Question

1. If you were flying a model airplane on a wire so that it traveled in a circle around you, would you hear a Doppler shift in the sound from the airplane? Explain.

2. In Olympic sprint events, the sound of the starter gun is sent to individual speakers at each runner’s starting block rather than relying on the sound traveling through the air. Why?

3. Does the speed of sound waves through air depend on the frequency of the sound? Justify or illustrate your answer.

4. Explain in your own words what resonance is.

5. Draw a fifth harmonic standing wave on a string with fixed ends.

   a. If the length of the string is 100 cm, what is the wavelength of the fifth harmonic? Of the first harmonic? Explain.

   b. If the frequency of the fifth harmonic is 400 Hz, what is the frequency of the first harmonic? Explain.

Answer 1

As the listener and the plane's separation never changes while the plane flies in a circle, so no dopler shift will be heard by the listener at the center.

A starting signal has to be perfectly simultaneous in Olympic level games where fraction of a second means winning or loosing. Relying on sound propagation comes with a limitation that challenges this aspect, which is speed of the sound itself. Speed of sound in air is finite and for this participates standing in front of the starter gun will receive the sound first and this will make the game unfair. Electricity is as fast as light, it's practically simultaneous. All individual speakers will go off at the same time if the gun's sound is broadcasted through them. This makes the starting time equal for all participants and the game fair.

Frequency of the sound is "how intensely the pressure of sound changes at a single point". Wave propagate because vibrating molecules transfer energy to its nearby molecules, the quickness of this transfer determines the propagation of the wave, and this is determined by material properties like bulk modulus. Frequency has no effect in wave propagation. A proof of this is while standing far from a concert stage the high of guitar riffs and low of the bass don't sound like out of sync.

Suppose a swing is swinging, now to increase its swing we have two options, one, to push it with a great amount of energy once. Or two, push it slightly each time it comes back and repeat, after a while as we are dumping energy on the swing "in sync" with its swing it's amplitude (how wide it swings) will increase.
In a vibratory mechanical system, if an external vibration pushes the system "in sync" (we call it having the same frequency), then the amplitude of the original wave quickly increases. We call this phenomenon "resonance".

a fifth harmonic standing wave on a string with fixed ends

A)

If the wavelength is L₅ then 5*L₅/2 = 100 ,so L₅ = 40cm

first harmonic has one antinode, so 1*L₁/2 = 100, so L₁ = 200cm for the first harmonic

B)

Let the speed of the standing wave be v,

so the frequency of fifth harmonic = 400 = v/L₅ = v / 40\\

So v = 16000cm/s

Now say the frequency of the first harmonic is f₁ ,

so, f₁*L₁= 16000,

=> f₁ = 16000/L₁ = 16000/200 = 80Hz