Exercises

Exercise 1

Construct a solution to the drumhead problem with zero initial displacement and initial velocity (d/dt) u(r,0) = u1(r) = r^2 - 1.

Exercise 2

Does a square drumhead sound the same as a round drumhead? Prove your answer.

Exercise 3

Try to find an initial condition that would make the first overtone large compared to the fundamental tone.

Hint: If you make the initial condition approximately equal to one of the terms in the solution, then the other terms in the solution will make small contributions. This is because the differential equation is linear.

Exercise 4

Solve the damped vibrating drumhead problem,

(d/dt)^2 u + 2 a (d/dt) u = Div[c^2 Grad[u]],

u(0,t) is bounded, u(1,t) = 0,

u(r,0) = r^2 - 1, (d/dt) u(r,0) = 0

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