Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 13E / i

The string is initially at rest, so $y(x,0)=y_t(x,0)=0$. Taking the Laplace transform in $t$ converts the wave equation intoFor $\mathrm{Re}p>0$, the solution that decays as $x\to \infty$ is $\hat{y}=A(p)e^{-px/c}$. The transformed boundary condition $\hat{y}(0,p)=\hat{f}(p)$ fixes $A=\hat{f}$, and therefore