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

0.5
0.4
0.3
0.2
η
0.1
0
−0.1
−0.2
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0 10 20 30 40 50 60
ξ
Figure 4.9: Pulse propagating over a synthetic periodic slowly-varying topography.
Dotted line: numerical solution for the LCM using RK4 for t=36.3247 and
N= 1024, dashed line: initial condition, solid line: flat bottom exact solution.
and the initial condition. The other parameters areρ 1 = 1,ρ 2 = 2, N= 1024,
∆ξ=2l/N= 0.0491,∆t=∆ξ=0.0491.
A detailed analysis of Fig. 4.9 shows that twice the period of the bottom oscillations
(2.5133) is in very good agreement with the reflected wavelength, as
expected from Bragg’s phenomenon theory [12]. See Fig. 4.10 where vertical
bars marking spatial intervals of size 2.5133 fall together with the end of each
period of the reflected signal. A comparison between the solutions for the flat and
periodic bottoms suggests that the attenuation in the wave amplitude is mainly
due to the dispersive term. It is also patent that the oscillations behind the pulse
correspond to the reflected wave due to the topography.
Example 4.5. The present example adds the nonlinearity ingredient to the previous
example. We consider again the periodic slowly-varying coefficient M(ξ)
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