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

thenω 0 satisfies the Neumann problem
⎧
⎪⎨
⎪⎩
ω 0ξξ +ω 0ζζ = 0, on− h 2
≤ζ≤ 0,
L
( )
ω 0ζ = M(ξ) u 1xt x(ξ, 0), t , atζ= 0,
ω 0ζ = 0, atζ=− h 2
L ,
with tangential derivativeω 0ξ (ξ, 0) = T [ M(˜ξ)u 1xt
( x(˜ξ, 0), t ) ]
(ξ). Thusω0x =
ω 0ξ /M(ξ). The subsequent term isω 1 = 0 because both boundary conditions are
homogeneous to O(ε). Nextω 2 satisfies
⎧
⎪⎨
⎪⎩
ω 2ξξ +ω 2ζζ = 0, on− h 2
≤ζ≤ 0,
L
ω 2ζ = 0, atζ= 0,
ω 2ζ − h 2
L h′′( ε x(ξ, 0) ) ω 0 = 0, atζ=− h 2
L .
Therefore
ω=ω 0 + O(ε 2 ).
(A.2)
If we establish a relation betweenα,βandεof the typeε 2 = O(β q ), with q≥ 1 2 ,
Eq. (A.2) leads to
(
1
M(ξ) T[ ] )
M(˜ξ)u 1t = 1
x
M(ξ) T[ ] (√ )
M(˜ξ)u 1tx + O β ,
which justifies the approximation done in Eq. (2.39).
90