When one note has twice the frequency of another note we say

When one note has twice the frequency of another note, we say that the notes are an octave apart. Imagine an organ in a church which is tuned in the summer, when the temperature of the sanctuary is 70° F. Two pipes are tuned to 300 Hz and 600 Hz respectively. Ignoring length contraction of the pipes, how far off are the two pipes from one octave in winter, when the sanctuary is only 55° F?

Solution

Speed of the sound varies with temperature, as we are ignoring the change in the length of the organ pipes, the wave length of the two sound will remain same in summer and winter, as the speed of the sound changes, the frequencies change

speed of sound vT , where T is absolute temperature

if f_s and f_w are the frequencies in summer and winter then

f_s/f_w = v_s/v_w =( T_s/T_w)^1/2

summer temperature T_s = 70⁰ F = (70-32)*100/180 = 21.11⁰ C

                                                          = 273+21.11 = 294.11 K

winter temperature   T_w   = 55⁰ F = 12.78⁰C = 285.78 K

f_s/f_w =( 294.11/285.78)^1/2 = 1.014

summer frequencies = 600Hz, 300Hz

winter frequencies = 600/1.014 ,300/1.014

                               = 591.72 , 295.86

the pipes produce different notes in winter

600-591.72 = 8.28 Hz and

300-295.86 = 4.14 Hz