Free waves
161
where ˆ
η(y) is the (still unknown) y-structure of the waves. Substitution of (7.17)
into (7.16) gives
ˆ
η
′′ +
σ 2 − f 2
C 2
0
− k
2
η =0 ,
(7.18a)
g ˆ
η
′ (0) + f
k
σ
ˆ
η(0) = 0 ; g ˆ
η
′ (L)+f
k
σ
ˆ
η(L)=0.
(7.18b)
where the primes indicate differentiation to y.
The solution of (7.18a) is
ˆ
η(y)=A sin αy + B cos αy
(7.19a)
α
2 =
σ 2 − f 2
C 2
0
− k
2 ,
(7.19b)
and the two homogeneous boundary conditions (7.18b) lead to a system of two
homogeneous equations for A and B, i.e.,
αgA +
fk
σ
B =0
(7.20a)
A(αg cos αL +
fk
σ
sin αL)+B(
fk
σ
cos αL − αg sin αL)=0. (7.20b)
This system has only a nontrivial solution when the coefficient determinant is
zero. This provides the eigenvalues σ as zeroes of
(σ
2 − f
2 )(σ
2 − C
2
0 k
2 )sinαL =0.
(7.21)
When σ is determined from (7.21), then α 2 follows immediately from (7.19b) and
as the equations for A and B are a dependent system, the eigenfunction ˆ
η(y) can
be calculated from (7.19a).
From (7.21) several free wave types are obtained. These are summarized below:
(i) Kelvin waves
One of the solutions of (7.21) is
σ = ±C 0 k,
(7.22)
and the phase speed of these so-called Kelvin waves is σ/k = ±C 0 ;K e l v i n
waves are hence nondispersive waves. Consider a wave that moves in the
positive x-direction, i.e. with σ = C 0 k. With α 2 = −f 2 /C 2
0 it follows that
α = if /C 0 and with (7.19a) the eigenfunction ˆ
η(y) is
ˆ
η(y)=−iA e
−
yf
C 0 .
(7.23)