Selected Solutions $$

1. To calculate how far away the mountain is, we need to know the speed of sound. Since the air is at T = 22 ^oC = 295 K, we know from the book that

   v = 331 m/s (1 + (T - 273 K)/273 K)^1/2 = (331 m/s)(1 + (295 K - 273 K)/273 K)^1/2 = 344 m/s

   Please note that the equation in the book is poorly written and should be re-written as above. Since the echo took 3.0 s to return, this means that the sound took 1.5 s to reach the mountain. At 344 m/s, this means that the mountain is

   d = (344 m/s)(1.5 s) = 516 m

2. Since the frequency of the sound that is heard by the bicyclist is lower in frequency than the original sound, we know that the car must be moving away from the bicyclist, i.e. the bicyclist is behind the car, moving at a speed that is 1/3 that of the cars. Behind the car, we know that the wavelength of the sound is given by

   l' = l + v_s/f

   This corresponds to a frequency that is given by

   f' = v/l' = f v/(v + v_s)

   This is the frequency relative to still air. As the bicyclist moves through this wave, she hears a frequency that is increased above this still air frequency by

   f'' = f'(v + v_o)/v = f (v + v_o)/(v + v_s)

   If we use the fact that 3v_o = v_s, we get

   f'' = f (v + v_o)/(v + 3v_o)

   Solving this for v_o, we get

   f'' (v + 3v_o) = f (v + v_o)
   v_o (3f'' - f) = v (f - f'')
   v_o = v (f - f'')/(3f'' - f) = 345 m/s (25/805) = 10.7 m/s

   Since the car is moving 3 times faster, the speed of the car is 32.1 m/s.

3. For a pipe that is open on both ends, we know that the resonant frequencies are given by

   f_n = n v/(2L)

   where v is the speed of sound, n=1,2,3,..., and L is the length of the tube. For the fundemental frequency, we set n = 1. This means that the length of the pipe can be found by solving for L, i.e.

   L = v/(2f₁) = (345 m/s)/(523.2 Hz) = .659 m

   When we close off one end, we impose a node at one end and change the resonant frequencies by

   f_n = n v/(4L)

   If f₃ = 261.6 Hz, then we can solve for the length of the pipe by

   L = 3v/(4f₃) = (1035 m/s)/(1046 Hz) = .989 m.