S-2  Velocity of Sound in Air

OBJECTIVE:

To calculate the velocity of sound from measurement of the wavelength in air for sound of a certain frequency.

APPARATUS:

Resonance tube with arrangement for varying water level (use only distilled water); rubber tipped hammer; tuning fork; Hg thermometer.

INTRODUCTION:

For a closed tube, resonance occurs at tube lengths of an odd multiple of one-fourth wavelength, i.e. at $\lambda$/4, 3$\lambda$/4, 5$\lambda$/4 etc.

SUGGESTIONS:

1. Find the positions of the water level in the tube for the first three of these resonances. Use these readings to calculate the speed of sound, $v = \vert\vec{v}\vert$. Initially have enough water that you can raise the level above the first resonance position. The tuning fork frequency is on the fork.
   
   Since the effective end of the resonance tube is not at the tube's end, do not use the position of the tube's top in your calculations, but rather take differences between the other readings.
2. Sound waves in gases have a speed $v = \sqrt{ \gamma RT/M }$. (Recall the formula for the speed of sound on a string, $v = \sqrt{T/\mu}$ (e.g., Lab SC-1)). Correct your value of $v$ to 0$^o$C (T = 273.16 K) and compare with that accepted for dry air at 0$^o$C: 331.29 $\pm$ .07 m/s, [Wong, J. Acoust. Soc. Am., 79, 1559, (1986)]. For humid air see 3. below.
3. We quantify proportions in gas mixtures by the pressure each gas contributes to the total pressure. This is called the ``partial'' or ``vapor'' pressure. Think of the speed as resulting from an average $<\gamma/M>$,
   $<\gamma/M> = \left[ (\gamma _{a}/M_{a})P_{a} +  (\gamma _{w}/M_{w})P_{w}\right]/ (P_{a} + P_{w}) ,$
   so that
   $v_{dry} \cong  v_{humid}\sqrt{(\gamma _{a}/M_{a})/< \gamma /M>} ,$

   where $\gamma _{air} = 1.40, \gamma _{w} = 1.33, P_{a}$ is the partial pressure of air, P$_{w}$ is the vapor pressure of water, M$_{a}$ $\sim$ 29 kg and M$_{w}$ = 18 kg.

   How should the v.p. of water, P$_{w}$, in the tube affect the speed?

   OPTIONAL: Humidity changes will affect tuning of what musical instruments?

4. What effect does atmospheric pressure have on the velocity of sound in dry air? (Assume air at these pressures is an ideal gas.)
5. Viscosity and heat conduction in the tube may reduce v by $\sim$0.1%. See N. Feather, ``The Physics of Vibrations and Waves'', Edinburgh Univ. Press, (1961), p. 110-120; this reference also has a delightful historical account (including Newton's famous goof).