Question

(15 pts) Bessel functions and the vibration of a circular drum In polar coordinates, the Laplacian is just like the Laplacian for the cylinder, but with the removed part เอ The structure of the Laplacian is what we call separable because the r and 0 terms are separate this allows us to solve certain physics problems on the disc by searching for solutions of the form f(r,0)-ar)b() The vibration of a circular drum head is described by 02t where u is the displacement of the membrane. Because the membrane is fastened to the edge of the drum, which we take to have radius 1, we have the boundary condition u(1,0,t) 0 for all 0 and t .(2 pts) Suppose that u is radially symmetric and that it has a constant frequency w in time, i.e that u(r, 0,ta(r)cos(wt -) where Show that is some phase .(2 pts) The solutions of d2 dr2 dr are known as Bessel functions of order 0. The solution with y(0) 1 and y'(0) 0 has a special name and is written as Jo(r). It is the only bounded solution in the disc (up to multiplication by a constant). Show that a(r)-aJo)solves (1) (1 pt) Suppose that z is a root of Jo, i.e. Jo(z)0. Find a value of w so that a(r)Jo(wr/c) satisfies the boundary condition a(1)0. (Note that this means that only certain frequencies give radially symmetric vibrating modes, which you can determine based on the roots of Jo!) (5 pts) Show that satisfies (2) and that Jo(0) = 1 and 0)-0. (You are allowed evaluate the derivative of the sum by differentiating term-by-term.) . (2 pts) If we allow vibrations that oscillate in the θ direction, we can have solutions of the form ) sin(n0). Show that such an a satisfies u(r, θ, t) = a(r) cos(wt- Tt drr dr c2 r2 .(2 pts) Show that if y solves (2), its derivative, which we'll alldy/dr, satisfies (1 pt) Let v solve (4). Show that a) solves (3) with 1.

Answer 1

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