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Fourier Sine Series Solution - Wave Equation, Interval and Boundary Conditions

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Please see attachment for complete questions (for the below "..." indicates equation to be found in attachment). Thanks!

(a) Write down the Fourier (sine) series solution u(x,t) of the wave equation ... on the interval ... satisfying the boundary conditions ... and the initial conditions ...

(b) Use the identity ... to show that the above series solution u(x,t) can be transformed into the form ... where F(x) is the odd periodic extension of f(x) ...

(c) The last result is no surprise. Why not?

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A Fourier sign series is investigated. The solution is detailed and well presented.

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