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Wavy Strings

 

Figure: A small segment of a taut string.

One system that exhibits wave motion is the taut string. Picture a string with a uniform mass per unit length  $\mu$  under tension  $F$. Ignoring any effects of gravity, the undisturbed string will of course follow a straight line which we label the $x$ 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 $x$ axis, which we will label the $y$ direction.

The sketch in Fig. 14.2 shows a small string segment of length  $d\ell$  and mass   $dm = \mu \, d\ell$  which makes an average angle  $\theta$  with respect to the $x$ axis. The angle actually changes from   $\theta - d\theta/2$ at the left end of the segment to   $\theta + d\theta/2$ at the right end. For small displacements  $\theta \ll 1$  [the large $\theta$ shown in the sketch is just for visual clarity] and we can use the SMALL-ANGLE APPROXIMATIONS

$$dx = d\ell \, \cos \theta \approx d\ell$$

$$dy = d\ell \, \sin \theta \approx \theta \, d\ell$$

$${dy \over dx} = \tan \theta \approx \theta$$ (14.11)

Furthermore, for small $\theta$ the net force

$$dF = 2F \, \sin (d\theta/2) \approx 2F \, (d\theta/2) = F \, d\theta$$ (14.12)

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

$$dF \approx dm \, a_y = \ddot{y} \, dm \qquad \mbox{\rm or}$$

$$F \, d\theta \approx \ddot{y} \, \mu \, d\ell \qquad \mbox{\rm or}$$

$$\left(\partial^2 y \over \partial t^2\right)_x \; \approx \ . . . . . . pprox \; {F \over \mu} \, \left( d \theta \over dx \right) .$$ (14.13)

Referring now back to Eq. (11) we can use   $\theta \approx dy/dx$  to set

$$\left( d\theta \over dx \right) \; \approx \; \left( \partial^2 y \over dx^2 \right)_t$$ (14.14)

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

$$\left(\partial^2 y \over \partial t^2\right)_x \; - \; {F . . . . . . \, \left( \partial^2 y \over dx^2 \right)_t \; \approx \; 0$$ (14.15)

which is the WAVE EQUATION with

$${1 \over c^2} \; = \; {\mu \over F} \quad \mbox{\rm or} \quad c \; = \; \sqrt{F \over \mu} .$$ (14.16)

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