Waves on a String The One Dimensional Wave Equation

The easiest path to understanding waves is by way of the simplest example: waves on a string under tension (Fig. 3.1). In its equilibrium position the string lies along the x-axis, but if the string is displaced in the y-direction it will vibrate because of opposition between the inertia of the string and a restoring force provided by the tension. Consider a segment of string lying between x and x + Ax. If a is the mass per unit length of the string (assumed constant), the equation of motion of this segment is aAx^-=Fy, (3.11)

where Fy is the y-component of the force on Ax, a consequence of the tension, assumed constant. This assumption requires that the lateral displacement of the string be small. We also assume that the string always lies in the xy-plane. The total force acting on Ax by the string on both sides of it is

where T is the tension in the string and tf is the angle between the string and the x-axis. Unless the tension is very small, the force of gravity is negligible. Because of the assumption of small displacements sint? « tantf = (3.13)

Figure 3.1: The (small) ^-displacement of a uniformly tense string stretched along the rc-axis varies in space and time. The angle $ between the tangent to the string at any point and the c-axis is shown greatly exaggerated.

Divide both sides of this equation by Ax and take the limit as Ax ^ 0:

dt2 dx2 We also can write this equation as

1 d2y d2y v2 dt2 dx2'

where

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