crucial element is whether, in the neighbourhood of the critical depth, more

¬‚uid is overtaking the wave than lagging behind it. This is exactly what the

the quantity ‚ 2 U /‚y 2 measures.

7.3 Non-linear Waves

Non-linear e¬ects become important when some dimensionless measure of

the amplitude of the disturbance, say ∆P/P for a sound wave, or ∆h/» for

a water wave, is no longer 1.

200 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

7.3.1 Sound in Air

The simplest non-linear wave system is one-dimensional sound propagation

in a gas. This problem was studied by Riemann.

The one dimensional motion of a ¬‚uid is determined by the mass conser-

vation equation

‚t ρ + ‚x (ρv) = 0, (7.64)

and Euler™s equation of motion

ρ(‚t v + v‚x v) = ’‚x P. (7.65)

In a ¬‚uid with equation of state P = P (ρ), the speed of sound, c, is given by

dP

c2 = . (7.66)

dρ

It will in general depend on P , the speed of propagation being usually higher

when the pressure is higher.

Riemann was able to simplify these equations by de¬ning a new thermo-

dynamic variable π(P ) as

P1

π= dP, (7.67)

P0 ρc

were P0 is the equilibrium pressure of the undisturbed air. The quantity π

obeys

dπ 1

=. (7.68)

dP ρc

In terms of π, Euler™s equation divided by ρ becomes

‚t v + v‚x v + c‚x π = 0, (7.69)

whilst the equation of mass conservation divided by ρ/c becomes

‚t π + v‚x π + c‚x v = 0. (7.70)

Adding and subtracting, we get Riemann™s equations

‚t (v + π) + (v + c)‚x (v + π) = 0,

‚t (v ’ π) + (v ’ c)‚x (v ’ π) = 0. (7.71)

7.3. NON-LINEAR WAVES 201

These assert that the Riemann invariants v ± π are constant along the char-

acteristic curves

dx

= v ± c. (7.72)

dt

This tell us that signals travel at the speed v ± c. In other words, they

travel, with respect to the ¬‚uid, at the speed of sound c. Using the Riemann

equations, we can propagate initial data v(x, t = 0), π(x, t = 0) into the

future by using the method of characteristics.

t

P

B

C’

A

C+

x

A B

Characteristic curves.

A

In the ¬gure, the value of v + π is constant along the characteristic curve C+

which is the solution of

dx

=v+c (7.73)

dt

B

passing through A, while the value of v ’ π is constant along C’ which is

the solution of

dx

=v’c (7.74)

dt

passing through B. Thus the values of π and v at the point P can be found if

we know the initial values of v + π at the point A and v ’ π at the point B.

Having found v and π at P we can invert π(P ) to ¬nd the pressure P , and

hence c, and so continue the characteristics into the future, as indicated by

the dotted lines. We need, of course, to know v and c at every point along

A B

the characteristics C+ and C’ in order to construct them, and this requires

us to to treat every point as a “P”. The values of the dynamical quantities

at P therefore depend on the initial data at all points lying between A and

B. This is the domain of dependence of P

202 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

A sound wave caused by a localized excess of pressure will eventually

break up into two distinct pulses, one going forwards and one going back-

wards. Once these pulses are su¬ciently separated that they no longer inter-

act with one another they are simple waves. Consider a forward-going pulse

propagating into undisturbed air. The backward characteristics are coming

from the undisturbed region where both π and v are zero. Clearly π ’ v is

zero everywhere on these characteristics, and so π = v. Now π + v = 2v = 2π

is constant the forward characteristics, and so π and v are individually con-

stant along them. Since π is constant, so is c. With v also being constant,

this means that c + v is constant. In other words, for a simple wave, the

characteristics are straight lines.

This simple-wave simpli¬cation contains within it the seeds of its own

destruction. Suppose we have a positive pressure pulse in a ¬‚uid whose

speed of sound increases with the pressure.

t

?