P

x

Simple wave characteristics.

The ¬gure shows that the straight-line characteristics travel faster in the high

pressure region, and eventually catch up with and intersect the slower-moving

characteristics. When this happens the dynamical variables will become

multivalued. How do we deal with this?

7.3.2 Shocks

Let us untangle the multivaluedness by drawing another set of pictures. Sup-

pose u obeys the non-linear “half” wave equation

(‚t + u‚x )u = 0. (7.75)

The velocity of propagation of the wave is therefore u itself, so the parts of

the wave with large u will overtake those with smaller u, and the wave will

7.3. NON-LINEAR WAVES 203

“break”.

u u

a) b)

u u

c) d)

?

A breaking non-linear wave.

Physics does not permit such multivalued solutions, and what usually hap-

pens is that the assumptions underlying the model which gave rise to the

nonlinear equation will no longer be valid. New terms should be included in

the equation which prevent the solution becoming multivalued, and instead

a steep “shock” will form.

u

d™)

Formation of a shock.

Examples of an equation with such additional terms are Burgers™ equation

2

(‚t + u‚x )u = ν‚xx u, (7.76)

and the Korteweg de-Vries (KdV) equation (4.11), which, by a suitable rescal-

ing of x and t, we can write as

3

(‚t + u‚x )u = δ ‚xxx u. (7.77)

Burgers™ equation, for example, can be thought of as including the e¬ects of

thermal conductivity, which was not included in the derivation of Riemann™s

204 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

equations. In both these modi¬ed equations, the right hand side is negligeable

when u is slowly varying, but it completely changes the character of the

solution when the waves steepen and try to break.

Although these extra terms are essential for the stabilization of the shock,

once we know that such a discontinuous solution has formed, we can ¬nd

many of its properties ” for example the propagation velocity ” from general

principles, without needing their detailed form. All we need is to know what

conservation laws are applicable.

Multiplying (‚t + u‚x )u = 0 by un’1 , we deduce that

1n 1

un+1 = 0,

‚t u + ‚x (7.78)

n n+1

and this implies that

∞

un dx

Qn = (7.79)

’∞

is time independent. There are in¬nitely many of these conservation laws,

one for each n. Suppose that the n-th conservation law continues to hold even

in the presence of the shock, and that the discontinuity is at X(t). Then

d X(t) ∞

n

un dx = 0.

u dx + (7.80)

dt ’∞ X(t)

This is equal to

X(t) ∞

™ ™

un (X)X un (X)X n

‚t un dx = 0,

’ + ‚t u dx + (7.81)

’ +

’∞ X(t)

where un (X) ≡ un (X’ ) and un (X) ≡ un (X+ ). Now, using (‚t +u‚x )u = 0

’ +

in the regions away from the shock, where it is reliable, we can write this as

n n

X(t) ∞

™

(un un ) X ‚x un dx ’ ‚x un dx

’ =’

+ ’

n + 1 ’∞ n+1 X(t)

n

(un+1 ’ un+1 ).

= (7.82)

+ ’

n+1

The velocity at which the shock moves is therefore

(un+1 ’ un+1 )

n

™ + ’

X= . (7.83)

n n

(u+ ’ u’ )

n+1

7.3. NON-LINEAR WAVES 205

Since the shock can only move at one velocity, only one of the in¬nitely many

conservation laws can continue to hold in the modi¬ed theory!

Example: Burgers™ equation. From

2

(‚t + u‚x )u = ν‚xx u, (7.84)

we deduce that

12

u ’ ν‚x u = 0,

‚t u + ‚x (7.85)

2

so that Q1 = u dx is conserved, but further investigation shows that no

other conservation law survives. The shock speed is therefore

2 2

™ = 1 (u+ ’ u’ ) = 1 (u+ + u’ ).

X (7.86)

2 (u+ ’ u’ ) 2

Example: KdV equation. From

3

(‚t + u‚x )u = δ ‚xxx u, (7.87)

we deduce that

12 2

u ’ δ ‚xx u

‚t u + ‚x = 0,

2