Doppler Shift for an Extended Axis Rotating Source: R-d Ratio Dependance

An conductible experimental model to predict the doppler shift experienced by an observer from an extended axis rotating source.

Introduction

For my lab group’s last lab in physics we are invited to design, test and analyze our own experiment utilizing one or more of the topics that we learned over the quarter. Truthfully, I didn’t put much effort into developing this lab. I was not that fond of the topic chosen; the Doppler affect. Although it is a fascinating phenomenon, at the time I didn’t see much of a way to derive ample insight into the topic given the allotment of time, resources and the inevitable occurrence of final exams.

As a quick overview, our experimental procedure was as follows:

1. Attach a frequency emitting device (like our phones) to a ceiling fan.

2. Record the period of motion of the fan’s circular motion. This time (T), along with the radius of the orbit (R) can be used to derive the velocity of the source.

3. Let the fan orbit while different known frequencies are emitted.

4. Place a recording device some distance (d) from the fan and record the emitted frequency.

5. Analyze the data using peak frequency spectrum visualizer software.

Besides, not already being that fond of the topic I became doubtful very quickly due to the method chosen to determine the velocity for the source. The method settled on to determine the velocity was the equation,

\(v_s=\frac{2{\pi}R}{T}=wR\)

For quick experimental purposes and to gain some general insight into the phenomenon this approach is fine. However, I found some trouble in it. This approach arguably doesn’t determine the correct velocity at the points in time for which the source velocity vectors along a straight line away and towards the observer. Only at those respective points in time, with their associated velocities, do I see it justifiable to utilize the Doppler shift equation we were using to test the phenomenon.

Under the suspicion that the data would be marginally off due to this simplification (along with other unaccounted for factors as well). I started by constructing my own model.

Construction

To start I want to clarify the form of the Doppler equation that my group and therefore I will use in order to determine the frequency shift given the different velocities I seek.

\(f_{\pm} = f_0\left(1 \pm \frac{v_s}{v_m}\right)\)

Above, the plus indicates the source moving towards the observer and minus indicates the source moving away from the observer.

The end goal is to obtain an expression that holds a more precise form for the velocity of the source at the points of it’s orbit for which we can generalize that the source is moving along a straight line away or towards the observer (in reality the path is circular). A preferable form of the formulation would be if it contained only parameters limited to by our experimental setup. This is preferential for two reasons, 1) I have already obtained some data with with these parameters and 2) further replication using similar parameters allows for easy assessment of the theory and formulation.

To begin I will determine the position vector of the source of a centered at the observer. This gives,

\(\vec{r}_{SO}(t) = R \cos(wt) \hat{i} + (R \sin(wt) + d) \hat{j} + h \hat{k}\)

The equation is of a similar form to the one that describes the motion of a particle in circular motion around a stationary axis. The only difference is the constant distance and height is being accounted for.

After obtaining the position vector it is easy to obtain the velocity and acceleration vectors, where the constant distance and height disappear.

\(\vec{v}_{SO}(t) = -wR \sin(wt) \hat{i} + wR \cos(wt) \hat{j}\)

\(\vec{a}_{SO}(t) = -w^2R \cos(wt) \hat{i} - w^2R \sin(wt) \hat{j}\)

Now that I have determined the set of parametric functions that describe the motion of the source with respect to the observer, two things must be determined; 1) the position of the source in it’s orbit for which the source’s velocity vector is directed along a straight line away from the observer and 2) the point in time that the source is at this position in it’s orbit.

The answer to 1) is easy to see if you extend a ray that passes through the origin and a left sided point on the source’s circular orbit where the ray is also tangent to the circular orbit. At this point the source’s velocity vector will be parallel to the ray and point towards the observer (this also depends on the direction of the rotation of the source).

I will find and utilize only the left tangent point in the orbit for this solution tactic. It is implied by symmetry and by taking the angular velocity to be constant that the velocities at both symmetrical points are of the same magnitude but opposing in direction. This is pertinent to the overarching goal of my solution and the observations made.

Source Rotating about and Axis Extended From the Observer

You may note that I have labeled my theorized points of maximum and minimum frequency shifting. This is for reference in the consideration section below. For now it may be set aside.

Devising the definitive point in space with respect to the experimental parameters takes a bit more thought along with some trigonometry.

After doing this it can be seen that the angle for which the source is in it’s orbit at this point in space is related to the radius of orbit (R) and the distance to the center of the orbit (d) by,

\(\bar{\theta} = \frac{\pi}{2} - \arccos\left(\frac{R}{d}\right)\)

Given that my current parametric forms are periodic in time I will insert the angle in with an adjustment to account for the transition to the time domain.

The point in time of straight line motion towards the observer is,

\(\bar{t}_1 = \frac{\bar{\theta}}{w} = \frac{1}{w} \left( \frac{3\pi}{2}-\arccos\left(\frac{R}{d}\right) \right), \\0< \\t \\ \\{\le} \\2\pi\)

and the point in time of straight line motion away from the observer is,

\(\bar{t}_2 = \frac{\bar{\theta}}{w} = \frac{1}{w} \left( \frac{3\pi}{2}+\arccos\left(\frac{R}{d}\right) \right), \\0< \\t \\ \\{\le} \\2\pi\)

Now that I have obtained the time at which the source’s velocity vector is directed along the ray it is easy to obtain the vector form of the velocity by substituting it in the formulated time.

As stated previously I will only evaluate the point in time of the source’s velocity vector pointing in a straight line towards the observer (respective to the the chosen direction of rotation). I will then utilize symmetry and constant angular velocity assumptions in order to infer the velocity of the source moving away.

\(\vec{v}_{SO}({\bar{t}}) = -wR \sin(w\bar{t}) \hat{i} + wR \cos(w\bar{t}) \hat{j}\)

which implies,

\(\vec{v}_{SO}(t) = -wR\sin\left(\frac{3\pi}{2} + \arccos\left(\frac{R}{d}\right)\right)\hat{i} + wR\cos\left(\frac{3\pi}{2} + \arccos\left(\frac{R}{d}\right)\right)\hat{j}\)

utilizing Ptolemy’s angle sum and difference identities along with other identity relations gives,

\(\vec{v}_{SO}(t) = -wR\left(\frac{R}{d}\right)\hat{i} - wR\left(\sqrt{d^2 - R^2}\right)\hat{j}\)

This is the vector form of the velocity. From here I can evaluate it’s magnitude. The magnitude of the source’s velocity is,

\(|\vec{v}_{SO}(t)| = \frac{wR\sqrt{R^2 + d^4 - R^2d^2}}{d}\)

Given this I can substitute this expressional form of the source’s velocity into the original Doppler shift equation to obtain,

\(f_{\pm} = f_0\left(1 \pm \frac{wR\sqrt{R^2 + d^4 - R^2d^2}}{v_md}\right)\)

Considerations

It is worth noting that this solution is not valid for instances where the radius of the orbit of the source (R) is larger than the distance from the center of the orbit to the observer (d). This is due to the fact that I utilized the inverse cosine function which is undefined in cases where the numerator is greater than the denominator. Other geometrical relations would have to obtained to retrieve a similar model.

Furthermore, when I ran the experiment myself I noticed that the data suggested that these points in time aren’t directly correlated to the points in time for which the observer would experience the maximum or minimum frequency shifting. Not to mention, the recorded minimum frequency shifting is still seen to be larger than even this model would predict.

My theory is that the actual points in space and therefore time are not accurately portrayed in a circular orbit. Initially, I had toyed around with the idea that the orbital velocity as seen by the observer would be more accurately modeled as a rotated elliptical orbit where the orbit’s eccentricity was a consequence of the relativistic principles associated with velocity. More specifically, I theorized that the orbit’s eccentricity was proportional to the ratio of the square of the radius of the orbit (R^2) over the distance (d) to the center of the orbit. Unfortunately, I am not that experienced with Galilean transformations, relativity, etc. and couldn’t quite come to a place where I was happy with the math.

As an aside to my original theory I marked the points on the orbit for which I believe are most closely related to the time in which the event of the maximum or minimum frequency would be emitted. These are the points that suggested a rotated elliptical orbit. These points are suspected due to idealized models for which the wave propagates out at some point along the orbit and increases by some measure of wavelength over time. These points are related to that effect and the points in the orbit for which the acceleration of the motion is zero. In most Doppler simulators similar propagation can be observed. For now I will contend with making the observations and holding off on the math. The formulation of this experimental model was still a lot of fun to make.

Interactive Tool for Experimentation

Doppler Shift Model

Above I have included a link to a model that predicts the maximum and minimum frequency shifts if you were to recreate the experiment yourself with a similar setup.

I tried to make it as user friendly as possible, so that all you would have to input was the period of the orbit of the source, the radius of the orbit of the source and the distance from the center of orbit to the observer. I have also included an environmental temperature parameter to account for the change in the speed of sound with respect to temperature. It is initially set at 74 degrees Fahrenheit, but it can be adjusted, however the parameter is set to Celsius.