# d'Alembert's solution of wave equation

#### pdestud

##### New member
Find the solution of

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

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

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

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

#### Subhotosh Khan

##### Super Moderator
Staff member
Find the solution of

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

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

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

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