We will show that a wave on a stretched string obeys this equation
Consider a string which is under a tension of t
and has a linear mass density m. The figure
shows a
small segment of this string which has a mass m. When there is no wave on
the string the segment has a length Dx. Thus
m = mDx. When there is a wave on the string, the
segment may stretch more and it will move up and down but it will not move
horizontally. Let us apply Newton's second law on the segment.
Newton's second law along the x-axis.
Since the segment does not move along the x-direction, the sum of the x-component of the external forces on it should be zero
where t is the tension on the string.