The wave equation provides an analytic description of wave motion over time and through a spatial medium.
This mathematical equation can subsequently be used to evaluate and model wave motion along the string and at boundaries.
In analyzing the one-dimensional (1D) string section illustrated in Fig. 1 below, we make the following assumptions:
The mass per unit length of the string is constant and the string is perfectly elastic (there is no resistance to bending).
The tension caused by stretching the string before fixing it at its endpoints is so large that gravitational forces on the string are negligible.
The string moves only in the transverse direction and these deflections are small in magnitude.
Figure 1:
A short section of a stretched, deformed string.
The mass of the short string section (length ) is
, where is the mass per unit length of the string.
Since there is no horizontal motion, the two horizontal components of tension must be constant:
.
The net vertical force on the section is
.
By Newton's Second Law:
.
Making use of the horizontal tension components, we obtain:
(1)
The expressions
and
are the slopes of the string at the points and :
Making these substitutions in Eq. 1 and dividing by ,
Letting approach zero, we obtain the linear partial differential equation
where
is the speed of wave motion on the string. This is the one-dimensional wave equation that describes small amplitude, lossless transverse waves on a stretched string.