Michelson-Morley Experiment question

[superspartan9]

Homework Statement

Consider an apparatus for performing a Michelson-Morley experiment to measure the speed of sound in the laboratory. A sound wave of frequency 3,600 Hz replaces light. The speed of sound in air is 330 m/s. The arms of the interferometer are 2m long, and the apparatus is placed in front of a large fan, which blows air along one of the arms at 8 m/s. Estimate the frequency of the beats that occur because of the interference of the waves reflected along the two arms of the interferometer. (hint: Be careful! You are measuring the Doppler shift for sound for the case of a moving medium, and both the speed and wavelength change.)

Homework Equations

f' = f_gamma[1 - cos(θ)(v/c)]
f' / f = (1 + v/c) / (√(1-(v/c)²))
u' = (u-v) / (1 - (v/c²)u)
These are all the equations in the corresponding section.

The Attempt at a Solution

Honestly, no idea where to start since I don't understand how these components are being put together, but I'm willing to bet that we need to find the relative velocity of the sound wave in the moving air medium. From there, we can make θ = 0 if they are moving apart or ∏ if they're moving together. At that point, you can find the adjusted frequency of the wave, but again, the equations will change depending on which way the sound and air are moving relative to each other (i.e. if the sound source is moving against, with, or perpendicular to the air.)

 

[mark726]

First, let's start with the basics. The Michelson-Morley experiment is typically used to measure the speed of light, but in this case, we are using it to measure the speed of sound. The setup of the experiment involves an interferometer, which is a device that splits a wave into two paths and then recombines them to create interference patterns. In this case, the wave we are using is a sound wave, which has a frequency of 3,600 Hz.

Now, let's consider the effect of the moving air on the sound wave. The speed of sound in air is 330 m/s, but in this case, we also have a fan blowing air at 8 m/s along one of the arms of the interferometer. This means that the sound wave will experience a Doppler shift due to the moving air. The equations you listed are correct, but we need to make some adjustments for the case of a moving medium.

First, let's define some variables. Let u be the speed of the sound wave in the stationary air, and v be the speed of the air blowing along one of the arms of the interferometer. We also need to consider the direction of the sound wave relative to the moving air. Let's say that the sound wave is moving perpendicular to the direction of the air flow. This means that the relative velocity between the sound wave and the air is u-v.

Now, let's consider the first equation you listed, f' = fgamma[1 - cos(θ)(v/c)]. This equation gives us the Doppler shift for a moving source. In this case, the source is the air, not the sound wave. So we need to make a small adjustment to the equation. Instead of using the speed of the air (v), we need to use the speed of the sound wave in the air (u). This gives us f' = fgamma[1 - cos(θ)(u/c)].

Next, let's consider the second equation, f' / f = (1 + v/c) / (√(1-(v/c)2)). Again, we need to make a small adjustment for the moving medium. Instead of using the speed of the air (v), we need to use the relative velocity between the sound wave and the air (u-v). This gives us f' / f = (1 + (u-v)/c) / (√(1-((u-v