How to understand Duhamel's principle? I have difficulty about the explanation of Duhamel's principle on my book. Here is what's written on my book:
Take wave equation as an example. Consider the equation:
\begin{cases}
\frac{\partial^2u}{\partial t^2} = a^2\frac{\partial^2u}{\partial x^2} + f(t,x) &t>0, -\infty < x < +\infty\\
u|_{t=0} = 0, \frac{\partial u}{\partial t}|_{t=0} = 0
\end{cases}
where $f(t,x)$ is the external force over a unit of mass of the string. $u(t,x)$ is the displacement at time $t$. By independent action principle, $u(t,x)$ is the summation of displacement $w(t,x;\tau)d\tau$ caused by the impulse $f(\tau, x)d\tau\ (0\leq\tau\leq t)$, thus $u(t,x) = \int_{0}^{t}w(t,x;\tau)d\tau$. Clearly, when $t<\tau$, $w(t,x;\tau) \equiv 0$. When $t>\tau$, the instantaneous impulse at time $\tau$ is the increase of momentum from $t=\tau-0$ to $t=\tau+0$. So $w(t,x;\tau)$ should satisfy the equation:
\begin{cases}
\frac{\partial^2w}{\partial t^2} = a^2\frac{\partial^2w}{\partial x^2} &t>\tau, -\infty < x < +\infty\\
w|_{t=\tau} = 0, \frac{\partial w}{\partial t}|_{t=\tau} = f(\tau,x)
\end{cases}
and 
$$
u(t,x) = \int_{0}^{t}w(t,x;\tau)d\tau
$$
I checked that such $u$ is the solution to the inhomogeneous equation. But I just cannot understand the derivation of the Duhamel's principle. In the explanation, $w$ is also the displacement, its dimension is length. $\tau$ is the time. Why $u(t,x) = \int_{0}^{t}w(t,x;\tau)d\tau$? The dimension of $u$ becomes length times the time?
Actually I didn't quite understand other parts. Could anyone help me understand Duhamel's principle by explaining it in more detail?
Any help is appreciated!
 A: *

*A linear system of DEs with higher time-derivatives can be converted to a linear system 
$$u_t -Lu ~=~ f, \qquad u|_{t=0}~=~0, \tag{1}$$ 
of DEs with only first-order time derivatives by introducing more variables in the standard fashion, so it is enough to study the latter, cf. the Wikipedia page for Duhamel's principle.
Here the active variable $u$ is a velocity profile in spacetime $(t,x)$;  the external source $f$ is a force; and $L$ is a differential operator that only involves $x$. (Mass is set to 1 in this picture.) 

*Instead of solving the inhomogeneous problem (1), we can solve the corresponding homogeneous problem
$$ w_t -Lw ~=~ 0, \qquad w|_{t=s}~=~f|_{t=s} ,\tag{2}$$ 
for the acceleration profile $w$.
For fixed fiducial time $s$, the acceleration profile $w$ is the retarded solution, evaluated at time $t$, representing the effect, at the later time $t$, of an infinitesimal impulse $f ( x , s ) d s$   administered at an earlier time $s$. 

*The velocity profile for the inhomogeneous problem (1) can be reconstructed as
$$u ~=~\int_0^t ds~w ,\tag{3}$$ 
because of the superposition principle for linear systems. One may check that eqs. (2) & (3) lead to a solution to (1).
