Solutions of the wave equation
We now can see whether this equation really does describe the essential properties of sound waves in matter. We want to deduce that a sound pulse, or disturbance, will move with a constant speed. We want to verify that two different pulses can move through each other—the principle of superposition. We also want to verify that sound can go either to the right or to the left. All these properties should be contained in this one equation. We have remarked that any plane-wave disturbance which moves with a constant velocity $v$ has the form $f(x - vt)$. Now we have to see whether $\chi(x,t) = f(x - vt)$ is a solution of the wave equation. When we calculate $\ddpl{\chi}{x}$, we get the derivative of the function, $\ddpl{\chi}{x} = f'(x - vt)$. Differentiating once more, we find \begin{equation} \label{Eq:I:47:15} \frac{\partial^2\chi}{\partial x^2} = f''(x - vt). \end{equation} The differentiation of this same function with respect to $t$ gives $-v$ times the derivative of the function, or $\ddpl{\chi}{t} = -vf'(x - vt)$, and the second time derivative is \begin{equation} \label{Eq:I:47:16} \frac{\partial^2\chi}{\partial t^2} = v^2f''(x - vt). \end{equation} It is evident that $f(x - vt)$ will satisfy the wave equation provided the wave velocity $v$ is equal to $c_s$. We find, therefore, from the laws of mechanics that any sound disturbance propagates with the velocity $c_s$, and in addition we find that \begin{equation*} c_s = \kappa^{1/2} = (dP/d\rho)_0^{1/2}, \end{equation*} and so we have related the wave velocity to a property of the medium. If we consider a wave travelling in the opposite direction, so that $\chi(x,t) = g(x + vt)$, it is easy to see that such a disturbance also satisfies the wave equation. The only difference between such a wave and one travelling from left to right is in the sign of $v$, but whether we have $x + vt$ or $x - vt$ as the variable in the function does not affect the sign of $\partial^2\chi/\partial t^2$, since it involves only $v^2$. It follows that we have a solution for waves propagating in either direction with speed $c_s$. An extremely interesting question is that of superposition. Suppose one solution of the wave equation has been found, say $\chi_1$. This means that the second derivative of $\chi_1$ with respect to $x$ is equal to $1/c_s^2$ times the second derivative of $\chi_1$ with respect to $t$. Now any other solution $\chi_2$ has this same property. If we superpose these two solutions, we have \begin{equation} \label{Eq:I:47:17} \chi(x,t) = \chi_1(x,t) + \chi_2(x,t), \end{equation} and we wish to verify that $\chi(x,t)$ is also a wave, i.e., that $\chi$ satisfies the wave equation. We can easily prove this result, since we have \begin{equation} \label{Eq:I:47:18} \frac{\partial^2\chi}{\partial x^2}  = \frac{\partial^2\chi_1}{\partial x^2} + \frac{\partial^2\chi_2}{\partial x^2} \end{equation} and, in addition, \begin{equation} \label{Eq:I:47:19} \frac{\partial^2\chi}{\partial t^2}  = \frac{\partial^2\chi_1}{\partial t^2} + \frac{\partial^2\chi_2}{\partial t^2}. \end{equation} It follows that $\partial^2\chi/\partial x^2 = (1/c_s^2)\,\partial^2\chi/\partial t^2$, so we have verified the principle of superposition. The proof of the principle of superposition follows from the fact that the wave equation is linear in $\chi$. We can now expect that a plane light wave propagating in the $x$-direction, polarized so that the electric field is in the $y$-direction, will satisfy the wave equation \begin{equation} \label{Eq:I:47:20} \frac{\partial^2E_y}{\partial x^2} = \frac{1}{c^2}\,\frac{\partial^2E_y}{\partial t^2}, \end{equation} where $c$ is the speed of light. This wave equation is one of the consequences of Maxwell’s equations. The equations of electrodynamics will lead to the wave equation for light just as the equations of mechanics lead to the wave equation for sound. 