Wave Equation (Purpose of $c$) Consider the sound wave equation: $\dfrac{\partial^2 p}{\partial x^2} = \dfrac{1}{c^2}\dfrac{\partial^2 p}{\partial t^2}$.
Imagine air molecules vibrating. The equation describes the behaviour of particles in the $x$ and $t$ coordinates. But air particles don't vibrate at the speed of sound. So why does speed of sound $c$ appear in the equation?
I know that 'information' travels down the wave at the speed of sound $c$. But I just feel like that is a physical property of the wave. How does an abstract equation relating behaviour of particles in $x$ and $t$ direction know about how fast information travels down the wave?
 A: The short answer is to give the right units. More deeply you should look at the solutions to this equation. These solutions can always be factored as
$$ p(x,t) = R(x-ct) + L(x+ct)$$
For see this you can look at d'Alambert's formula, but the fundamental reason is that the wave equation in one-dimension is factored as
$$ \frac{\partial^2 p}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2p}{\partial t^2} = \left[  \frac{\partial}{\partial x} - \frac{1}{c}\frac{\partial}{\partial t}\right]\left[  \frac{\partial}{\partial x} + \frac{1}{c}\frac{\partial}{\partial t}\right] p(x,t) = 0 \; . $$

Velocity of waves
The $R$ and $L$ are functions of only one variable. Thus it has only one "shape" for all times. For example, thought $R$ as having a sharp pike in the origin, thus $R(x-ct)$ will have a pike at the position and time such that $x-ct=0$, i.e. at the position $x=ct$. This means that the wave has moved the distance $ct$ in the time $t$ and it gives the common meaning that $c$ is the velocity of the wave.
Note that the $L$ is a wave that moves with the same velocity $c$ but in the opposite direction. $R$ is sometimes called a wave that propagates to the right and $L$ to the left. When you sum the two partial waves this fixed shape is somewhat lost for general solutions and indeed they don't need to propagate with velocity $c$, they can even be standing wave solutions.
A: From the dimensionality considerations, a coefficient with dimensions of speed is necessary, since one derivative is in respect to position and the other is in respect to time:
$$
[\frac{\partial^2 p}{\partial x^2}]=\frac{[p]}{L^2}, [\frac{\partial^2 p}{\partial t^2}]=\frac{[p]}{T^2}, [c]=\frac{L}{T}.
$$
One could then ask a more precise question: why is this coefficient exactly the speed of sound? The reason is that this equation defines the speed of sound! Indeed, the traditional relation $\omega=ck$ (or $c=\lambda f$) trivially follows from solving the wave equation in terms of plane waves.
