Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 12B (Complex Methods)

For $\mathrm{Re}s>0$,This is the Laplace transform time-shift rule in its simplest form.

Fourier transformation givesFor $x>0$, close the inverse-transform contour in the upper half-plane and take the residue at $k=im$; for $x<0$, close it in the lower half-plane. This gives the one-dimensional modified Helmholtz Green functionThe same formula works for complex $m$ with $\mathrm{Re}m>0$: it decays at both ends, is continuous at zero, and its derivative has the jump $G^{\prime }(0+)-G^{\prime }(0-)=-1$ required by the delta source.

Convolution therefore yields

Let $U(s,x)={\mathcal{L}}_t\{u(t,x)\}$. The initial data turn the wave equation intoUsing the Green function from part (ii) with $m=s$ givesSincewe obtain the D'Alembert formula with initial velocity