Maths problem

[pErvinalia]

1. When the mirror is parallel to the direction of train travel, it will take 15.52 real seconds to register a second (using my parameters above). The sound wave will be travelling at 20m/s relative to the mirror/train. To get to the reflector 300 metres away it will take 15 seconds. On the return journey it is travelling at 580m/s second relative to the reflector/train. That 300m will be covered in 0.52 seconds.

True to form this is wrong too. The correct answer is actually half that. I used a distance between reflectors in the clock of 300m, but it should actually be 150m, as it needs to make the return journey as well to tick off one second.

[pErvinalia]

Ok, I think i've got this modelled correctly now.

Assumptions:
Train travelling at 30m/s
Speed of sound: 300m/s
Clock reflectors 1.5m apart, therefore registering ticks in one one-hundredths of a second (0.01 seconds).

Case 1: Clock parallel to train travel
Sounds is moving at 270m/s relative to the clock reflectors and train. It will take 0.0055555555555 seconds to reach the clock reflector.
On return the sound is moving at 330m/s relative to the clock/train. It will take 0.045454545454545 seconds to reach the initial reflector/emitter.
Therefore total time to complete travel and tick off one one-hundredth of a second is 0.0101010101010101 real seconds.
Therefore the clock is running at 0.01/0.010101010101 = 0.99 = 99% of real time.

Case 2: Clock perpendicular to train travel
Given that sound is moving at 10x the speed of the train, it will cover 10x the distance in the same time period. If you visualise this as a graph, you get a triangle. (will try and make a diagram later if people need). That is, the hypotenuse is 10x the length of the x-axis.
(edit, graph:)

Untitled.jpg (17.34 KiB) Viewed 2025 times

To determine the angle that the sound has to move relative to the direction of travel, we can use this ratio and the cos function. Cos = adjacent/hypotenuse. Hypotenuse/adjacent = 10, as explained. Therefore cos (@) = 0.1
Therefore @ = 84.261deg
Now we need to work out what the actual x-axis distance is for a y-axis distance of 1.5m (the distance between the reflectors; i.e. when the sound hits the reflector and completes half a tick). x-axis = adjacent = distance travelled by the train in half a tick of the sound clock (i.e. 0.005sec on the sound clock).
To get the x-axis distance we use the angle from above and the tan function. Tan (84.261) = opposite/adjacent = 1.5m/adjacent.
Therefore adjacent = 0.15076m
So for the sound clock to tick 0.01 seconds, the sound blip needs to travel 2x the hypotenuse, with the hypotenuse being 10x the distance of the x-axis (adjacent) distance = 1.5076
Therefore the sound blip needs to travel 3.0151m. At 300m/s that is 0.01005 real seconds.
Therefore, the clock is running at 0.01/0.01005 = 0.994987. Note, this is the actual value without rounding throughout the calculations. = 99.5% of real time.

So from this it seems that the clock slows down more as moves from perpendicular to parallel with train direction of travel. This goes against what I predicted earlier, and has me scratching my head a bit. No doubt these calculations are all wrong as well.

[Brian Peacock]

Yeah, but you're still measuring distance becuase sound is a compression wave which travels at a fixed rate in a stable medium. My model demonstrates that the points of emmission, reflection, and reception are fixed in time and therefore the clock's timing is a function of basic geometry. The model I posted was only to show geometric relationship and how the angle and direction of travel effected the return time. If I get the time I'll try and make a more detailed model that makes the trigometric relationships more explicit.

Remember, here the speed of sound is not multiplied by the speed of the train becuase the medium it travels through is not moving - periodic compression or expansion of a sound's wavelengths (frequency shifting) only occurs for a static observer when a sound source is moving toward or away from them.

[pErvinalia]

The medium could be moving. Depends on the construction of the clock.

[mistermack]

To shed more light on the matter, I made a comment on the science forum about the invariance of the speed of light, saying that something similar happened with sound waves, and it sort of rolled on from that.
What the example does show, is that a sound clock gets time dilation, like a clock that works on light, measuring the speed of light.
Someone claimed that the difference would be that a light clock would run at the same rate, wherever you pointed it, whereas a sound clock would not. But they didn't provide evidence for that, so I wondered if it was right.

What he is now saying is that a light clock would ALSO give different values depending on where you point it, if it wasn't for length contraction.

But he (Janus) has given his workings so here is a link : (scroll down to the bottom)

http://www.scienceforums.net/topic/1021 ... ght/page-3

His workings don't copy and paste in the same format, so you need to look at the site.

[Brian Peacock]

I'm on the phone now but I'll take a look at that later.

The medium could be moving. Depends on the construction of the clock.

MrM didn't mention if it was a windy day.

[pErvinalia]

MM, just skimmed through it. Looks like he did more or less the same calculation as I did, although with ye olde trigonometry in place of the cos and tan.

[pErvinalia]

By the way, I don't think it is correct to say that a sound clock experiences time dilation. It doesn't physically experience time differently to another inertial frame. It simply runs slower due to the sound wave having to travel further.

[mistermack]

By the way, I don't think it is correct to say that a sound clock experiences time dilation. It doesn't physically experience time differently to another inertial frame. It simply runs slower due to the sound wave having to travel further.

It's dilation compared to another frame in the still air.
The sound clocks in each different frame run at different rates.

[pErvinalia]

That's not what dilation means in Relativity.

[mistermack]

That's not what dilation means in Relativity.

Tis

[pErvinalia]

Oh, ok then. But back in Normaltown, time dilation is a physical phenomenon that effects the perception of time regardless of whether there is a clock involved. A sound clock will experience time dilation exactly as another other physical object would in an inertial frame. But at low speeds like in the train example the effect of time dilation is minuscule on the train (and a stationary observer). The reason the sound clock runs slow is that the sound simply has to travel further. There's absolutely no exceptional physical phenomenon going on there.

[pErvinalia]

To put it another way, once the tiny relativistic dilation effects are taken out, the clock on the train reads the same speed to an observer on the train as to an observer on the train platform.

[Brian Peacock]

Time is no time dilating for either the emitter or reflector as they are always locked into the same frame of reference. All that happens is that the distance the ping travels varies with reference to the angle in relation to the direction of the train. That's why this is a 2d geometry problem not a 4d GR problem.

[pErvinalia]

Time dilation is irrelevant within the same intertial frame. It is only a coherent concept when we are talking about observations between intertial frames. Which is why the concept exists independent of whether it is a sound clock, a light clock, or a pendulum.