A source and receiver of EM waves move relative to each...

A source and receiver of EM waves move relative to each other at velocity $v$. Let $v$ be positive if the receiver and source are moving apart from each other. The source emits an EM wave at frequency $f_{\mathrm{s}}$ (in the source frame). The time between wavefronts as measured by the source is $T_{\mathrm{s}}=1 / f_{\mathrm{s}}$. (a) In the receiver's frame, how much time elapses between the emission of wavefronts by the source? Call this $T_{r}^{\prime}$. (b) $T_{r}^{\prime}$ is not the time that the receiver measures between the arrival of successive wavefronts because the wavefronts travel different distances. Say that, according to the receiver, one wavefront is emitted at $t=0$ and the next at $t=T_{\mathrm{o}}^{\prime}$. When the first wavefront is emitted, the distance between source and receiver is $d_{\mathrm{r}}$. When the second wavefront is emitted, the distance between source and receiver is $d_{\mathrm{r}}+v T_{\mathrm{r}}^{\prime}$. Each wavefront travels at speed $c$. Calculate the time $T_{\mathrm{r}}$ between the arrival of these two wavefronts as measured by the receiver. (c) The frequency detected by the receiver is $f_{\mathrm{r}}=1 / T_{\mathrm{r}}$. Show that $f_{\mathrm{r}}$ is given by $$ f_{\mathrm{r}}=f_{\mathrm{s}} \sqrt{\frac{1-v / c}{1+v / c}} $$

A source and receiver of EM waves move relative to each other at velocity $v$. Let $v$ be positive if the receiver and source are moving apart from each other. The source emits an EM wave at frequency $f_{\mathrm{s}}$ (in the source frame). The time between wavefronts as measured by the source is $T_{\mathrm{s}}=1 / f_{\mathrm{s}}$. (a) In the receiver's frame, how much time elapses between the emission of wavefronts by the source? Call this $T_{r}^{\prime}$. (b) $T_{r}^{\prime}$ is not the time that the receiver measures between the arrival of successive wavefronts because the wavefronts travel different distances. Say that, according to the receiver, one wavefront is emitted at $t=0$ and the next at $t=T_{\mathrm{o}}^{\prime}$. When the first wavefront is emitted, the distance between source and receiver is $d_{\mathrm{r}}$. When the second wavefront is emitted, the distance between source and receiver is $d_{\mathrm{r}}+v T_{\mathrm{r}}^{\prime}$. Each wavefront travels at speed $c$. Calculate the time $T_{\mathrm{r}}$ between the arrival of these two wavefronts as measured by the receiver. (c) The frequency detected by the receiver is $f_{\mathrm{r}}=1 / T_{\mathrm{r}}$. Show that $f_{\mathrm{r}}$ is given by
$$
f_{\mathrm{r}}=f_{\mathrm{s}} \sqrt{\frac{1-v / c}{1+v / c}}
$$

Key Concepts

Relativistic Doppler Effect

This concept describes how the frequency of electromagnetic waves changes due to the relative motion between the source and the observer. It extends the classical Doppler effect by incorporating the effects of special relativity, particularly when the relative speeds approach the speed of light. The effect arises because there is both a time dilation effect and a change in the distance each wavefront travels depending on the motion, leading to the specific formula for the frequency shift.

Time Dilation

Time dilation is a phenomenon from special relativity where a moving clock runs slower when observed from a stationary frame. In the context of electromagnetic waves, the period between wave emissions (and subsequently between their receptions) is lengthened when measured in a frame in which the source is moving, playing a critical role in determining the observed frequency of the waves.

Invariance of the Speed of Light

One of the fundamental postulates of special relativity is that the speed of light in a vacuum is constant and independent of the motion of its source or observer. This invariance underpins the derivation of the relativistic Doppler effect because it ensures that the propagation speed of wavefronts remains c in all inertial frames, affecting how time intervals and distances translate between frames.

Lorentz Transformation

Lorentz transformations are the mathematical relationships that connect the space and time coordinates of events as measured in different inertial frames moving at a constant relative velocity. They are essential for deriving how time intervals (such as the emission period of wavefronts) change between the source and receiver frames, thereby supporting the derivation of the relativistic Doppler shift formula.

Transcript

00:01 In this problem, we're going to derive the doppler shift for life formula.

00:06 We have a frame as the source frame moving away from frame r, the receiver frame with speed v.

00:14 The source to rest in s, source frame.

00:18 Now, you remember that two, if you are looking between two crests in a wave, then in a picture, that would be considered wavelength.

00:29 And the time of admission of two crests then, by definition, is the period.

00:36 Ts is the time where an observer in s, at rest in s, will say that the period of the wave is.

00:48 Now, this is the proper time.

00:55 This will be the dilated time.

01:02 Now, do not interpret tr prime as the period of the wave received by the receiver.

01:11 Is not.

01:13 This is just the, what does he say, the admission between, say, crest 1 and crest 2 is, the time.

01:21 It's always saying, that's not the time of reception, the time difference between receiving crest 1, say, and crest 2.

01:32 It's not the same.

01:33 That's the whole purpose of this problem, and what you see down here in these diagrams to find that actual time.

01:39 But in terms of the emission, when would tr prime say the emission of crest 1 is and then crest 2? that's what we're going to get here, and that's just the time dilation formula.

01:52 Tr prime is equal to t .s.

01:55 Times gamma.

01:58 Now we come down here, we come down here, and t equals zero, the first crest is given off.

02:07 Moving with speed c.

02:11 Now, the distance between the source and the receiver at that point is going to be d suburb.

02:19 And that's important in a couple minutes.

02:22 Now, the receiver says that the next crest, crest 2, will be emitted at tr prime time.

02:32 Now, your question makes mention of this t not prime because he wanted you to think.

02:37 This is kind of like a period that the receiver is attributing to the source, but it's still not the period of the wave that he receives.

02:49 It's a period he's giving to the source, not to the wave.

02:56 That's what the t -0 prime represents, a period for the source of emissions, but not for what he experiences due to the fact the source is moving, as we're going to see in a minute.

03:12 So now, crest 1 is moving to speed c to him.

03:17 Crest 2 is now on its way.

03:21 So we need the time between the reception of crest 1 and the reception of crest 2.

03:32 So tr.

03:34 His clock will read tr prime, which is the time when he says crest 2 was admitted, plus the time it takes for light to travel from the position of the source at that point to him.

03:49 And that is dr plus v, tr prime over c.

03:59 That's the time it'll take for crest 2 to get there.

04:02 So this is the time on his clock when quest 2 will arrive.

04:08 Minus.

04:09 Now, remember, we want the time between receiving crest 1 and crest 2.

04:14 So we've got to subtract off the clock reading when crest 1 got there, which would have been dr over c.

04:21 X barr over c...

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