Doppler effect with no given numbers

[rissa_rue13]

Homework Statement

A source of sound mvoes along one straight line between listeners Abigail and Bertha. The wavelength of the sound waves heard by Abigail is about 1/2 the wavelength heard by Bertha. What is the ratio of of the speed of the source to the speed of sound? Assume that the air carrying the sound waves is stationary.

Homework Equations

http://www.kettering.edu/~drussell/Demos/doppler/Eq1.gif

The Attempt at a Solution

So I know that the wavelength that Abigail observes is 1/2 the wavelength that Bertha observes. Neither of them are moving and the speed of sound is 343 m/s. I don't know how to tie these thoughts together to solve this problem.

 

[Doc Al]

That image you linked to is not viewable.

In any case, you need to apply the Doppler formula (for the case of a moving source) twice: once for Abigal, once for Bertha. Hint: Call the original wavelength emitted by the source "w" and compare the two equations you end up with.

 

[rissa_rue13]

I've tried that and just ended up confusing myself, so I'm not sure if that's right. I reviewed some examples from class and he solves for f_s and then substitutes that in, but the cancellations didn't work out doing it that way.

 

[tiny-tim]

So I know that the wavelength that Abigail observes is 1/2 the wavelength that Bertha observes. Neither of them are moving and the speed of sound is 343 m/s. I don't know how to tie these thoughts together to solve this problem.

Hi rissa_rue13!

Show us what equations you have tried.

(your image file isn't viewable)

 

[Doc Al]

Show what you've tried and I'll take a look.

By setting up the two equations as I suggested, you can eliminate f_s and solve for v_s in terms of the speed of sound.

 

[rissa_rue13]

So, I set up f_b and solve for f_s:
(v+v_s)/w_b=f_s
right?
Then, if I substitute that into f_a:
f_a=[(v+v_s)/w][v/(v-v_s)]
But, I'm trying to get v_s/v, so I don't get how that works?

 

[Doc Al]

Try this: Write an equation for Abigal's observed frequency (f_a) in terms of f_s and the other variables. Then write a similar equation for Bertha's observed frequency (f_b). Then you can compare them more easily.

 

[rissa_rue13]

But f_a and f_b are different because the wavelength Abigail observes is 1/2 that of the wavelength Bertha observes.

 

[rissa_rue13]

f_a=v(1/2w)
f_b=vw
so wouldn't that make f_s different for Abigail and Bertha?

 

[Doc Al]

But f_a and f_b are different because the wavelength Abigail observes is 1/2 that of the wavelength Bertha observes.

Sure, and we will make use of that key fact. But first write out the two equations. (Hint: The only difference between the equations will be the sign of the source velocity.)

 

[rissa_rue13]

f_a=f_s[v/(v+v_s)] -> v/(1/2w)=f_s[v/(v+v_s)]
f_b=f_s[v/(v-v_s)] -> v/w=f_s[v/(v+v_s)]

 

[Doc Al]

f_a=f_s[v/(v+v_s)] -> v/(1/2w)=f_s[v/(v+v_s)]
f_b=f_s[v/(v-v_s)] -> v/w=f_s[v/(v+v_s)]

Good. But all you need is the left side version of those formulas. Now take the next step. Hint: What's f_a/f_b?

 

[rissa_rue13]

So,
f_a/f_b=[f_s(v/v+v_s)]/[f_s(v/v-v_s)]
right?
Then, wouldn't that reduce to:
f_a/f_b=[v(v-v_s)]/[v(v+v_s)] ?

 

[Doc Al]

So,
f_a/f_b=[f_s(v/v+v_s)]/[f_s(v/v-v_s)]
right?
Then, wouldn't that reduce to:
f_a/f_b=[v(v-v_s)]/[v(v+v_s)] ?

Absolutely. Replace the left hand side with a specific numerical value, based on information given in the problem; simplify the right hand side a bit. Once you've done that you can solve for v_s in terms of v.

 

[rissa_rue13]

The only number I'm given is that w_a=1/2w_b. So would I substitute that in by doing:
(2v/w)/(v/w) which would reduce to 2.
And then put that in to:
2=[(v-v_s)/(v+v_s)] ?
Wouldn't that make v=v_s ?

 

[Doc Al]

I just noticed that in your formulas you have f_a and f_b reversed. Abigal is the one who hears the smaller wavelength--and thus larger frequency--sound since the source is moving towards her. So fix that first.

The only number I'm given is that w_a=1/2w_b. So would I substitute that in by doing:
(2v/w)/(v/w) which would reduce to 2.

Good, except as noted above.

And then put that in to:
2=[(v-v_s)/(v+v_s)] ?
Wouldn't that make v=v_s ?

No, it wouldn't. If v = v_s, that right hand side would equal 0. (Again, first correct the error noted above.)

 

[rissa_rue13]

So f_a/f_b should still equal 2, right? It was just the other part of the equation that was wrong?

2 = (v+v_s)/(v-v_s)
2v-2v_s=v+v_s
v=3v_s
v/3=v_s​

assuming v=343 m/s (speed of sound in air):
v_s= 114 m/s
Therefore,
v_s/v = (114 m/s)/(343 m/s) = 0.333
right?

 

[Doc Al]

All good! (Just a tip: No need to actually calculate the speed. Once you found that v_s = v/3, it immediately follows that v_s/v = 1/3.)