u ’ δu‚xx u + δ(‚x u)2

2

‚t u + ‚x =0

2 3 2

.

.

.

where the dots refer to an in¬nite sequence of (not exactly obvious) conserva-

tion laws. Since more than one conservation law survives, the KdV equation

cannot have shock-like solutions. Instead, the steepening wave breaks up

into a sequence of solitons. A movie of this phenomenon can be seen on the

course home-page.

Example: Hydraulic Jump, or Bore

v

h 2

2

h1 v1

A Hydraulic Jump.

206 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

A stationary hydraulic jump is a place in a stream where the ¬‚uid abruptly

increases in depth from h1 to h2 , and simultaneously slows down from super-

critical (faster than wave-speed) ¬‚ow to subcritical (slower than wave-speed)

¬‚ow. Such jumps are commonly seen near weirs, and whitewater rapids4 . A

circular hydraulic jump is easily created in your kitchen sink. The moving

equivalent is the the tidal bore. A link to pictures of hydraulic jumps and

bores is provided on the course web-site.

The equations governing uniform (meaning that v is independent of the

depth) ¬‚ow in channels are mass conservation

‚t h + ‚x {hv} = 0, (7.88)

and Euler™s equation

‚t v + v‚x v = ’‚x {gh}. (7.89)

We could manipulate these into the Riemann form, and work from there, but

it is more direct to combine them to derive the momentum conservation law

1

‚t {hv} + ‚x hv 2 + gh2 = 0. (7.90)

2

From Euler™s equation, assuming steady ¬‚ow, v = 0, we can also deduce

™

Bernoulli™s equation

12

v + gh = const., (7.91)

2

which is an energy conservation law. At the jump, mass and momentum

must be conserved:

h1 v1 = h2 v2 ,

1 1

h1 v1 + gh2 = h2 v2 + gh2 ,

2 2

(7.92)

21 22

and v2 may be eliminated to ¬nd

1 h2

2

v1 = g (h1 + h2 ). (7.93)

2 h1

A change of frame reveals that v1 is the speed at which a wall of water of

height h = (h2 ’ h1 ) would propagate into stationary water of depth h1 .

4

The breaking crest of Frost™s “white wave” is probably as much as an example of a

hydraulic jump as of a smooth downstream wake.

7.3. NON-LINEAR WAVES 207

Bernoulli™s equation is inconsistent with the two equations we have used,

and so

12 12

v1 + gh1 = v2 + gh2 . (7.94)

2 2

This means that energy is being dissipated: for strong jumps, the ¬‚uid down-

stream is turbulent. For weaker jumps, the energy is radiated away in a train

of waves “ the so-called “undular bore”.

Example: Shock Wave in Air: At a shock wave in air we have conservation

of mass

ρ1 v1 = ρ2 v2 , (7.95)

momentum

2 2

ρ1 v1 + P1 = ρ2 v2 + P2 . (7.96)

In this case, however, Bernoulli™s equation does hold5 , so

12 12

v1 + h1 = v2 + h2 . (7.97)

2 2

Here, h is the speci¬c enthalpy (E + P V per unit mass). Entropy, though, is

not conserved, so we cannot use P V γ = const. across the shock. From mass

and momentum conservation alone we ¬nd

P2 ’ P1

ρ2

2

v1 = . (7.98)

ρ2 ’ ρ1

ρ1

For an ideal gas with cp /cv = γ, we can use energy conservation to to elimi-

nate the densities, and ¬nd

γ + 1 P2 ’ P1

v1 = c0 1 + . (7.99)

2γ P1

Here, c0 is the speed of sound in the undisturbed gas.

5

Recall that enthalpy is conserved in a throttling process, even in the presence of dissi-

pation. Bernoulli™s equation for a gas is the generalization of this thermodynamic result to

include the kinetic energy of the gas. The di¬erence between the shock wave in air, where

Bernoulli holds, and the hydraulic jump, where it does not, is that the enthalpy of the gas

keeps track of the lost mechanical energy, which has been absorbed by the internal degrees

of freedom. The Bernoulli equation for channel ¬‚ow keeps track only of the mechanical

energy of the mean ¬‚ow.

208 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

7.3.3 Weak Solutions

We want to make mathematically precise the sense in which a function u

with a discontinuity can be a solution to the di¬erential equation

1n 1

un+1 = 0,

‚t u + ‚x (7.100)

n n+1

even though the equation is surely meaningless if the functions to which the

derivatives are being applied are not in fact di¬erentiable.

We could play around with distributions like the Heaviside step function

or the Dirac delta, but this is unsafe for non-linear equations, because the

product of two distributions is generally not meaningful. What we do is

introduce a new concept. We say that u is a weak solution to (7.100) if

n

dx dt un ‚t • + un+1 ‚x • = 0, (7.101)

n+1

R2

for all test functions • is some suitable space T . This equation has formally

been obtained from (7.100) by multiplying it by •(x, t), integrating over

all space-time, and then integrating by parts to move the derivatives o¬ u,

and onto the smooth function •. If u is assumed smooth then all these

manipulations are legitimate and the new equation (7.101) contains no new

information. A conventional solution to (7.100) is therefore also a weak

solution. The new formulation (7.101), however, admits solutions in which u

has shocks.

Let us see what is required of a weak solution if we assume that u is

everywhere smooth except for a single jump from u’ (t) to u+ (t) at the point