The Doppler effect (quantitative)

The first version of the qualitative post contained a paragraph at the end in which I did some real math (I have since removed that paragraph, as it appears in a different form in this post). My mother loved the bit about the tennis and thought she had really grasped the general idea; alas, when confronted with a paragraph containing algebraic variables, she felt somewhat bewildered and lost because I hadn't given it enough of an introduction. I was reminded that she hasn't really done any advanced math in several years. Mom, I apologize, and I'm going to take some time now to talk a bit about the philosophy of mathematics in physics because I do plan on using math in this blog whenever it's applicable (which will be, presumably, often, as this is a physics blog).

First, I've titled these two posts as "qualitative" and "quantitative." This is a standard and very useful division in physics. When discussing a subject "qualitatively," one is really trying to grasp the basic idea or gain a sense of intuition. Then, after gaining that first level of understanding, one begins to approach a problem "quantitatively" by actually working the math. I would not be surprised if many of the subjects I broach in this blog will be handled in a similar fashion.

Next, I want to say that one purpose of introducing abstract variables in a mathematical treatment of a problem is to generalize the solution. For example, in the last post, I tried to explain the Doppler effect using a very specific situation. If my mom is hitting tennis balls every two seconds, then decides to run towards the ball machine, she will have to hit tennis balls at a faster rate. We've solved the Doppler effect for that one case. But that situation doesn't apply if we wished to talk about listening to a police siren on a street corner or the rotational speeds of galaxies in an argument about dark matter. That's why the abstraction of math is so useful in physics - we can deal with the problem in such a way that we can use the same language and solution in each instance.

Finally, I'm going to be using variables to represent various parameters in the problem. Specifically, I'm going to use the following:
f₀: the frequency at which the balls are emitted by the ball machine
v: the speed or velocity of the tennis balls
v_r: the speed or velocity at which my mom runs
f: the frequency at which she hits each tennis ball.










The choice of these variables is completely arbitrary; I could have used anything to represent the quantities of interest. In general, however, we try to use variables that are easily associated with what they represent, like "v" for velocities and "f" for frequencies. The subscripts are then used to delineate different quantities of the same type. f₀ is given a "0" because in a sense, it is the original frequency or the initial frequency. My mom's velocity is given the subscript "r" because she is the receiver, in contrast to the source (if the ball machine were moving, I would have described its velocity as "v_s").

If there are any questions or comments about this introductory stuff, please do comment below.

Onto the problem. Given the variables defined above, I want to define a couple more terms. If the machine spits out balls at a frequency f₀, then the time between each ball, T₀, is 1/f₀ (in the example, I said that the time between each ball was 2 seconds, so the frequency is then 1/2 per second).

In the time elapsed before the next ball is fired, the previous ball has traveled a distance equal to its speed times the time elapsed, or v*1/f₀. This is the distance between each ball.

To determine the time between each ball that my mom observes, we will need the absolute distance between balls (v*1/f₀) and the speeds of both my mom and the balls. At the instant when my mom hits a ball, she is v*1/f₀ away from the next ball. However, she is still running towards that ball, and the ball is still moving towards her. Therefore, that distance will be covered by the combination of her running towards the ball and the ball moving towards her, i.e. with a velocity equal to the sum of her velocity and the ball's velocity, v+v_r. Therefore the time it takes for her to see the next ball is the total distance divided by the total velocity,


T = v * 1/f₀ * 1/(v+v_r).


To calculate the frequency at which she observes each ball, we invert that time, so


f = (v+v_r)/v*f₀.


In the example, f₀ = 1 per 2 seconds, v = 12 m/s, and v_r = 12 m/s, so f = (12+12)/12 * 1/2 = 1 per second, which is exactly what we saw. However, this equation is now general for any similar situation. The equation can be generalized still further to take into account a moving source as well, in which case


f = (v+v_r)/(v+v_s)*f₀.


Now we can talk about listening to sirens on a sidewalk or galactic rotation curves and use the same equation to represent all 3 situations.

The Doppler Effect (qualitative)

In my first post, I talked about how the Doppler effect is a shift in the observed frequency of a wave caused by the relative motion of a source and an observer. In this, my first detailed post, I will try to explain how that actually works. As my mom plays tennis, and this blog is ostensibly aimed at her, I'm going to use a rather tortured tennis analogy.

Suppose my mother is using a ball machine to practice her ground strokes. The ball machine spits out a tennis ball every 2 seconds and each ball moves at the same speed. To use some real numbers, a tennis court is about 24 meters long from baseline to baseline, and let's assume that the balls take 2 seconds to go from the machine to my mother standing at the other baseline (or the ball takes 1 second to get to the net, and then another second to get to my mom). Therefore, as my mom hits a ball, the ball machine is in the process of shooting the next one. As my mom continues to practice, she hits a stroke every 2 seconds.


In this video, my mom hits ground strokes every 2 seconds, and the next ball is released as she hits the previous stroke (this is some pretty great animation, huh? Unfortunately, the timing is not quite right, so it's not 2 seconds in real time).

Now, suppose she wanted to work on her volleying and therefore decides to run to the net. She hits a stroke (a nice approach shot, presumably), and then starts sprinting to the net as the ball machine spits out the next ball. assuming she runs really fast (like Usain Bolt), she can make it to the net in 1 second. When she gets there, the next ball from the ball machine will already be there to meet her (remember, the ball takes 1 second to get to the net, so while my mom was running in, the next ball was also heading towards the net). Instead of hitting a ball every 2 seconds, she'll hit this one after only 1 second, because she was moving relative to the ball machine. Then, once she's stationary at the net, she will once more see a ball every 2 seconds. In a sense, this is the Doppler effect. When my mom, the observer, was moving relative to the ball machine, the source, the frequency with which she hit balls changed (from once every 2 seconds to once every second). Then, when she was no longer moving, the frequency returned to its usual value.


In this one, my mom hits a ground stroke and runs to the net, at which point she is confronted with the next ball after only 1 second, instead of the usual 2. This is caused by the relative difference between her speed and the ball machine, or the Doppler effect (again, the timing isn't quite right, but you get the idea).