\(\displaystyle \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \qquad (t \geq 0, -\infty < x < \infty) \)

\(\displaystyle u(x,x) = \phi(x) \qquad (-\infty < x < \infty) \)

\(\displaystyle \frac{\partial u}{\partial x} (x, -x) - \frac{\partial u}{\partial t} (x,-x) = \psi(x) \qquad (-\infty < x < \infty) \)

where \(\displaystyle \phi(x) \) and \(\displaystyle \psi(x) \) are twice continuously differentiable.