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# Wavy Strings

One system that exhibits wave motion is the taut string. Picture a string with a uniform mass per unit length    under tension  . Ignoring any effects of gravity, the undisturbed string will of course follow a straight line which we label the axis. There are actually two ways we can "perturb" the quiescent string: with a "longitudinal" compression/stretch displacement (basically a sound wave in the string) or with a "transverse" displacement in a direction perpendicular to the axis, which we will label the direction.

The sketch in Fig. 14.2 shows a small string segment of length    and mass     which makes an average angle    with respect to the axis. The angle actually changes from   at the left end of the segment to   at the right end. For small displacements    [the large shown in the sketch is just for visual clarity] and we can use the SMALL-ANGLE APPROXIMATIONS

 (14.11)

Furthermore, for small the net force
 (14.12)

acting on the string segment is essentially in the direction, so we can use Newton's SECOND LAW on the segment at a fixed location on the string:

 (14.13)

Referring now back to Eq. (11) we can use     to set
 (14.14)

-- i.e. the curvature of the string at time . Plugging Eq. (14) back into Eq. (13) gives

 (14.15)

which is the WAVE EQUATION with
 (14.16)

We may therefore jump right to the conclusion that waves will propagate down a taut string at this velocity.

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Next: Polarization Up: WAVES Previous: The Wave Equation
Jess H. Brewer - Last modified: Sun Nov 15 21:20:11 PST 2015