# Is the speed of a wave dependent on the speed of the source?

1. Jan 23, 2014

### fuserofworlds

So, I am reading my Modern Physics textbook (Tipler & Llewellyn), in the chapter on relativity. In discussing the Michelson-Morley experiment, the book states, "The speed of the waves depends on the properties of the medium and is derived relative to the medium. For example, the speed of sound waves in air, i.e., their absolute motion relative to still air, can be measured." Some further examples and this statement suggest to me that the speed of a wave relative to the medium is dependent on the speed of the source relative to the medium.

However, later in the book it states quite clearly, "the speed of sound waves does not depend on the motion of the sound source. When an approaching car sounds its horn...the speed of the waves traveling through the air does not depend on the speed of the car."

Am I missing something here? Which statement is correct?

2. Jan 23, 2014

### vela

Staff Emeritus
I don't see how that statement suggests that the speed of the wave depends on the motion of the source. There is no mention of a source at all.

3. Jan 23, 2014

### fuserofworlds

vela, sorry for the confusion. The book makes it clear that the waves must be measured relative to still air. To me, this implies to me that if the air (the medium) were moving, you would measure a different speed. Because inertial reference frames are identical, you would likewise measure a different speed if the sound source were moving relative to the air.

There is an example discussing how light would behave classically if it propagated through the "ether":
"Light source, mirror, and observer are all moving with speed v relative to the ether. According to classical theory, the speed of light c, relative to the ether, would be c - v relative to the observer for light moving from the source toward the mirror, and c + v for light reflecting from the mirror back toward the source."

Since sound does have a medium and behaves classically, does this mean that its speed is dependent on the speed of the source?

4. Jan 23, 2014

### AlephZero

In newtonian mechanics, the speed of sound measured relative to the medium does NOT depend on the speed of the source.

If you are getting confused by thinking about the Doppler effect, be careful about exactly what is going on. For example, if you are in a car driving at 100 mph and listening to the radio, the air inside the car is moving at the same speed as the car, so the speed of the car relative to the earth makes no difference to what you hear. On the other hand, if the car passes somebody standing on the road who is listening to the (loud) car radio, the air around them is not moving at the same speed as the car..

5. Jan 23, 2014

### fuserofworlds

So, are you saying that in special relativity, the speed of sound (or of any wave) measured relative to the medium DOES depend on the speed of the source?

6. Jan 23, 2014

### Dick

No, nobody is saying the speed of waves in a medium (i.e. relative to the state of motion frame of the medium) is dependent on the speed of the source. They are saying the opposite. Why do you think it does?

Last edited: Jan 24, 2014
7. Jan 24, 2014

### fuserofworlds

This is the example in my book. I understand that we now know light has the same speed in any reference frame; but the book seems to imply that this is only because light does not travel through a medium. What am I getting from the book is that, for a wave traveling in a medium, if the source is moving relative to the medium, than the speed of the wave measured in the reference frame of the source will be different from the speed of the wave measured in the reference frame of the medium. QED, the speed of a wave (or one that propagates through a medium) is dependent on the speed of the source, relative to the medium.

If this is not the case, and the speed of a wave is NOT dependent on the speed of the source, then my questions are:
1. Why is this the case/how does one show this?
2. What is the reasoning behind the book's example? Is it just to demonstrate outdated thinking, from before Einstein's postulates?

8. Jan 24, 2014

### ehild

The book speaks about the speed of light relative to the observer. The observer moves with speed v relative to the "ether". The light moves with speed c relative to the ether, either in + or - direction. Within classical theory, the relative speed of light with respect to the observer is c-v and c+v.

ehild

9. Jan 24, 2014

### vela

Staff Emeritus
The speed of the wave relative to the medium is always the same, regardless of whether the source is moving relative to the medium. If, in still air, a car honks its horn when it's 340 m away from me and I'm not moving, I will hear the sound one second later, regardless of whether the car is driving toward or away from me or is stationary.

An observer moving relative to the medium will, of course, see the wave propagate at different speed. This is just plain old Galilean relativity. If I'm moving toward the aforementioned car, the sound will have to travel less than 340 m before it gets to me. From my perspective, the sound was emitted 340 m away from me and took less than a second to get to me, so it moved faster than 340 m/s. Note that this does not imply that the sound isn't moving at 340 m/s relative to the still air. It still does; the sound, however, moves faster than 340 m/s relative to me.

If a moving source also happens to be the observer, the source, like any other observer moving relative to the medium, will see the sound propagate at a different speed.

10. Jan 24, 2014

### j824h

We need to clarify what the speed of wave is.
Generally, speed of wave is the constant $c$ which makes the wave equation
$$\frac{\partial^2\psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2\psi}{\partial t^2},$$
holds in any reference frame.

And you should note that it's the reference frame what varies in each case.

Mechanical wave in Newtonian mechanics: the one specific frame static to the medium.
Light in SR: any inertial frame.​

With the definition above, let me try to explain.

In both cases, the speed of wave is a constant, which doesn't change by movement of source or observer.

1. It's the definition.
2. To apply the argument involving medium for light is to demonstrate the outdated. But the argument itself isn't outdated. It works very precisely for mechanical wave, where we usually find the speed of wave much smaller than the speed of light thus the Newtonian mechanics doesn't show significant error.

11. Jan 24, 2014

### fuserofworlds

vela and j824h, thank you, that makes sense!