# Duhamel's principle

## Main Question or Discussion Point

Hi, I am having difficulty understanding and applying Duhamel's principle. (I'm not great with math but somehow I found myself in this graduate math class of death)...

From my text its stated that

Ux1x1+Ux2x2+...+Uxnxn - Utt = f(x,t) (for x an element of all real), t>0

u(x,0)=0, ut(x,0) = 0 (for x an element of all real)

Or in words, the laplacian of u minus the second time derivative of u = a function of x and t.
The initial conditions are zero displacement and zero velocity.

Next we can assume some v(x,t;tow) is the solution of a homogeneous wave eqn

vx1x1+vx2x2+...+vxnxn-vtt = 0 (for x an element of all real), t>tow

v(x,tow;tow) = 0, vt(x,tow;tow) = -f(x,tow)

Or in words, the homogeneous wave eqn for t larger than some time tow, with the initial conditions of this equation being zero position and -f(x,tow) velocity.

How on earth does this work?

I know the solution is

u(x,t) = int( v(x,t;tow)) dtow from zero to t, but i have no idea how i can derive this.

I think i lack a major understanding of this principle. Can someone explain to me in simple terms what this is saying?