Question

6. (10 pts) The traveling of a wave is governed by the one-way wave equation, диди Ət +Qax = 0, u(x,0) = f(2). Assuming that a > 0, this equation describes a wave, whose form is given by f(x), travels to the right at speed a. In this exercise, we use Fourier transform methods to solve this equation. (a) (4 pts) Apply Fourier transform (with regard to x) to the equations above to derive the initial value problem of uw,t) in the frequency space. (b) (4 pts) Find the solution ûw,t) of the initial value problem. (c) (2 pts) Apply inverse Fourier transform on û(w,t) to recover the solution u(x, t) of the original initial value problems in the physical space. (Hint: The time shifting rule for FT might come handy for this problem.)

Answer 1

solution Given a. ucro) = f(x) we know that f u ext)) = a û wit) f u(xt)) = iw û (wit) Substituting #hede vakoseasing an o kapeziget then fourier transform of the above en D is důcwt) et ailw û c wit) to at û cw, o) = f(w) dů taiw û to dů - aiw (dů = -faiw at du Ců (Wit)) = - aint + c ů (Wit) = A e aiut (b. given initial value is û cw,o) = f(w) û cw,o) = A A = f(W) = son of the initial value problem Iůcw, t) = f (w) e lawt/

(C). Now of taking the û (wit) inverse to Fourier transform get rolh ucx,t) ucxrt) = £ 1 fras). û (lo, t) . e ' wx dw -jawtiwa dw ucait)= iwla-aut) dw ucxit) = woxt) gets fuse