Doppler Effect

DEFINITION

Change of Wave Frequency depending on relative Velocity between a Source & a Receiver.

DEFINITION - CLASSIC

Change of Frequency between Source & Receiver for Mechanical Wave that are propagating through a Medium (considered @rest), following Galilean Transformation.
Usually applied to Sound. #TODO

Doppler Effect - Classic

$\tilde{f} = f \cdot \frac{v - v_{R}}{v - v_{S}}$

$\tilde{f}$ [#h-G $H z$ ]: Frequency of Wave as heard by Receiver
$f$ [#h-G $H z$ ]: Frequency of Wave produced by Source
$v$ [#h-G $\frac{m}{s}$ ]: Wave Velocity in Medium
$v_{R}$ [#h-G $\frac{m}{s}$ ]: Velocity of Receiver w/ respect to propagation Medium
$v_{S}$ [#h-G $\frac{m}{s}$ ]: Velocity of Wave-Source w/ respect to propagation Medium

Example 1: Sound Wave & Ambulance

I'm not moving ( $v_{R} = 0$ ) while an Ambulance w/ sirens is moving towards me ( $v_{S} > 0$ ) ⇒ $\tilde{f} = f \cdot \frac{v_{R}}{v - V_{S}} > f$ → I hear ↑pitched sirens
The ambulance is in front of me
I'm still not moving ( $v_{R} = 0$ ) while the Ambulance w/ sirens is moving away from me ( $v_{S} < 0$ ) ⇒ $\tilde{f} = f \cdot \frac{v_{R}}{v - V_{S}} < f$ → I hear ↓pitched sirens

DEFINITION - SPECIAL RELATIVITY

Evolution of Classi Approach that takes in consideration the fact that Light is a Wave BUT doesn't travel in a Medium.

Doppler Effect - Special Relativity

${\begin{cases} ω_{R} = ω_{S} \cdot γ \cdot (1 + \frac{v}{c} \cdot \cos (θ_{S})) \\ \tan (θ_{R}) = \frac{c \cdot \sin (θ_{S}) \cdot \sqrt{1 - \frac{v^{2}}{c^{2}}}}{c \cdot \cos (θ_{S}) + v} \end{cases}$

Example 1: Galactic Velocities

Star emits Light in $\forall$ directions, even WHEN moving.
IF Star is moving away FROM Earth → Light Angle = $θ_{S}$ = $π$ .
Apply Special Relativity Doppler Effect → $ω_{R}$ = $\dots$ = $ω_{S} \cdot \sqrt{\frac{1 - \frac{v}{c}}{1 + \frac{v}{c}}}$ < $ω_{S}$ ⇒ Proves Star in moving way FROM us.
That's a starting point to find Star Velocity relative to us.

A common notation is:

"Red Shift" = WHEN $ω_{R} < ω_{S}$ = Received Light is "closed to the red" than Source
"Blue Shift" = WHEN $ω_{R} > ω_{S}$ = Received Light is "closed to the blue" than Source

Example 1: Blue-Shift

#TODO

"RAZIO"

"RAZIO" - CLASSIC

Use Galilean Reference Frame Transformation.
0. Take Spherical Wave ⇒ Their Wave Front are equally-spaced points on 1D system

We have:
- 1 Source $S$ moving w/ Velocity $v_{S}$
- 1 Receiver $R$ moving w/ Velocity $v_{R}$
- @Time $t = t_{1}$ $S$ is @distance $L$ from $R$ .
The experiment goes as described:
1. @ $t = t_{1} = 0$ $S$ emits 1 Spherical Wave that starts travelling toward $R$
2. @ $t = t_{2}$ , $R$ has received the Wave that has travelled $L + v_{R} \cdot t_{2}$
3. @ $t = {\tilde{t}}_{1} > t_{2}$ , $S$ emits another Spherical Wave that starts travelling toward $R$
4. @ $t = {\tilde{t}}_{2} > {\tilde{t}}_{1}$ , $R$ has received the Wave that has travelled $(L - v_{S} \cdot t_{2}) + v_{R} \cdot \tilde{t_{2}}$
1° Wave Velocity (w/ respect to Medium that's @rest) $v$ = $\frac{L + v_{R} \cdot t_{2}}{t_{2}}$ → $t_{2} = \frac{L}{v - v_{R}}$
2° Wave Velocity (w/ respect to Medium that's @rest) $v$ = $\frac{(L - v_{S} \cdot t_{2}) + v_{R} \cdot \tilde{t_{2}}}{{\tilde{t}}_{2} - {\tilde{t}}_{1}}$ → ${\tilde{t}}_{2} = \frac{L}{v - v_{R}} + {\tilde{t}}_{1} \cdot \frac{v - v_{S}}{v - v_{R}}$
The Period between the Wave-emission is $\tilde{T}$ = ${\tilde{t}}_{2} - t_{2}$ = ${\tilde{t}}_{1} \cdot \frac{v - v_{S}}{v - v_{R}}$ → Frequency $\tilde{f}$ = $f \cdot \frac{v - v_{R}}{v - v_{S}}$

"RAZIO" - SPECIAL RELATIVITY

Use Lorentz Reference Frame Transformation
0. Take Spherical Wave ⇒ Their Wave Front are equally-spaced points on 1D system

We have:
- 1 IRF $R$
- 1 IRF $\tilde{R}$ w/ Boost on $x$ -axis, moving w/ Velocity $\vec{v}$
- They are in Standard Configuration
- Source $S$ is always in origin of $\tilde{R}$ → $S$ is moving
- Receiver $R$ is anywhere BUT @ rest
@ $t = t_{1}$ T $S$ emits light that starts travelling toward $R$
Remember Phase is invariant under Lorentz Transformation^[1] → Phase in $R$ = Phase in $S$ → $ω_{S} \cdot (\tilde{t} - \frac{\tilde{r}}{c}) = ω_{R} \cdot (t - \frac{r}{c})$
Use some geometry to express: ${\begin{cases} \vec{\tilde{r}} = \tilde{x} \cdot \cos (θ_{S}) + \tilde{y} \cdot \sin (θ_{S}) \\ \vec{r} = x \cdot \cos (θ_{R}) + y \cdot \sin (θ_{R}) \end{cases}$
Now we can apply Restricted Lorentz Transformation → ${\begin{cases} ω_{S} \cdot (\tilde{t} - \frac{\tilde{r}}{c}) = ω_{R} \cdot (t - \frac{r}{c}) \\ \vec{\tilde{r}} = \tilde{x} \cdot \cos (θ_{S}) + \tilde{y} \cdot \sin (θ_{S}) \\ \vec{r} = x \cdot \cos (θ_{R}) + y \cdot \sin (θ_{R}) \\ \tilde{t} = γ \cdot (t - \frac{v}{c^{2}} \cdot x) \\ \tilde{x} = γ \cdot (x - v \cdot t) \\ \tilde{y} = y \end{cases}$ → #ASk → $ω_{S} [γ \cdot (t - \frac{v}{c^{2}} \cdot x) - \frac{1}{c} \cdot [γ \cdot (x - v \cdot t) \cdot \cos (θ_{S}) + y \cdot \sin (θ_{S})]] = ω_{R} [t - \frac{1}{c} \cdot (x - \cos (θ_{R}) + y \cdot \sin (θ_{R})]$
For this equation to be true $\forall x, y, t$ , their coefficients on both members must be equal
- Components of $t$ : $ω_{S} \cdot (γ + γ \cdot \frac{v}{c} \cdot \cos (θ_{S})) = ω_{R}$ → $$\omega_R=\omega_S \cdot\gamma \cdot \left(1+ \frac{v}{c}\cdot\cos(\theta_S)\right)$$
- Components of $x$ : $- ω_{S} \cdot γ \cdot \frac{v}{c^{2}} - ω_{S} \cdot \frac{γ}{c} \cdot \cos (θ_{S}) = - \frac{ω_{R}}{c} \cdot \cos (θ_{R})$ → $$\cos(\theta_R)= \frac{\frac{v}{c}+\cos(\theta_S)}{1+ \frac{v}{c}\cdot\cos(\theta_S)}$$
- Components of $y$ : $- \frac{ω_{S}}{c} \cdot \sin (θ_{S}) = - \frac{ω_{R}}{c} \cdot \sin (θ_{R})$ → $$\sin(\theta_R)=\frac{\sqrt{1- \frac{v^2}{c^2}}}{1+ \frac{v}{c}\cdot\cos(\theta_S)}\cdot\sin(\theta_S)$$
Notice components of $x$ & $y$ show the angle change → $\frac{\sin (θ_{R})}{\sin (θ_{R})} = \tan (θ_{R})$ gives the same equation found in Boost: $$\tan(\theta_R)= \frac{c\cdot\sin(\theta_S)\cdot\sqrt{1- \frac{v^2}{c^2}}}{c\cdot\cos(\theta_S)+v}$$

QUESTIONS

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#h-t LINKED

https://www.youtube.com/watch?v=lz5yiVBtPvs

CAUSE Phase just mean seeing the peak of a wave, invariant for 2 observers ↩︎