Question

4.(10pts) Write Laplaces' equation in cylindricaol co-ordinates(p527 ex.3,use pinstead ofr) Assume the solution, e, φ, z), n can be written φ (p, φ, z)s u(p, φ)e-kz and Show that the equation for u is the two dimensional wave equation; Written in polar co-ordinates:xpcosp,y psinp For a plane wave traveling in a direction defined by:4-kcosce, ky-kinα Show that the plane wave solution can be written; look for a solution u z(x)en (2-n212,-0 And the equation for Z, is Bessels equation:Zh "x2 + xZ Using the plane wave solution; and the fact that a sum of solutions is also a solution; By changing variables to:c-ω + φ, ωο-β-φ, ai-r-φ Obtain the following integral representation ofZn showRe(icosw) sinhasinp use this to obtain acceptable contours forag, Wt-o

Answer 1

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