City-Data Forum How do I solve a wave equation that has a time-dependent right-hand side?
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Today, 07:10 AM
 149 posts, read 56,295 times Reputation: 88

I am practicing the 3 most common second-order PDEs at the moment (diffusion equation, wave equation and Laplace equation), and right now I am stuck on a form of the wave equation that has a time-dependent right-hand side, which looks like this:

uₜₜ - c²⋅uₓₓ = (BI/ρ)⋅sin(ωt)

The problem involves a string that is affected by a magnetic field B (which is directed "into" the picture, judging from the cross symbols), I is related to the length of the string (I think) and ρ is the mass per length.
The term BI⋅sin(ωt) has the units N/m.
The string is horizontal.

I know how to solve an equation like this when the right-hand side is position-dependent:
simply find the stationary equation uₛ(x) and make the substitution v(x,t) = u(x,t) - uₛ(x), et cetera.
But what about the case when there is a time dependency?
My book is terrible at explaining things and just kinda makes this random ansatz out of nowhere without justifying it, and I don't know what's happening there.
Apparently I am supposed to be able to use (1/ρ)⋅f(x,t) = 1(x)⋅(BI/ρ)⋅sin(ωt) in some way, but it's exactly that part that isn't justified.
Could you please explain how I am supposed to think here?