Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

Question

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad w_t(x,0) = \psi(x) $$ is given by $$ (*) \quad w(x,t) = \frac{1}{2c}\int_{x - ct}^{x+ct} J_0(\lambda s) \psi(y) dy, \quad s^2 = c^2 t^2 - (x-y)^2, $$ where $J_0$ denotes the $0$-th Bessel function with formula $$ J_0(z) = \frac{2}{\pi} \int_0^{\pi/2} \cos(z \sin \theta) d\theta. $$ Prove (*) with the approach $u(x_1, x_2, t) = \cos(\lambda x_2) w(x_1, t)$ and the solution formula for the Cauchy-Problem of the two-dimensional wave equation.