(a) Show that the Laplace transform of the Heaviside step function $H(t-a)$ is

$$\int_{0}^{\infty} H(t-a) e^{-p t} d t=\frac{e^{-a p}}{p}$$

for $a>0$.

(b) Derive an expression for the Laplace transform of the second derivative of a function $f(t)$ in terms of the Laplace transform of $f(t)$ and the properties of $f(t)$ at $t=0$.

(c) A bar of length $L$ has its end at $x=L$ fixed. The bar is initially at rest and straight. The end at $x=0$ is given a small fixed transverse displacement of magnitude $a$ at $t=0^{+}$. You may assume that the transverse displacement $y(x, t)$ of the bar satisfies the wave equation with some wave speed $c$, and so the tranverse displacement $y(x, t)$ is the solution to the problem:

$$\begin{array}{cl} \frac{\partial^{2} y}{\partial t^{2}}=c^{2} \frac{\partial^{2} y}{\partial x^{2}} & \text { for } 0<x<L \text { and } t> \\ y(x, 0)=\frac{\partial y}{\partial t}(x, 0)=0 & \text { for } 0<x<L, \\ y(0, t)=a ; y(L, t)=0 & \text { for } t>0 . \end{array}$$

(i) Show that the Laplace transform $Y(x, p)$ of $y(x, t)$, defined as

$$Y(x, p)=\int_{0}^{\infty} y(x, t) e^{-p t} d t$$

is given by

$$Y(x, p)=\frac{a \sinh \left[\frac{p}{c}(L-x)\right]}{p \sinh \left[\frac{p L}{c}\right]}$$

(ii) By use of the binomial theorem or otherwise, express $y(x, t)$ as an infinite series.

(iii) Plot the transverse displacement of the midpoint of the bar $y(L / 2, t)$ against time.