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I am confused about my solutions to the following governing equation:

[tex] u_{tt}-c^2u_{xx}=F_{xx} [/tex]

For [tex]F=A(x)sech^2\left(\frac{x-c_gt}{B}\right) [/tex]

Where [itex]c,c_g,B \in \mathbb{R} [/itex] and [itex] A(x)[/itex] is a linear function. Also, we have [itex]c_g<c[/itex]. Substituting physical values for the parameters, I can solve the equation using a 2-d integrator to implement the method of Duhamel. My solutions are unclear to me though.

I expect to see the intensity of the response increase with position, as it does along the ray traveling at [itex]c_g[/itex]. I do not, however, see the same response out of the ray traveling at [itex]c[/itex]. I also do not see a ray traveling at [itex]-c[/itex], as I naively would expect. In the plot attached, the ray traveling at speed c is the lower one, since this is an x-t diagram.

I also would think that the ray traveling at speed c would constantly be generated at the location of the forcing, otherwise it is unclear to me how the information from the forcing would get to the faster wave traveling at speed c.

I think my numerical implementation is fine. Perhaps the fact that I'm working on the half domain, [itex] x \ge 0 [/itex] is messing things up?

Any help is appreciated,

Nick

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# Causality and the inhomogeneous wave equation with a moving source

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