Fourier transform for the wave problem Solving the wave problems l(x,t)→>U(k,t) a2u=0 q(x)→>Φ(k) U +aku=0 y(x)→>Y(k) u 1=09( l12=0=/(x) U=Φ coskat expat y(s)ds +psin kat 2a Jx-atFourier Transform for the wave problem      =  =  + = = = 0 0 2 2 | | 0 t t t tt U U U a k U  + − + = + + − x at x at s ds a u x at x at ( ) 2 1 [ ( ) ( )] 2 1    Solving the wave problems      = = − = = = | ( ) | ( ) 0 0 0 2 u x u x u a u t t t tt xx   kat U kat ka sin cos + 1  =  ( ) ( ) ( ) ( ) ( , ) ( , ) x k x k u x t U k t →  →  →    u = U k t e dk ikx ( , )