Use the joint Laplace and Hankel transform to solve the initial-boundary value problem
$\qquad c^2 \left( u_{rr} + \frac{1}{r} u_r + u_{zz} \right) = u_{tt}, \qquad 0 < r < \infty, \quad 0 < z < \infty, \quad t > 0$,
$\qquad u_z(r, 0, t) = H(a - r)H(t), \qquad 0 < r < \infty, \quad t > 0$,
$\qquad u(r, z, t) \to 0 \quad \text{as} \quad r \to \infty \quad \text{and} \quad u(r, z, t) \to 0 \quad \text{as} \quad z \to \infty$,
$\qquad u(r, z, 0) = 0 = u_t(r, z, 0)$,
and show that
$\qquad u_t(r, z, t) = -acH \left( t - \frac{z}{c} \right) \int_0^\infty J_1(ak) J_0 \left\{ ck\sqrt{t^2 - \frac{z^2}{c^2}} \right\} J_0(kr) dk.$
