# 3.1: A Classical Wave Equation

The easiest way to find a differential equation that will provide wavefunctions as solutions is to start with a wavefunction and work backwards. We will consider a sine wave, take its first and second derivatives, and then examine the results. The amplitude of a sine wave can depend upon position, $$x$$, in space,

$A (x) = A_0 \sin \left ( \frac {2 \pi x}{\lambda} \right ) \label {3-3}$

or upon time, $$t$$,

$A(t) = A_0\sin(2\pi \nu t) \label {3-4}$

or upon both space and time,

$A (x, t) = A_0 \sin \left ( \frac {2 \pi x}{\lambda} - 2\pi \nu t \right ) \label {3-5}$

We can simplify the notation by using the definitions of a wave vector, $$k = \frac {2\pi}{\lambda}$$, and the angular frequency, $$\omega = 2\pi \nu$$ to get

$A(x,t) = A_0\sin(kx − \omega t) \label {3-6}$

When we take partial derivatives of A(x,t) with respect to both $$x$$ and $$t$$, we find that the second derivatives are remarkably simple and similar.

$\frac {\partial ^2 A (x, t)}{\partial x^2} = -k^2 A_0 \sin (kx -\omega t ) = -k^2 A (x, t) \label {3-7}$

$\frac {\partial ^2 A (x, t)}{\partial x^2} = -\omega ^2 A_0 \sin (kx -\omega t ) = -\omega ^2 A (x, t) \label {3-8}$

By looking for relationships between the second derivatives, we find that both involve A(x,t); consequently an equality is revealed.

$k^{-2} \frac {\partial ^2 A (x, t)}{ \partial x^2} = - A (x, t) = - \omega \frac {\partial ^2 A (x, t)}{\partial x^2} \label {3-9}$

Recall that $$\nu$$ and $$λ$$ are related; their product gives the velocity of the wave, $$\nu \lambda = v$$. Be careful to distinguish between the similar but different symbols for frequency $$\nu$$ and the velocity v. If in ω = 2πν we replace ν with v/λ, then

$\omega = \frac {2 \pi \nu}{\lambda} = \nu k \label {3-10}$

and Equation $$\ref{3-11}$$ can be rewritten to give what is known as the classical wave equation in one dimension. This equation is very important. It is a differential equation whose solution describes all waves in one dimension that move with a constant velocity (e.g. the vibrations of strings in musical instruments) and it can be generalized to three dimensions. The classical wave equation in one-dimension is

$\frac {\partial ^2 A (x, t)}{\partial x^2} = \nu ^{-2} \frac {\partial ^2 A (x, t)}{\partial t^2} \label {3-11}$

Example $$\PageIndex{1}$$

Complete the steps leading from Equation $$\ref{3-5}$$ to Equations $$\ref{3-7}$$ and $$\ref{3-8}$$ and then to Equation $$\ref{3-11}$$.