Sound Summary

• Speed of Sound in a fluid
  

  $$v = \sqrt{\frac{B}{\rho}}$$ (153)

  
  
  where $B$ is the bulk modulus of the fluid and $\rho$ is the density. These quantities vary with pressure and temperature.

• Speed of Sound in air is $v_a \approx 340$ m/sec.

• Doppler Shift: Moving Source
  

  $$f' = \frac{f_0}{(1 \mp \frac{v_s}{v_a})}$$ (154)

  
  
  where $f_0$ is the unshifted frequency of the sound wave for receding (+) and approaching (-) source.

• Doppler Shift: Moving Receiver
  

  $$f' = f_0(1 \pm \frac{v_s}{v_a})$$ (155)

  
  
  where $f_0$ is the unshifted frequency of the sound wave for receding (-) and approaching (+) receiver.

• Stationary Harmonic Waves
  

  $$y(x,t) = y_0 \sin(kx) \cos(\omega t)$$ (156)

  
  
  for displacement waves in a pipe of length $L$ closed at one or both ends. This solution has a node at $x = 0$ (the closed end). The permitted resonant frequencies are determined by:
  

  $$kL = n\pi$$ (157)

  
  
  for $n = 1,2...$ (both ends closed, nodes at both ends) or:
  

  $$kL = \frac{2n-1}{2}\pi$$ (158)

  
  
  for $n = 1,2,...$ (one end closed, nodes at the closed end).

• Beats If two sound waves of equal amplitude and slightly different frequency are added:
  

  $$s(x,t) = s_0 \sin(k_0 x - \omega_0 t) + s_0 \sin(k_1 x - \omega_1 t)$$ (159)

  $$= 2 s_0 \sin(\frac{k_0+k_1}{2} x - \frac{\omega_0 + \omega_1}{2} t) \cos(\frac{k_0-k_1}{2} x - \frac{\omega_0 - \omega_1}{2} t)$$ (160)

  
  
  which describes a wave with the average frequency and twice the amplitude modulated so that it ``beats'' (goes to zero) at the difference of the frequencies $\delta f = \vert f_1 - f_0\vert$.