Good day,
In standard PDE literature, there are a lot of solution approaches for the 1-D wave equation initial-boundary value problem. One way is with separation of variables. Below I describe the problem, the standard solution, and then I ask about a more complicated problem.
Given:
Wave Equation utt = c2uxx.; x ∈ (0, 1), t> 0 with
c=1,
u(0, t) = u(1, t)=0, t> 0,
u(x, 0) = f(x),
ut(x, 0) = g(x),
x ∈ (0, 1).
Solution:
u(x, t) =summation over k=1..∞ of [ sin (k π x) (ak cos (k π t) + bk k π sin (k π t)) ]
ak, bk determinted by Fourier analysis of the functions f(x), g(x) .
Question:
My question is about an obstacle.
Let's say there is an obstacle of infinitesimal length of time and space, at x=x2 and t=t2, such that u(x2,t2)=d2.
1. How does the solution u(x,t) change after time t2?
2. How does the answer to 1 change if the obstacle is extended for a finite time?
3. How does the answer to 1 change if the obstacle is extended for a finite space?
4. How does the answer to 1 change if d2 = 0?
5. Can you point me to some references where I can find the solutions for 1-4? In my opinion, this is related to string vibration as well as electromagnetic wave propagation, for example if a propagating wave encounters a conductor. I hope to be able to find at least some elementary references on this topic.
Thank you in advance..
I believe it's related to the topic of nonhomogeneous PDEs, so it appears that one way to solve this problem is to add a forcing term to the wave equation. What if I do not add the forcing term, can the answer be calculated first without the obstacle, and then modified upon the wave encountering the obstacle at time t2 ?
It could be solved by a number of methods:
(1). Method of separation of variables,
(2). Method of trial functions,
(3). Method of direct integration, and so forth
You dont need separationof variables.
Any function of the form
A f(x-ct)+ B g(x+ct)
But then of course you adapt to initial conditins and /or boundary conditions.
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