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What does an extra term mean $a \phi$ in the following equations and what is the physical meaning of these two?

$$\phi_{tt} - c^2 \nabla^2 \phi - a \phi = 0$$

$$\phi_{tt} - \nabla^2 \phi - a \phi = 0$$ Solution is in the form: $$\phi = A e^{ik (x-ct)} \cos( \frac{2 \pi y}{L})$$

How do I solve them? After I plugged everything in?

worship_the_ocean
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  • In quantum field theory, $a=m^2$ is the mass of a particle. – md2perpe Dec 11 '20 at 00:33
  • "physical meaning" is a question better asked on physics stack exchange. – K.defaoite Dec 11 '20 at 00:44
  • Up to constants, this is the Klein-Gordon equation and, as it was pointed out, $a$ is normally interpreted as mass. – penovik Dec 12 '20 at 09:45
  • Up to the sign in front of the coefficient $a$, this is the 'damped wave equation', aka. 'telegrapher's equation' or 'telegraph equation'. See various posts on this site where boundary value problems are solved using separation of variables and Fourier series. – EditPiAf Dec 12 '20 at 13:15

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