 Solution

Rewrite the ratio of hyperbolic sines asFor $\mathrm{Re}p>0$, expand the denominator by the geometric series. Part (a) inverts each delayed exponential:At each finite time only finitely many terms are nonzero, so this reflected-step solution of the wave equation is locally a finite sum. The first front travels from $x=0$; the second is its sign-reversed reflection from the fixed endpoint, with repeated round trips. At $x=L$ the delays coincide and cancel. At $x=0$ the sum telescopes to $a$ for $t>0$.

An equivalent standing-wave Fourier series isIts static ramp has exactly the sine coefficients $2a/(n\pi )$, so cancellation gives the zero initial displacement in the interior. Both representations solve the equation away from the fronts and in the distributional sense across them; the convention $H(0)=1/2$ agrees with Fourier midpoint values.