Subsituting the expression (7.47) into (7.50), and keeping only terms of ¬rst

order in ψ, gives

d2 d2

2

’ k2 ψ + ikψ‚y (’‚y U ) = 0,

’iω ’ k ψ + iU k

dy 2 dy 2

2

The physical stream function is, of course, the real part of this expression.

7.2. MAKING WAVES 197

or

d2 ‚2U 1

2

’k ψ’ ψ = 0. (7.51)

dy 2 ‚y 2 (U ’ ω/k)

This is Rayleigh™s equation 3 . If only the ¬rst term were present, it would

have solutions ψ ∝ e±ky , and we would have recovered the results of section

7.1.1. The second term is signi¬cant, however. It will diverge if there is a

point yc such that U (yc ) = ω/k. In other words, if there is a depth at which

the ¬‚ow speed coincides with the phase velocity of the wave disturbance, thus

allowing a resonant interaction between the wave and ¬‚ow. An actual in¬nity

in (7.51) will be evaded, though, because ω will gain a small imaginary part

ω ’ ωR + iγ. A positive imaginary part means that the wave amplitude is

growing exponentially with time. A negative imaginary part means that the

wave is being damped. With γ included, we then have

U ’ ωR /k

1 γ

≈ δ U (y) ’ ωR /k

+ iπ sgn

(U ’ ωR /k)2 + γ 2

(U ’ ω/k) k

’1

U ’ ωR /k γ ‚U

δ(y ’ yc ).

= + iπ sgn

(U ’ ωR /k)2 + γ 2 k ‚y yc

(7.52)

To specify the problem fully we need to impose boundary conditions on

ψ(y). On the lower surface we can set ψ(0) = 0, as this will keep the ¬‚uid

at rest there. On the upper surface y = h we apply Euler™s equation

v2

v+v—„¦ =’

™ P+ + gh = 0. (7.53)

2

We observe that P is constant, being atmostpheric pressure, and the v2 /2 can

be neglected as it is of second order in the disturbance. Then, considering

the x component, we have

k2

t

’ = ’g‚x vy dt = ’g

x gh ψ (7.54)

iω

on the free surface. To lowest order we can apply the boundary condition on

the equilibrium free surface y = y0 . The boundary condition is therefore

k2

1 dψ k ‚U

+ = g 2, y = y0 . (7.55)

ψ dy ω ‚y ω

3

Lord Rayleigh. On the stability or instability of certain ¬‚uid motions. Proc. Lond.

Math. Soc. Vol. 11 (1880)

198 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

We usually have ‚U /‚y = 0 near the surface, so this simpli¬es to

k2

1 dψ

= g 2. (7.56)

ψ dy ω

That this is sensible can be con¬rmed by considering the case of waves on

still, deep water, where ψ(y) = e|k|y . The boundary condition then reduces

to |k| = gk 2 /ω 2, or ω 2 = g|k|, which is the correct dispersion equation for

such waves.

We ¬nd the corresponding dispersion equation for waves on shallow ¬‚ow-

ing water by computing

1 dψ

, (7.57)

ψ dy y0

from Rayleigh™s equation (7.51). Multiplying by ψ — and integrating gives

d2 ‚2U 1

y0

— 2

|ψ|2 .

’k ψ+k

0= dy ψ (7.58)

dy 2 ‚y 2 (ω ’ U k)

0

An integration by parts then gives

y0

‚2U

— dψ dψ 1

y0

2 2

|ψ|2 .

+ k |ψ| +

ψ = dy (7.59)

‚y 2 (U ’ ω/k)

dy dy

0

0

The lower limit makes no contribution, since ψ— is zero there. On using (7.52)

and taking the imaginary part, we ¬nd

’1

‚2U

dψ γ ‚U

ψ— |ψ(yc )|,

Im = sgn π (7.60)

‚y 2

dy k ‚y

y0 yc yc

or

’1

‚2U |ψ(yc )|2

1 dψ γ ‚U

Im = sgn π . (7.61)

‚y 2 |ψ(y0)|2

ψ dy k ‚y

y0 yc yc

This equation is most useful if the interaction with the ¬‚ow does not sub-

stantially perturb ψ(y) away from the still-water result ψ(y) = sinh(|k|y),

and assuming this is so provides a reasonable ¬rst approximation.

If we insert (7.61) into (7.56), where we approximate,

k2 k2 k2

≈g ’ 2ig

g γ,

2 3

ω2 ωR ωR

7.3. NON-LINEAR WAVES 199

we ¬nd

3

ωR 1 dψ

γ= Im

2gk 2 ψ dy y0

’1

3

‚2U |ψ(yc )|2

γ ωR ‚U

= sgn π . (7.62)

2gk 2 ‚y 2 |ψ(y0)|2