Internal Undular Bores in the Coastal Ocean
37
The Whitham modulation theory allows this cnoidal wave to vary slowly with
𝜏, X, that is the wavenumber k, modulus m and mean level d vary slowly with
𝜏, X. The Whitham modulation equations describing this variation can be obtained
by averaging conservation laws, the original Whitham method, see [28, 29] or by
exploiting the integrability of the constant-coefficient KdV equation, see [21] for
instance. Because here we are concerned with the case when 𝛼 = 𝛼(𝜏) varies slowly
with 𝜏, and so the variable-coefficient KdV equation (13) is not integrable, we will
use the original Whitham method, readily adapted to this present case. A similar
strategy was used by [24] for a frictionally perturbed KdV equation. An alternative
method developed by [22] for a perturbed KdV equation is not available here because
to use it one must make a change of variable in (13) ̃
U = 𝛼U to generate a KdV equation for ̃
U with a perturbation term of the form 𝛼 𝜏 ̃
U∕𝛼. But as one of our concerns is
with the situation when 𝛼 passes through zero, this approach cannot be used here.
As three modulation equations are needed, we supplement (15, 16) with the equation for conservation of waves,
k 𝜏 + (kV) X = 0 .
(30)
The remaining two modulation equations are obtained by inserting the cnoidal wave
solution into the conservation laws (15, 16) and averaging over the phase 𝜃. The
outcomes are
d 𝜏 + 𝛼M X = 0 , M = <
U
2
2
> ,
(31)
M 𝜏 + P X = 0 , P = <
𝛼U
3
3
−
3U
2
X
2
> ,
(32)
where the < ⋯ > denotes a 2𝜋-average over 𝜃. The expression M is given by
M =
d 2
2
+
a 2
2
{C 4 − b
2
} ,
C 4 =
1
3m 2 K(m)
{3m
2 K(m) − 5mK(m) + 4mE(m) + 2K(m) − 2E(m)} ,
(33)
while that for P is given by
P = 𝛼{−
2d
3
3
+ 2dM + a
3
{−
2b
3
3
+
(1 − m)b
2m
+ (b +
1 − 2m
2m
) C 4 +
5
6
C 6 } ,
C 6 =
1
15m 3 K(m)
{15m
3 K(m) − 34m
2 K(m) + 23m
2 E(m)
+27mK(m) − 23mE(m) − 8K(m) + 8E(m)} .
(34)