I Find period of circular-orbiting source based on max observed freq

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The discussion focuses on deriving an equation to determine the period of a circular-orbiting source based on the maximum observed frequency and parameters like the observer's distance from the center, the radius of the orbit, and the speed of the wave in the medium. The Doppler effect is analyzed, particularly how the observed frequency shifts as the observer moves closer to the source's orbit, leading to a sinusoidal frequency variation. The derived equations approach the solution but struggle when the observer's distance approaches the orbit radius, complicating the relationship between the observed frequency and the source's period. The challenge lies in isolating the angular frequency in the equations, which may require numerical methods for accurate solutions, especially when the observer's distance is comparable to the orbit radius. Ultimately, the discussion highlights the complexity of the problem and the necessity for numerical approaches in certain scenarios.
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Deriving an equation to calculate the time period of a circular-orbiting source from an observer at a distance away from the circle based on max frequency observed
This question refers to Doppler Effects observed in circular motion (at non relativistic speeds, so ##v\ll c##, ignoring transversal Doppler shifts).

Suppose there is a source emitting a frequency, ##f_s##. An observer at the center will experience no shift in observed frequency (##f_r##). As they move towards the circle of motion, ##f_r## starts to shift up and down "sinusoidally". I use quotes, as it starts go skew as the observer approaches the source's orbit, because ##f_r## gradually increases as the source approaches, and then instantly drop to a minimum after it passes the observer.

My goal here is to derive an equation that, given the parameters of this environment:

$$\begin{array}{l}{{X:Distance\;of\;observer\;from\;center}}\\ {{R:Radius\;of\;orbit\;of\;source}}\\ {{C:Speed\;of\;wave\;in\;medium}}\\ {{f_{\mathrm{max}}\;:Maximum\;observed\;frequency}}\\ {{f_{\mathrm{s}}\;:Source\;frequency}}\end{array}$$

I've tried my hand at deriving something, but it is *almost* correct. It just breaks down when ##X## approaches ##R## and the sinusoidal-look starts to break down and the actual Time period, ##T_s##, starts to deviate from the actual value. Here's my take:

The source moves in a circle of radius ##R##, with its position as a function of time:
$$\mathbf{r}_{s}(t)=(R\cos(\omega t),R\sin(\omega t)),$$
where ##ω=2π/T_s## is the angular frequency of the source, and ##T_s=1/f_s## is its time period.
And for the distance,
$$d(t)=\sqrt{(R\cos(\omega t)-X)^{2}+(R\sin(\omega t))^{2}}$$
$$d(t)=\sqrt{R^{2}-2R X\cos(\omega t)+X^{2}}$$
From the Doppler Effect equation,
$$f_{r}(t)=f_{s}\left(1-\frac{v_{\mathrm{rel}}(t)}{c}\right)$$
Substituting ##v_{rel}##,
$$f_{r}(t)=f_{s}\left(1+\frac{R\omega X\sin(\omega t)}{c\sqrt{R^{2}-2R X\cos(\omega t)+X^{2}}}\right)$$
Since the maximum ##f_r## occurs when ##sin(\omega t)=1##,
$$f_{\mathrm{max}}=f_{s}\left(1+{\frac{R\omega X}{c\sqrt{R^{2}+X^{2}-2R X\cos(\pi)}}}\right)$$
$$f_{\mathrm{max}}=f_{s}\left(1+\frac{R\omega X}{c(X+R)}\right)$$
Since ##\omega = 2\pi/T_s##,
$$f_{\mathrm{max}}=f_{s}\left(1+{\frac{2\pi R X}{c T_{s}(X+R)}}\right)$$
$$T_{s}=\frac{2\pi R X}{c(X+R)\left(\frac{f_{\mathrm{max}}}{f_{s}}-1\right)}$$
This equation is almost right. I do not know where to proceed from here. My go at it might also be completely incorrect. Additionally, here is the equation I used to check the "actual" values of T should be:
$$d\left(t\right)=\sqrt{\left(R^{2}-2RX\cos\left(\omega t\right)+X^{2}\right)}$$
$${\frac{f_{r}}{f_{s}}}={\frac{1-{\frac{v}{c}}\cos\theta}{\sqrt{1-{\frac{v^{2}}{c^{2}}}}}}$$
$$r_{rel}=-v\cos{\theta}=\frac{d}{dt}\left(d\left(t\right)\right)$$
$$\boxed{f_{r}=f_{s}\left(1-\frac{r_{rel}}{c}\right)}$$
With this form of the equation, ##\omega## cannot be separated to only one side of the equation, right? I don't suppose the above equation can be used to rearrange it to get what I want?
 
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We analyze the Doppler shift by tracking individual wave crests emitted by a source orbiting with angular frequency ##\Omega## at a radius ##R##. The source's position at emission time ##t## is ##\bigl(R\cos(\Omega t),\,R\sin(\Omega t)\bigr)##. The emitted wave has an angular frequency ##\omega_s=2\pi f_s##. We index the crests by integers ##n\in\mathbb{Z}##, with the nth crest emitted at

$$t_n=\frac{2\pi n}{\omega_s}.$$

The phase of the emitted wave at ##t=t_n## is thus ##2\pi n##. An observer is fixed at ##(X,0)##. The distance traveled by the nth crest is

$$d_n = \sqrt{(X-R\cos(\Omega t_n))^2 + (R\sin(\Omega t_n))^2}.$$

This crest arrives at the observer at time

$$\tau_n = t_n + \frac{d_n}{c}.$$

Therefore, consecutive crests arrive at times

$$\tau_n = \frac{2\pi n}{\omega_s} + \frac{1}{c}\sqrt{\,X^2-2XR\cos\left(\Omega\,\tfrac{2\pi n}{\omega_s}\right)+R^2}.$$

For large ##n##, we approximate the observed frequency by

$$f_r(n) \approx \frac{1}{\,\tau_{n+1}-\tau_n\,}.$$

Using the expression for ##\tau_n##, we find

$$\tau_{n+1}-\tau_n = \left(\frac{2\pi}{\omega_s}\right) + \frac{1}{c}\left(\sqrt{\,X^2-2XR\cos\left(\Omega\,\tfrac{2\pi(n+1)}{\omega_s}\right)+R^2} - \sqrt{\,X^2-2XR\cos\left(\Omega\,\tfrac{2\pi n}{\omega_s}\right)+R^2}\right),$$

which leads to

$$f_r(n) = \frac{1}{\displaystyle \frac{2\pi}{\omega_s} + \frac{1}{c}\left(d_{n+1}-d_n\right)}.$$

To determine the maximum of ##f_r(n)##, we treat ##n## as continuous and set the derivative with respect to ##n## equal to zero:

$$0 = \frac{d}{dn}\bigl[f_r(n)\bigr] = \frac{d}{dn}\left[\,\bigl(\tau_{n+1}-\tau_n\bigr)^{-1}\right].$$

Applying the chain rule gives

$$\frac{d}{dn}\bigl(\tau_{n+1}-\tau_n\bigr) = \frac{d}{dn}\left(\frac{2\pi}{\omega_s} + \frac{1}{c}\bigl(d_{n+1}-d_n\bigr)\right),$$

where

$$d_{n+1} = \sqrt{\,X^2-2XR\cos\left(\Omega\,\tfrac{2\pi(n+1)}{\omega_s}\right)+R^2}$$

and

$$d_n = \sqrt{\,X^2-2XR\cos\left(\Omega\,\tfrac{2\pi n}{\omega_s}\right)+R^2}.$$

Differentiating ##d_{n+1}-d_n## with respect to ##n## involves applying the chain rule to each square root. This gives a transcendental equation in ##\tfrac{2\pi n}{\omega_s}##. Let ##n_{\ast}## represent a real solution satisfying

$$\frac{d}{dn}\bigl(\tau_{n+1}-\tau_n\bigr)\Bigl|_{n=n_{\ast}}=0.$$

Thus, ##f_r(n)## reaches an extremum at ##n=n_{\ast}##. The peak observed frequency is then

$$f_{\max} = f_r(n_{\ast}) = \frac{1}{\,\tau_{n_{\ast}+1}-\tau_{n_{\ast}}\,}.$$

Note that the orbital phase ##\Omega\,\tfrac{2\pi n_{\ast}}{\omega_s}## is entangled within a trigonometric function inside a square root, preventing us from isolating ##\Omega## (or ##T_s=2\pi/\Omega##) in a closed form using elementary functions. We must solve for ##\Omega## numerically by iteratively adjusting ##\Omega## until the peak of ##\frac{1}{\,\tau_{n+1}-\tau_n\,}## matches the measured ##f_{\max}##. In the limit ##X\gg R##, ##d_{n+1}-d_n## becomes a small perturbation, simplifying the derivative test and leading to a far-field Doppler approximation where a closed-form relation between ##T_s## and ##\tfrac{f_{\max}}{f_s}-1## exists. However, when ##X## is on the order of ##R##, the transcendental nature of the exact discrete-crest condition prevents a purely algebraic solution, which means that numerical methods are needed.
 
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