MORE  SOUND

1.

If you stand in front of a corrugated pipe culvert at position A (see the figure at right) and clap your hands, creating a sharp pulse of sound, the reflections off the corrugations (spaced 10 cm apart) will return to you at different times, creating a distinctive sound.

Taking into account the lateral dimensions of the culvert, sketch the resultant wave amplitude  s(t)  detected at position A as a function of time  t  and calculate the dominant frequency of the reflected sound just before it becomes inaudible.

2.

A steel bar (produced by careful welding) is 1 km long and has a square cross section of dimensions 5 mm $\times$ 5 mm. The bar is placed under a tension of 5000 N. Compare the 1-way (end to end) travel times for a transverse vibrational pulse (as for a stretched string; not a transverse sound pulse, which might also be possible in a solid) and a longitudinal sound pulse. Neglect gravity.

3.

The figure shows a transmitter and receiver of waves contained in a single instrument. It is used to measure the speed  V  of a target object (idealized as a flat plate) that is moving directly toward the unit, by analyzing the waves reflected from it.

(a)

Apply the Doppler equations twice, first with the target as observer and then with the target as a source, and show that the frequency  $\nu_r$  of the reflected waves at the receiver is related to their source frequency  $\nu_s$  by   ${\displaystyle \nu_r \; = \; \nu_s \, \left( v + V \over v - V \right) }$  where  v  is the speed of the waves.

(b)

In a great many practical situations,  $V \ll v$. In this case, show that the equation above becomes ${\displaystyle {\nu_r - \nu_s \over \nu_s } \; \approx \; {2 V \over v}. }$

4.

Two trains are traveling toward each other at 100 ft/s relative to the ground. One train is blowing a whistle at 500 Hz.  (a) What frequency will be heard on the other train in still air?  (b) What frequency will be heard on the other train if the wind is blowing at 100 ft/s toward the whistle and away from the listener?  (c) What frequency will be heard if the wind direction is reversed?

5.

A plane flies with ${5 \over 4}$ the speed of sound. The sonic boom reaches a man on the ground exactly 1 minute after the plane passes directly overhead. What is the altitude of the plane? Assume the speed of sound to be 330 m/s.