I'll need some help and clarification about solving this equation.(adsbygoogle = window.adsbygoogle || []).push({});

After some non-dimensionalization, I can arrive at the following wave equation with a moving point source. The initial conditions are zero.

[itex]\Delta P - \frac{\partial^2 P}{\partial \tau^2} = - A \cos(\tau) \delta^3(\vec{r} - \vec{r}_s(\tau))[/itex]

Whether the source is a cosine, a sine, or something else that is periodic is unimportant.

[itex]\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}[/itex] or something to that effect. [itex]\vec{r}_s[/itex] is the location of the moving source.

I wanted to solve this equation, so I tried a change of variables to make the source stationary. I try [itex]\vec{r}' = \vec{r} - \vec{r}_s(t)[/itex]. In Cartesian coordinates this means [itex]x' = x - x_s(t)[/itex], [itex]y' = y - y_s(t)[/itex], [itex]z' = z - z_s(t)[/itex].

Sooo...

[itex]\Delta P = \frac{\partial^2 P}{\partial x^2} + \frac{\partial^2 P}{\partial y^2} + \frac{\partial^2 P}{\partial z^2} = \frac{\partial^2 P}{\partial x'^2} \left(\frac{\partial x'}{\partial x}\right)^2 + \frac{\partial^2 P}{\partial y'^2} \left(\frac{\partial y'}{\partial y}\right)^2 + \frac{\partial^2 P}{\partial z'^2} \left(\frac{\partial z'}{\partial z}\right)^2[/itex]

But [itex]\frac{\partial x'}{\partial x} = \frac{\partial y'}{\partial y} = \frac{\partial z'}{\partial z} = 1[/itex]. So [itex]\Delta P = \frac{\partial^2 P}{\partial x'^2} + \frac{\partial^2 P}{\partial y'^2} + \frac{\partial^2 P}{\partial z'^2}[/itex]

[itex]\frac{\partial^2 P}{\partial x'^2} + \frac{\partial^2 P}{\partial y'^2} + \frac{\partial^2 P}{\partial z'^2} - \frac{\partial^2 P}{\partial \tau^2} = - A \cos(t) \delta^3(\vec{r}')[/itex]

This seems simpler than I thought it would be. I would expect additional source terms because the source is moving. I'm convinced there's something fundamentally wrong with my change of variables. If this is wrong, what am I missing?

Also, I'll need some help solving the equation once I get it into a good form. I have Strauss's PDE book, which details the inhomogeneous wave equation with zero initial conditions in section 9.3, however, I've always found Strauss to be too terse to be useful.

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# Linear wave equation with moving point source

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