Lecture 30

Readings

http://umdberg.pbworks.com/w/page/125299340/Workout%3A%20Beats

http://umdberg.pbworks.com/w/page/107019360/Standing%20waves

http://www.compadre.org/books/SoundBook

Beats Activity

• How does a beat show up in sound waves?
• An oscillation of volume on top of main tone

• If the two original frequencies are close in frequency:

• If two original frequencies are farther apart:

Standing Waves

• Mathematically - derived by adding 2 waves going in opposite directions

  • Not super important to see that math

• More important is that they represent how a string oscillates when both ends are fixed

  • (or 1 end is fixed and the other is completely free)

• Although called standing waves , the waves don't go anywhere

  • They oscillate in place

• An infinite number of "normal modes" are possible:

• In music, these are called "harmonics" or "overtones"

• Each normal mode oscillates at a different frequency

• Mathematics:

  • $L$$L$ is length of string, fixed at both ends

  • Find $\lambda$$\lambda$ from stretch , express in terms of fraction of $L$$L$

  • $v_0$$v_0$ for wave fixed , still $WaveVelocity(v_0)=\sqrt{\frac{\mathrm{T}}{\mu }}$$Wave\ Velocity\ (v_0) = \sqrt{\frac{\Tau}{\mu}}$

  • So $v_0=\frac{\omega }{k}$$v_0 = \frac{\omega}{k}$ also , and

    • $\omega =2\ast \pi \ast f$$\omega = 2*\pi *f$
    • $k=\frac{2\ast pi}{\lambda }$$k = \frac{2*pi}{\lambda}$
    • $v_0=\lambda \ast f$$v_0 = \lambda * f$
    • $f=\frac{v_0}{\lambda }$$f = \frac{v_0}{\lambda}$

  • Example: for $n=3$$n = 3$ ( 3rd normal mode ) :

    • $\lambda =\frac{2\ast L}{3}$$\lambda = \frac{2*L}{3}$
    • $f=\frac{v_0}{\frac{2\ast L}{3}}=\frac{3\ast v_0}{2\ast L}$$f = \frac{v_0}{\frac{2*L}{3}} = \frac{3*v_0}{2*L}$