Doppler Effect Relative Motion

[anhhuyalex]

To calculate the detected frequency from the source frequency, we use this formula:

{ f }_{ D }=\frac { v\pm { v }_{ D } }{ v\pm { v }_{ s } } { f }_{ s }

where {v}_{s}, {v}_{D} are velocities of the source and the detector respectively with reference to the medium. My question is why do we consider motion relative to the medium? Wouldn't a detector moving at 3 m/s receiving sound from a source moving at 8 m/s in the same direction detect the same frequency as a stationary detector receiving sound from a source moving at 5 m/s? However, from the above equation, that is not true. Thanks for taking your time to answer.

 

[paisiello2]

The reason is that the waves appear to be hitting you at with a speed v + vD + vs when in fact relative to a stationary observer with no velocity the waves are only traveling with a speed v.

Suppose you were measuring the intervals of water waves coming at you on a lake. You are in one boat and your friend is in another boat a distance away. The lake is still and there are no waves on it except for the ones that your friend can create by hitting his oar paddle on the water.

You agree that if your friend moved toward you with a speed vs (say he had a small motor that generated negligible waves compared to the oar paddle ones) that the frequency of the paddle oar waves hitting you would increase as his speed increases, right? You would bob up and down even more.

Now if you in turn started your small motor and moved towards your friend with a speed vD then the waves would seem to be coming in even faster. I'm sure you have experienced this if you have ever been on a speed boat before.

This is the equivalent of an increased frequency caused by a Doppler shift. You actually can't tell how fast the waves are moving (v) unless you also know your friend's and your own speeds. This is because the apparent speed would be v + vs + vD = λ*fD where λ would be the measured length of the wave.

 

[dauto]

The formula is correct. The speeds are measure relative to the medium because the medium iks carrying the waves. The wave speed is constant relative to the medium. Your intuition is based on the relativity of motion which doesn't apply here since there is a medium

 

[anhhuyalex]

Why doesn't it apply just because of a medium? Since all motion are relative anyway, why is it that we need to consider the velocities from the reference of the medium instead of considering the velocities from the reference of the source or the detector?

 

[paisiello2]

You understand that the speed of a wave v is constant in the medium? If you accept that premise then the only thing that could possibly change the frequencies is the relative speeds between the source and detector.

 

[anhhuyalex]

What would happen if the speed of the wave is less than the speed of the source when the source is moving towards the detector? That would make a negative value for the denominator, wouldn't it, while the numerator is positive? Then, the frequency is going to be negative. Is that possible?

 

[paisiello2]

If something moves with a speed (vS) that is faster than the speed a wave propagates in a medium (v) then the object has broken the sound barrier and generates a shock wave with a speed =vS.

The Doppler shift equation does not apply in this case. My guess is that you would first feel the sonic boom then hear the unshifted regular sound waves moving at speed v with a frequency fS.

 

[dauto]

Why doesn't it apply just because of a medium? Since all motion are relative anyway, why is it that we need to consider the velocities from the reference of the medium instead of considering the velocities from the reference of the source or the detector?

It doesn't apply because the medium provides an absolute referential. Relativity of motion requires all the relevant elements of a problem to change their speed by the same amount. If you are in a moving car with closed windows it feels like you're sitting still. Everything is moving along at the same speed, including the air. Than relativity of motion applies. But if open the window than you can tell the difference between a moving car and a stopped car (Just feel the wind). In order for the relativity of motion to apply in the Doppler effect example you must carry the air along with you. But that's not the situation described by the formula.

 

[dauto]

What would happen if the speed of the wave is less than the speed of the source when the source is moving towards the detector? That would make a negative value for the denominator, wouldn't it, while the numerator is positive? Then, the frequency is going to be negative. Is that possible?

In that case you get a sonic boom AKA shock wave AKA bow wave like the one you see at the wake of a boat. The Doppler formula doesn't apply. There is no wave between you and the source if the source is moving faster than the wave.