question about wave equation Dalambert's formula for the  following BVP 
$$\hspace{35mm}u_{tt}=C^2u_{xx} ~~-\infty <x<\infty, t>0$$
$$u(x,0)=\phi(x)$$
$$u_t(x,o)=\gamma(x)$$
is that $$u(x,t)=\frac{1}{2}[\phi(x+ct)+\phi(x-ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}\gamma(x)$$
Know my question is that what is the behaviuor of the wave equation when $\phi(x)=0$,
Thanks.
 A: As an example, suppose we want to solve the wave equation $u_{t t}=u_{x x}$ subject to $u(x,0)=0$ and $u_t(x,0)=2x \left/\left(1+x^4\right)\right.$. We can interpret this as follows. We impart an initial velocity to a very long string at rest. The precise initial velocity is given by $g(x)=2x\left/\left(1+x^4\right)\right.$. We might do this by striking the string in two locations with equal, but opposite vertical forces. Here is the graph of this function:

We can use d'Alembert's formula to find the solution analytically.
$$u(x,t) = \frac{1}{2}\int_{x-t}^{x+t} \frac{2s}{1+\left(s^2\right)^2} \, ds=\frac{1}{2}\arctan \left(s^2\right)|_{x-t}^{x+t}  = \frac{1}{2}\left(\arctan
\left((x+t)^2\right)-\arctan \left((x-t)^2\right)\right)$$
Here's the graph of $\arctan \left(x^2\right)$.

The solution is the superposition of two waves half this size; one is shifted to the left and the other is shifted to the right and reflected. At time $t=0$, the two waves cancel each other out and we get zero - the initial condition. A short time later, a downward hump appears to the left and an upward hump appears to the right. These two waves travel in their respective direction as time progresses. Here's an animation of this process.

I generated the animation with the following Mathematica code.
g[x_] := 2 x/(1 + x^4); 
u[x_, t_] = 1/2 Integrate[g[s], {s, x - t, x + t},
  Assumptions -> {Element[x, Reals], t > 0}];
Animate[Plot[u[x, t], {x, -5, 5},
  PlotRange -> {{-5, 5}, {-1, 1}},
  AspectRatio -> 1/3], {t, 0, 4}]

