Given the wave equation:
$u_{tt} = c^2u_{xx} \quad x \in \mathbb{R}, t> 0$
$ u(x,0) = \phi(x), x \in \mathbb{R}$
$ u_{t}(x,0) = \psi (x), x \in \mathbb{R}$
The solution is:
$u(x,t) = \frac{1}{2} ( \phi(x - ct) + \phi(x + ct)) + \frac{1}{2c} \underbrace{\int_{x-ct}^{x+ct} \psi(\zeta) d\zeta}_{=:D}$
The first two term are very intuitive. These are simple the waves propagating with a speed of $c$. But what is the term marked with $D$? I've never seen it when solving the wave equation in physics. It there any effect physical that corresponds to this term?
Is there an intuitive explanation for this term?
Edit: to see what actually happens I plotted an example:
$g(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{\frac{-1}{2} (x/ \sigma)^2}$
$f(x,t) := \frac{1}{(t+1)} g(x)$
$\frac{d}{dt} f(x,t) = -\frac{1}{(t+1)^2} g(x)$
I've never seen anything like this. In real waves, even though $\psi $ is generally not 0