How Do You Solve the 1D Wave Equation with Gravity and Nonhomogeneous Terms?

[urista]

I have a wave equation Ytt=c^2 Yxx - g where g is a constant. The boundary conditions are Y(0,t)=Y(L,t)=0 with initial conditions Y(x,0)=0 and Yt(x,0)=0 I tried to solve it by Laplace transfoming the PDE in time and everything worked fine until I got to the point where I had to inverse the transform but things got ugly. Obviously, I have a nonhomogenous PDE with homogeneous boundary conditions. I was going to expand everything in terms of the related eigenfunctions sin(n Pi x/L)but it's not right to expand the constant g in terms of eigenfunctions. I can't do separation of variables because the PDE is inhomogeneous. What's the trick here to get me started without using integral transforms?

 

[HallsofIvy]

"I was going to expand everything in terms of the related eigenfunctions sin(n Pi x/L)but it's not right to expand the constant g in terms of eigenfunctions."

Why not? If you are restricted to a finite interval, say 0 to a, then it is fairly simple to expand a constant in a sine series by treating it as an odd function with period 2a. If you have an infinite interval, you will need to use a Fourier Transform anyway.

 

[urista]

Thanks HallsofIvy, so I should expand g=Sum form n=0 to infinity(gn sin(n Pi x/L)), and of course the gn coefficients can be found using inner product and the orthogonality of the eigenfunctions, correct?

 

[HallsofIvy]

Yes, just find the coefficients the way you would for any function. Because there is a discontinuity at 0 and L, there will be an inaccuracy for any finite truncation of the series but that shouldn't bother you .