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1

Yes, the two approaches are equivalent. As you've noted, there's only one physical effect going on here, interference, and the standard beat frequency / path length interference formulas are just special cases of the same thing. There are some restrictions. Both beat frequency and path length interference only make physical sense when the sources are ...


2

The exercise seems not too difficult. With $f_0 = 456 \text{Hz}$, $c$ the speed of sound an $v$ the speed of the observer, you just have to find the beat frequency by adding the wave equations for 2 different sound waves with frequencies $f_1=f_0(1-\frac{v}{c})$ and $f_2=f_0(1+\frac{v}{c})$. So this is just an exercise in applying the Doppler effect, and you ...


0

The question may be about the covariance or otherwise of temperature, in which case have a look here. As well, have a look at the paper "Temperature in special relativity" by J. Lindhard, Physica Volume 38, Issue 4, 5 June 1968, Pages 635-640.


3

In the kinetic theory of gases, you only really define the temperature for molecules that are in constant, random, and rapid motion. So if you have a container with a gas at temperature $T$ you don't change the internal energy of the gas by uniformly moving the container. Uniformly moving the container gives all the molecules a non-zero average motion, but ...


1

I think you can avoid all these troubles if you define the temperature as proportional to the variance of velocity, i.e. $$E[(v-\overline{v})\cdot(v-\overline{v})]=E(v\cdot v)-\overline{v}\cdot\overline{v}$$ Here $E$ means expected value, $v$ ranges over the velocities of the individual particles, and $\overline{v}=E(v)$. Clearly this is frame-...


1

All kinds of weird things happen if you try to define temperature in a moving object. The paradox to me (not a generalized accepted answer) resolves by realizing that temperature should only be defined as measured when the object is stationary. Not only is not a scalar but it is not even well defined for areference frame in relative motion. Is temperature ...


10

Temperature is related to kinetic energy in the rest frame of the fluid/gas. In non-relatvistic kinetic theory the distribution function is $$ f(p) \sim \exp\left(-\frac{(\vec{p}-m\vec{u})^2}{2mT}\right) $$ where $\vec{u}$ is the local fluid velocity. The velocity can be found by demanding that the mean momentum in the local rest frame is zero. Then $\vec{u}...


7

[June 19,2016: thoroughly revised, giving a more detailed, comparative presentation and better references] General case. In relativistic thermodynamics, inverse temperature $\beta^\mu$ is a vector field, namely the multipliers of the 4-momentum density in the exponent of the density operator specifying the system in terms of statistical mechanics, using the ...


2

I think it is wrong to define the temperature by the average energy of the molecule in all frames of reference. The reason for that is clear: take all of your particles and send them at $100 m/s$ to the north. This won't make the gas hotter, just like the fan does not cool/heat the air (another great mystery!). The organized movement does not participate in ...


2

The observed wavelength does change, and this is called the doppler effect. But the speed does not change. The statement "...the wavefront of a short pulse emitted like this will reach a given distance along y before it will along x" does not follow from any logical reasoning. Following the same logic you will conclude that when the source is next to the ...


2

According to all observational evidence (including the original Michelson-Morley experiment) speed of light is constant in all directions. The confusion comes from misinterpretation of the picture you attach. I propose to understand it as follows. Say, you have a lightsource that emits pulses at a given frequency. Each pulse propagates at a constant speed ...


0

I agree with Lubos Motl that your concern really seems to be the time delay between emission and detection (which depends on distance), rather than the Doppler Effect (change in frequency, which depends on relative speed between observer and object). As Lubos Motl suggests, using light instead of sound would solve your problems. The transit time is much ...



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