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I am trying to simplify the following equations to get a relationship involving just [itex] \eta [/itex]:

1) [itex] \nabla^2 \phi(x,z,t) = 0 [/itex]

for [itex] x\in [-\infty,\infty][/itex] and [itex] z\in [-\infty,0] [/itex], [itex] t \in [0,\infty] [/itex]

subject to the boundary conditions

2) [itex] \phi_t+g \eta(x,t) = f(x,z,t)[/itex] at z=0

3) [itex] \eta_t = \phi_z [/itex] at z=0

and

4) [itex] \phi \to 0 \ as \ z \to -\infty [/itex]

Here, g is a constant, [itex] \eta, \phi [/itex] are the dependent variables of the system and f represents a forcing function. Another important constraint is that for systems I'm interested in, f is non zero only for a small time interval.

For the case where f=0, one can find that

[itex] \eta_{tt}-\frac{g}{k} \eta_{xx} =0 [/itex]

where k is the wavenumber of the system.

I want to find an analogous relation when forcing is present.

Any help is appreciated,

Nick

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# Solving coupled PDEs

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