Solving a partial differential equation

[Haorong Wu]

Homework Statement

$(\partial_t^2-\partial_z^2) h(t,z)=A \cos (k(t-z))$

Relevant Equations

None

If the right-hand side is zero, then it will be a wave equation, which can be easily solved. The right-hand side term looks like a forced-oscillation term. However, I only know how to solve a forced oscillation system in one dimension. I do not know how to tackle it in two dimensions.

I have tried to generalize it into two dimensions by solving it pretending $h$ depends only on $t$ and $z$ separately, but I have no clues on how to carry on.

I have tried it in Mathematica. It gives no results.

Thanks ahead.

 

[Orodruin]

Try changing variables to $\xi = z-t$ and $\eta = z+t$.

 

[Delta2]

This is special case of the inhomogeneous wave equation or wave equation with source term. The so called source term is the right hand side. If the right hand side is zero, we have the homogeneous wave equation or simply wave equation.

 

[Haorong Wu]

Thanks, @Orodruin and @Delta2.

By changing variables with $t-z=\alpha$ and $t+z=\beta$, I found that the equation becomes $\partial_\alpha \partial_\beta=\frac A 4 \cos (k\alpha)$, which can be easily solved.