Physics organ pipe?

Question

(a) What should be the length of an organ pipe, closed at one end, if the fundamental frequency, at a temperature of 20.0 °C, is to be 264.1 Hz? (b) What is the fundamental frequency of the organ pipe of part (a) if the temperature drops to 0°C?

Answer 1

The standing wave in the air column inside the pipe has a displacement node (pressure antinode) at the stopped end and a displacement antinode (pressure node) at the fipple.  This means the pipe will have an odd number of quarter-waves inside it.  For an ideal gas,

v = √(γp/d)

where

γ is a constant dependent on the thermodynamic properties of the gas,

p is the gas pressure, and

d is the density of the gas.

For this problem, we can treat γ and p as constants.  For an ideal gas, the density will vary inversely with absolute temperature, so the velocity will vary directly with the square root of temperature.  According to my copy of Halliday and Resnick, _Physics, Part I_, 2nd ed, Table 20-1, the speed of sound in air at 0 deg C is 331.3 m/sec.  I assume this is at atmospheric pressure and for dry air.

If the wavelength remains constant, frequency also varies directly with velocity, so the frequency also varies directly with the square root of temperature.  That means we can answer part b simply by multiplying the frequency at 20.0 °C by the square root of the ratio of the absolute temperatures, which means we answer part b first with the information at hand.  The frequency at 0°C is therefore

264.1 Hz * √(273.15°K/293.15°K) = 254.9 Hz

To answer part a, we note that the wavelength is the speed of sound divided by the frequency and the pipe will be one quarter this long.  Therefore the pipe has a length of

331.3 m/sec / 254.9 Hz / 4 = 0.3248 m

This figure does not allow for end effects or changes in the density of the air because of changes in its composition, e.g., presence of humidity.