# About Duhamel Principle in Wave Equation

## Main Question or Discussion Point

$$\begin{array}{l} \frac{{\partial ^2 u}}{{\partial t^2 }} - a^2 \frac{{\partial ^2 u}}{{\partial x^2 }} = f(x,t) \\ t = 0:u = 0,\frac{{\partial u}}{{\partial t}} = 0 \\ \end{array}$$......(*)
$$\begin{array}{l} \frac{{\partial ^2 W}}{{\partial t^2 }} - a^2 \frac{{\partial ^2 W}}{{\partial x^2 }} = 0...(t > t_j ) \\ t = t_j :W = 0,\frac{{\partial W}}{{\partial t}} = f(x,t_j )\Delta t_j \\ \end{array}$$.......(**)

let u(x,t) be the solution of equation (*)，$$W(x,t;t_j ,\Delta t_j )$$be solution of (**)。why $$u(x,t) = \mathop {\lim }\limits_{\Delta t_j \to 0} \sum\limits_{j = 1}^l {W(x,t;t_j ,\Delta t_j )}$$ exist?