Symmetric solutions for wave equation

Problem 5. Waves can also appear in higher dimensional materials. For example, the surface of water. KdV describes this one way, but we can also use the spherically symmetric wave equation 02 20 K2 02 r or w2 0+2 u(r, t) = 0, that can be used to describe waves that propagate through space from a point source. (a) (5 pts.) Show that u(r, t) = Je (kraut) is a solution to this equation. (b) (2 pts.) We can take a portion (real part) of this solution by letting w(r, t) = = cos(kr + WT). Then, using the conversions = r cos 0 and y = r sind, convert w(r, t) to a function w(x, y, f). (c) (3 pts.) Plot the solution w(x, y, () as a surface for k = w = 1 and for t = 0, t = 1/2 and / = 1. (d) (3 pts.) Describe what happens to w(x, y, t) as we vary w and k. Note that w and k must be greater than zero. Plotting this may prove useful