B Does Motion Affect Perceived Wavelength and Light Speed?

[Speady]

[Mentors' note: This thread has been moved from the relativity forum, as it is a question about the classical behavior of waves]

Is the length of a beam of light the number of crests times the wavelength?

 

[Ibix]

If the radiation is coherent, yes. Otherwise the question may not be well posed.

 

[Nugatory]

Is the length of a beam of light the number of crests times the wavelength?

If you are considering a pulse of coherent monochromatic light that is long compared to the wavelength, then that is an excellent approximation.

 

[Speady]

If the wavelength varies with the speed of the source or observer, does the length of the pulse of coherent monochromatic light also varies caused by the number of wave crests times the wavelength?

 

[Ibix]

If the wavelength varies with the speed of the source or observer, does the length of the pulse of coherent monochromatic light also varies caused by the number of wave crests times the wavelength?

With the caveats mentioned above, the length of the pulse of light is always the number of wave crests (which is frame independent) times the wavelength as measured in whatever frame you are working in.

Remembering your last thread, note that the length of the pulse as measured in your frame is not necessarily the same as the wave speed (as measured in your frame) times the pulse duration if the clock used to measure the duration is moving in whatever frame you are working in.

 

[Speady]

So two different observers, each moving at a different speed relative to the pulse, simultaneously measure a different length for the same pulse?
How exactly can the position of the (same) pulse be determined? (Assuming that observers move towards and away from the source at 12 km / s, making relativistic differences of time and place negligible, but the length differences of the pulse could be significant).

 

[Ibix]

So two different observers, each moving at a different speed relative to the pulse, simultaneously measure a different length for the same pulse?

If it's a light pulse, there is no "different speed relative to the pulse". If you mean "two different observers moving relative to one another" then yes, of course.

Let's say I am at rest in some frame and I emit a light pulse for one second. In my frame, then, the light pulse is one light second long. Consider a pair of receivers that are one light second apart as measured in my frame, traveling at the same speed. Using my simultaneity convention, the nearer one stops receiving the pulse at the same time the further one starts receiving it. This is true whatever speed the receivers are traveling at. Clearly, then, the measured length of the pulse is the same for receivers moving at any speed low enough for length contraction to be negligible. But that doesn't mean that each receiver is illuminated for exactly one second.

If you want to use your naive "wave speed times duration" calculation to measure the pulse length then you need to work in the frame where the receiving clock is at rest. In your example, this means that the source is moving and the beam length is longer than one light second.

(Assuming that observers move towards and away from the source at 12 km / s, making relativistic differences of time and place negligible, but the length differences of the pulse could be significant).

See my previous paragraphs. Do note that your last thread got closed because you refused to even consider what I was saying here. If you don't understand, ask questions about what I'm saying. Don't just repeat things that are false.

 

[Nugatory]

So two different observers, each moving at a different speed relative to the pulse, simultaneously measure a different length for the same pulse?

To be precise, if two different observers moving relative to one another each choose to use the frame in which they are at rest to measure the length of the pulse they will get different results. That's a special case of the general fact that the length will always depend on the choice of frame.

 

[Speady]

Let's say I am at rest in some frame and I emit a light pulse for one second.

Suppose your pulse of a second has orange light, with 500,000,000,000,000 wave crests of 0.000000600 nm each, together 300,000 km long.
Suppose the observers each receive the start of the pulse at the same time.
The wavelengths change for the two observers by the speed of 12 km / s then to 0.000000600024 nm (slightly redder, moving away from the source) and to 0.000000599976 nm (slightly more yellow, moving towards the source).
The same wave crests times new wavelengths then simultaneously give at the same time of the same pulse an observed length of 300,012 km and 299,988 km.
If I calculate the length in the frames of the observers, then the lengths become each 24 cm shorter by length contraction (negligible compared to 12 km).
How can I explain this?

 

[Ibix]

0.000000600 nm

If you are going to write out numbers like this, use standard form. Then you are less likely to make mistakes like the above - it should be 600nm, or 6×10^-7m.

How can I explain this?

By accounting for the motion of the receiver or source. In the frame where the source is at rest, the satellites are moving. It should not be surprising that the time during which the receivers are illuminated is different from the emission time - you are quite literally moving the goal posts. And since you are moving the goalposts it would be a mistake to simply multiply that time by the wave speed and hope to get the pulse length. (Consider doing this with sound and a receiver moving at the speed of sound. The receiver will always be in the pulse - do you conclude that the pulse is infinitely long?)

In the frames where the satellites are at rest, on the other hand, the source is moving. Now you can use the satellite clocks to measure the pulse length in this frame and it will, indeed, be different from the length measured in the other frame. But this should not be surprising because of the movement of the source - the pulse will be shortened or lengthened by the Doppler factor (not the Lorentz factor), as expected.

If you cannot see this, you need to do the calculation properly. In some frame (probably the rest frame of the source), write down the coordinates of the events of the start and end of pulse emission and the equations of motion of the leading and trailing edges of the pulse. Then write down the equation of motion of the receiver and determine the coordinates of the start and end of pulse reception. Then Lorentz transform the coordinates into the frame where the receiver is at rest. This will allow you to calculate the pulse length and duration of illumination in either frame. You can also take the limit $v\ll c$ if you want.

Do you understand my explanation? If not, do you know how to do the above calculation?

 

[Ibix]

Length contraction is the wrong tool to be applying here, by the way. You get length contraction when you compare the length of something measured in its rest frame to the length measured in a frame where it is moving. But the light pulse is not at rest in either frame, so you should not expect the length contraction factor to be relevant here.

 

[Speady]

Length contraction is the wrong tool to be applying here

Thanks for your explanation

 

[Speady]

It should not be surprising that the time during which the receivers are illuminated is different from the emission time

That's right. This is a real observed time difference on the observers' clocks. From T = 0 at the start of the observation, the clock at the end of the observation will be at T = 0.99996 s for one observer and T = 1.00004 s for the other observer. (The clocks of the observers are, by the way, synchronized with the clock of the source. Any difference due to time dilation is negligible: the factor is 1.0000000008)

Now you can use the satellite clocks to measure the pulse length in this frame and it will, indeed, be different from the length measured in the other frame. But this should not be surprising because of the movement of the source - the pulse will be shortened or lengthened by the Doppler factor

That's right. The length is also the number of crests times the changed wavelength. It gives a changed length.
But given the two different places the two different lengths occupy (for one and the same pulse), one shorter and the other longer, my question: is the changed wavelength an apparently altered wavelength, while the actual wavelength remains the emitted 600 nm?

 

[Nugatory]

my question: is the changed wavelength an apparently altered wavelength, while the actual wavelength remains the emitted 600 nm?

There is no such thing as an "actual" or an "altered" wavelength. Different observers with different states of motion will measure different wavelengths and frequencies. No measurements is any more or less "actual" than any other, although there is a very useful convention for waves in a medium: when the speed, frequency or wavelength is stated without qualification, it means "as measured when both source and receiver are at rest relative to the medium".

But no matter our state of motion relative to the wave, its source, and the medium, the answer to the question in the post is yes: speed equals frequency times wavelength, pulse width equals number of crests times wavelength equals speed times the passage time.

Because this question has been answered we can close this thread.

There is an interesting followup question, namely how to reconcile the relationship between speed, frequency, wavlelength, and pulsewidth with the experimental fact that the speed of light is the same in all inertial frames. That question belongs in the relativity forum.