a reduced model for internal waves interacting with submarine ...

The problem in conformal coordinates is
⎧
φ ξξ +φ ζζ = 0, on− h 2
L
≤ζ≤ 0,
⎪⎨
⎪⎩
( )
φ ζ (ξ, 0, t)= M(ξ)η t x(ξ, 0), t , atζ= 0,
(2.23)
φ ζ = 0, atζ=− h 2
L ,
where the previous Neumann condition at the top is now modified by M(ξ)=
z ζ (ξ, 0) which is the nonzero element of the Jacobian of the conformal mapping at
the unperturbed interface. As shown in [24], its exact expression is:
M(ξ 0 )=1− π 4
L
∫∞
h 2
−∞
h ( Lx(ξ,−h 2 /L)/l )
cosh 2( πL
2h 2
(ξ−ξ 0 ) ) dξ.
Moreover, the Jacobian along the unperturbed interface is an analytic function.
Hence a highly complex boundary profile has been converted into a smooth
variable coefficient in the equations.
To obtain the Neumann condition at the unperturbed interface in problem
(2.23), consider
φ ζ =φ x x ζ +φ z z ζ
evaluated at z=0 (equivalentlyζ= 0):
φ ζ (ξ, 0, t)=φ x (x, 0, t) x ζ (ξ, 0)+φ z (x, 0, t) z ζ (ξ, 0).
The Cauchy-Riemann relations and the fact that z(ξ, 0)=0 and z ξ (ξ, 0)=0 imply
22