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v>c v=c

v<c

c-v

c c

c

v v

v

c+v

c+v

Figure 9.3: Waves on a water owing over a rock when v < c (left panel), v > c (middle

panel) and v = c (right panel.

denoted by v. At the bottom of the channel a rock is disrupting the ow. This rock

generates water-waves that propagate with a velocity c compared to the moving water.

When the ow velocity is less than the wave velocity (v < c, see the left panel of gure 9.3)

the waves propagate upstream with an absolute velocity c v and propagate downstream

;

with an absolute velocity c + v. When the ow velocity is larger than the wave velocity

(v > c, see the middle panel of gure 9.3) the waves move downstream only because the

wave velocity is not su ciently large to move the waves against the current. The most

interesting case is when the ow velocity equals the wave velocity (v = c, see the right

panel of gure 9.3). In that case the waves that move upstream have an absolute velocity

given by c v = 0. In other words, these waves do not move with respect to the rock that

;

generates the waves. This wave is continuously excited by the rock, and through a process

similar to an oscillator that is driven at its resonance frequency the wave grows and grows

until it ultimately breaks and becomes turbulent. This is the reason why one can see strong

turbulent waves over boulders and other irregularities in streams. For further details on

channel ow and hydraulic jumps the reader can consult chapter 9 of Whitaker 67]. In

general the advective terms play a crucial role steepening and breaking of waves and the

formation of shock waves. This is described in much detail by Whitham 66].

9.3 Geometric ray theory

Geometric ray theory is an approximation that accounts for the propagation of waves

along lines through space. The theory nds is conceptual roots in optics, where for a

long time one has observed that a light beam propagates along a well-de ned trajectory

through lenses and many other optical devices. Mathematically, this behavior of waves is

accounted for in geometric ray theory, or more brie y \ray theory."

Ray theory is derived here for the acoustic wave equation rather than for the prop-

agation of light because pressure waves are described by a scalar equation rather than

the vector equation that governs the propagation of electromagnetic waves. The starting

point is the acoustic wave equation (6.7) of section 6.3:

!2 p = 0:

1 + c2 (9.14)

r rp

For simplicity the source term in the right hand side has been set to zero. In addition,

the relation c2 = = has been used to eliminate the bulk modulus in favor of the wave

velocity c. Both the density and the wave velocity are arbitrary functions of space.

9.3. GEOMETRIC RAY THEORY 95

In general it is not possible to solve this di erential equation in closed form. Instead

we will seek an approximation by writing the pressure as:

p(r !) = A(r !)ei (r !) (9.15)

with A and real functions. Any function p(r !) can be written in this way.

Problem a: Insert the solution (9.15) in the acoustic wave equation (9.14), separate the

real and imaginary parts of the resulting equation to deduce that (9.14) is equivalent

to the following equations:

!2 A= 0

2 1 (r

2A A + (9.16)

| {z } | {z }

} |c(4) }

2

r ; jr j ; rA)

{z

| {z

(1) (2)

(3)

and

1 (r

) + Ar2 )A = 0 :

2 (rA (9.17)

r ; r

The equations are even harder to solve than the acoustic wave equation because they

are nonlinear in the unknown functions A and whereas the acoustic wave equation is

linear in the pressure p. However, the equations (9.16) and (9.17) form a good starting

point for making the ray-geometric approximation. First we will analyze expression (9.16).

Assume that the density varies on a length scale L , that the amplitude A of the

wave- eld varies on a characteristic length scale LA. Furthermore the wavelength of the

waves is denoted by .

Problem b: Explain that the wavelength is the length-scale over which the phase of

the waves varies.

Problem c: Use the results of section 9.1 to obtain the following estimates of the order

of magnitude of the terms (1) (4) in equation (9.16):

;

!2 A A (9.18)

A 2A A

1 (r

2A A

L2 2 c2 2

LAL

r jr j rA)

A

To make further progress we assume that the length-scale of both the density variations

LA and L.

and the amplitude variations are much longer than a wavelength:

Problem d: Show that under this assumption the terms (1) and (3) in equation (9.16)

are much smaller than the terms (2) and (4).

Problem e: Convince yourself that ignoring the terms (1) and (3) in (9.16) gives the

following (approximate) expression:

2 = !2 : (9.19)

c2

jr j

Problem f: The approximation (9.19) was obtained under the premise that 1= .

jr j

Show that this assumption is satis ed by the function in (9.19).

CHAPTER 9. SCALE ANALYSIS

96

Whenever one makes approximations by deleting terms that scale-analysis predicts to be

small one has to check that the nal solution is consistent with the scale-analysis that is

used to derive the approximation.

Note that the original equation (9.16) contains both the amplitude A and the phase

but that (9.19) contains the phase only. The approximation that we have made has thus

decoupled the phase from the amplitude, this simpli es the problem considerably. The

frequency enters the right hand side of this equation only through a simple multiplication

with !2 . The frequency dependence of can be found by substituting

(r !) = ! (r) : (9.20)

Problem g: Show that the equations (9.19) and (9.17) after this substitution are given

by:

(r)j2 = c12 (9.21)

jr

and

1 (r

) + Ar2 )A = 0 :

2 (rA (9.22)

r ; r

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