Beat frequency and velocity of observer

[RainMD]

Homework Statement

Two sound sources 15m apart producing identical 229 Hz sounds. As you move from one to the other, you hear beat frequency of 2.5. How fast are you moving?[/B]

Homework Equations

The question asks for the velocity of the observer Vo. The trick here is to use the standing wave model to get the solution, NOT the doppler reflection equation.

The Attempt at a Solution

I used the doppler reflection equation to get a speed of 1.87 m/s. This was to see if my answer is wrong or right.
I started by getting the different frequencies from the beat equation (assuming one is 229 and the other is below/higher than that), and then I got the wavelengths. I solved for the change in wavelengths and divided that by the reciprocal of the beat frequency (time) to get a velocity of 0.4 m/s. Any ideas what I'm doing wrong?

 

[TSny]

Hello RainMD. Welcome to PF!

Which source has the Doppler shift? Or would both have a shift?

 

[RainMD]

Hello RainMD. Welcome to PF!

Which source has the Doppler shift? Or would both have a shift?

The question didn't specify, so I assumed both have doppler shift and solved for the correct velocity. Keep in mind that the question asks for the velocity using the standing wave model.

 

[TSny]

OK, both sources will have a frequency shift. (In your first post it appears to me that you assumed only one of the sources is detected to have a shift in frequency.)

I don't get an answer of 0.4 m/s. I get the 1.87 m/s. If you'll show the details of your calculation, we can check your work.

 

[RainMD]

OK, both sources will have a frequency shift. (In your first post it appears to me that you assumed only one of the sources is detected to have a shift in frequency.)

I don't get an answer of 0.4 m/s. I get the 1.87 m/s. If you'll show the details of your calculation, we can check your work.

Okay, so I went with both sources shifting. That means f2= 230.25 and f1= 227.75 (in order to get 2.5 as the result of their differences)
Now I get the wavelength using the speed equation y = V/f , with v = 343 m/s (speed of sound). So 343/230.25 = 1.506
Same with the second frequency, 343/227.75 = 1.48.
So, the change in wavelength is 0.02 m. Is this correct?

 

[TSny]

stts

Okay, so I went with both sources shifting. That means f2= 230.25 and f1= 227.75 (in order to get 2.5 as the result of their differences)
Now I get the wavelength using the speed equation y = V/f , with v = 343 m/s (speed of sound). So 343/230.25 = 1.506
Same with the second frequency, 343/227.75 = 1.48.
So, the change in wavelength is 0.02 m. Is this correct?

If you walk between the sources you will not observe any change in wavelengths of the waves, but you will observe frequency shifts of the two individual waves.

But, this is kind of beside the point. You are asked to use a standing wave model rather than using the Doppler effect. So, we need to be clear about how to look at the problem from the point of view of a standing wave. As the two waves from the sources interfere in the region between the sources a standing wave is produced. The standing wave forms a stationary pattern relative to the ground (i.e., the nodes and antinodes are at fixed locations relative to the ground.)

As you walk between the sources, you are walking relative to the standing wave. Do you see why you will hear beats as you walk?

 

[RainMD]

tts

If you walk between the sources you will not observe any change in wavelengths of the waves, but you will observe frequency shifts of the two individual waves.

But, this is kind of beside the point. You are asked to use a standing wave model rather than using the Doppler effect. So, we need to be clear about how to look at the problem from the point of view of a standing wave. As the two waves from the sources interfere in the region between the sources a standing wave is produced. The standing wave forms a stationary pattern relative to the ground (i.e., the nodes and antinode are at fixed locations relative to the ground.)

As you walk between the sources, you are walking relative to the standing wave. Do you see why you will hear beats as you walk?

So, the beats represent a relationship between nodes and antinodes? Since I'm walking from a high frequency to a lower one? I don't get how this will help since beats are just a difference.

 

[TSny]

As you walk along the standing wave, what will you hear at the instant you pass a node? An antinode?

 

[RainMD]

As you walk along the standing wave, what will you hear at the instant you pass a node? An antinode?

Antinodes = high pitch sound
Nodes = low or normal pitch sounds
Is that correct?

 

[TSny]

No. You will hear essentially only one pitch. But what about the loudness of that pitch when you are at a node? An antinode?

 

[RainMD]

No. You will hear essentially only one pitch. But what about the loudness of that pitch when you are at a node? An antinode?

A loud pitch at antinodes and a low one in nodes, since they represent constructive and destructive interference point (?)
I'm really sorry about all the confusion I'm causing.

 

[TSny]

"Pitch" refers to frequency. As you walk along the standing wave, you do not hear any change in frequency. But you are right that at the antinodes the sound will be loud while at the nodes the sound will be soft. (Ideally, there would be no sound at the nodes.) So, as you walk along the standing wave and go from node to anti-node to node to anti-node to...etc. you will hear soft, loud, soft, loud, ...etc. Beats! How often you hear beats depends on how fast you are walking. So, there should be a way to deduce your speed of walking from the given beat frequency. But, you'll need to know the distance between the nodes or antinodes.

 

[RainMD]

"Pitch" refers to frequency. As you walk along the standing wave, you do not hear any change in frequency. But you are right that at the antinodes the sound will be loud while at the nodes the sound will be soft. (Ideally, there would be no sound at the nodes.) So, as you walk along the standing wave and go from node to anti-node to node to anti-node to...etc. you will hear soft, loud, soft, loud, ...etc. Beats! How often you hear beats depends on how fast you are walking. So, there should be a way to deduce your speed of walking from the given beat frequency. But, you'll need to know the distance between the nodes or antinodes.

Thank you so much! I solved the problem with your help. Have a great day!

 

[TSny]

Great!