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

tio of h 2 /h 1 , where the effects of bottom topography are more pronounced. This
might be a justification for using a higher-order weakly nonlinear model and will
be thoroughly explored in the near future.
3.3 Dispersion relation for the higher-order model.
Comparison with the previous model
To obtain the dispersion relation for the improved model, consider its linearization
around the undisturbed state so that
⎧
⎪⎨
⎪⎩
η t = u 1ξ ,
u 1t +
(
1− ρ )
2
η ξ = ρ √
2 βT[u1 ] ξt + β ρ 1 ρ 1 3 u 1ξξt,
in the presence of a flat bottom.
Taking derivatives once in t,ηcan be eliminated from the second equation,
u 1tt +
(
1− ρ )
2
u 1ξξ − ρ √
2 βT[u1 ] ξtt − β ρ 1 ρ 1 3 u 1ξξtt= 0.
Let u 1 = Ae i(kx−ωt) . Substituting above and using again thatT [e ikx ]=i coth ( )
kh 2
L e
ikx
we get
(
ω 2 1+ β 3 k2 + ρ ( )) ( )
√
2 kh2 ρ2
β k coth = − 1 k 2 ,
ρ 1 L ρ 1
that is,
ω 2 =
(
ρ2
ρ 1
− 1 ) k 2
1+ β 3 k2 + ρ √ (
2
ρ 1
β k coth
kh2
L
). (3.10)
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