This notebook is not to be shared by any other person besides the author. No person outside the physics 223 course can have a copy of this notebook.
Please report any error or typos. This is a first draft, so typos are expected.
Students should ignore the python code in this notebook, but you should pay attention to the text, animations, and plots.
You most likely have experienced the doppler shift when an ambulance passes by you with the siren on. As the ambulance approaches you, you hear a constant high pitched siren. When the ambulance passes you and moves away from you, you hear a constant lower pitched siren.
The doppler shift is the alteration of the observed frequency due to either the source moving, or the observer moving, or both source and observer moving. The observed shift in frequency is dependent on the relative velocity of the source and observer.
What is quality of the wave that is associated with the pitch of a soud wave?
Juan stands at the 35 yard line and thows a football at 25 mph to Melissa who is running towards Juan at 10 mph. What is the velocity of the ball in Melissa's frame of reference?
Huy runs to the right at a constant 20 mph. Robert runs to the right at a constant 23 mph. What is the relative velocity between Huy and Robert?
Beth runs to the left at a constant 20 mph. Abdul runs to the right at a constant 23 mph. What is the relative velocity between Beth and Abdul?
import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
from matplotlib import animation
from IPython.display import HTML
from wave_animations import animate_wavefront, animate_wavefront_doppler
Supose we have a wave source at $x=0~m$. Suppose Maria stands at $x=10~m$. Maria will measure the frequency of the wave by measuring the time it takes for the wave to repeat itself. Maria can mark any point on the wave to observe, but a natural point on the wave for Maria to observe is the wave crest. The animaition below illustrates the situation. The top panel shows a wave traveling to the right. The location of two wave crests is marked with a blue and red vertical line. The source of the wave is at $x=0$, while Maria stands at $x=10~m$. The second panel (bottom) removes the wave and leaves only the two marked wave crests. (We remove the wave, because the wave is continous and it is beneficial to just examine one particular spot of the wave.) Maria is marked with a red diamond and the source is marked with a green diamond. Maria will start her stop watch when the wave crest marked in blue passes her. She will stop her stop watch when the wave crest marked in red passes her. The time she records will be the period. She will calculate the frequency as $f_o=1/T$, where the o subscript stands for observer.
When will Maria make her recordings
Suppose the wave crest marked in blue is emitted from the source at $t=0$. Maria will observe this wave creast at $d/v$ where $d=10~m$ and $v$ is the wave speed, this is the time it takes the signal to travel from the source to the observer. The wave crest marked in red will be emitted in one period after the first, so the wave crest in red is emitted at $t=T$. Maria will record the second wave crest at $T + d/v$.
In general the time that an observer measures a signal from a moving source is $t + d/v$, where $d$ is the distance from the source to the observer and $v$ is the speed of the wave. This time is called the retarded time. You will most likely come accross this concept in later courses.
A wave moves to the right with wave parameters: $A=1 ~ mm$, $\lambda=10~cm$, $T=0.5~s$. The wave is sourced at $x=0~m$. Yu stands at $x=2~m$ from the source. The first wave crest is emitted at $t=0$. At what time will Yu observe the first and second wave crests?
What is the wave number and angular frequency is problem 5?
anim=animate_wavefront()
HTML(anim.to_html5_video())
Note. Our everyday experience and intuition informs us that the observed frequency is larger (higher pitch) than the source. So, we should expect that result in our derivation (one way to check if we did it right).
Maria stands at a distance $d$ away from the source, in the animation $d=10~m$. The source is travleing toward Maria at a speed $v_s$. Maria measures the frequency of the observed wave by marking the time between observations of each wave crest. The experiment is set up so that when the source crosses $x=0$ the first wave crest is generated at time $t=0$.
Maria will then take the difference in the two times to compute the observed period, we will call this $T_o$
\begin{align} \begin{split} T_o & = t_{2,o} - t_{1,o} \\ & = T + \frac{d-v_s T}{v} - \frac{d}{v} \\ & = T\left( 1 - \frac{v_s}{v} \right) \\ & = T\left( \frac{ v - v_s}{v} \right) ~. \end{split} \end{align}Maria computes the observed frequency by taking the reciprocal of the observed period \begin{align} f_o = \frac{ v}{v-v_s} f ~. \end{align} Note that the observed frequency is greater than the source frequency $f$. Also, the book by knight denotes the above expression as $f_+$.
The situation is illustrated in the animation below. Again, the wave is removed and we only concentrate on the wave crests. The green square represents the moving source. The red diamond represents the stationary observer. The wave variables are the same as the animation above, i.e. same period, wave length, and amplitude. You should observe that the time between arrivals of the red and blue wave crests is shorter than the animation above.
anim=animate_wavefront_doppler()
HTML(anim.to_html5_video())
Note. Our everyday experience and intuition informs us that the observed frequency is smaller (lower pitch) than the source.
Maria stands at a distance $d$ away from the source. The source is travleing away from Maria at a speed $v_s$. Maria measures the frequency of the observed wave by marking the time between observations of each wave crest. The experiment is set up so that when the source crosses $x=0$ the first wave crest is generated at time $t=0$.
Maria will then take the difference in the two times to compute the observed period, we will call this $T_o$
\begin{align} \begin{split} T_o & = t_{2,o} - t_{1,o} \\ & = T + \frac{d+ v_s T}{v} - \frac{d}{v} \\ & = T\left( 1 + \frac{v_s}{v} \right) \\ & = T\left( \frac{ v + v_s}{v} \right) ~. \end{split} \end{align}Maria computes the observed frequency by taking the reciprocal of the observed period \begin{align} f_o = \frac{ v}{v+v_s} f ~. \end{align} Note that the observed frequency is less than the source frequency $f$. Also, the book by knight denotes the above expression as $f_-$.
In this experiment the source is stationary, but Maria is moving toward the source with velocity $v_o$. We don't need to do much work here to get the correct expression. The first thing we should note, is that the observed frequency will be larger than the source frequency, that is we expect Maria to hear a higher pitch.
So, how do we obtain our expression without doing much work? Well, we move to Maria's frame of referecne. In Maria's frame of reference, she is stationary, but the source is moving toward her at velocity $v_o$. But, there is one more catch. Maria doesn't observe the wave to be traveling at $v$ but at $v + v_o$! Let's consider another example, your buddy throws a ball toward you at 30 mph, while you run toward you buddy at 10 mph. In your frame of reference the ball is moving toward you at 40 mph. Hopefully, that cleard it up for you.
Ok, so all we have to do is substitute $v_s \to v_o$ and $v \to v + v_o$ in the expression for $f_+$ \begin{align} f_0 = \frac{ v + v_o}{v} f~, \end{align} this is our result for an observer moving toward a stationary source.
There are a few more examples we could work out. To work these examples it is useful to move to Maria's frame of reference, i.e. the perspecitve of Maria as if she was stationary. You will need to combine this with the expressions for $f_+$ and $f_-$ to derive the correct expression. Be sure you first determine if the observed frequency is going to be less than or greater than the source.
1) Suppose the source is stationary and Maria is moving away from the source at $v_o$. Show that the observed frequency is \begin{align} f_0 = \frac{ v - v_o}{v} f~. \end{align}
2) Suppose the source is moving toward Maria with velocity $v_s$ and Maria is moving toward the source at $v_o$. Show that the observed frequency is \begin{align} f_0 = \frac{ v + v_o}{v-v_s} f~. \end{align}
3) Suppose the source is moving toward Maria with velocity $v_s$ and Maria is moving away from the source at $v_o$. Show that the observed frequency is \begin{align} f_0 = \frac{ v - v_o}{v-v_s} f~. \end{align}
4) We still have not exhausted all possible scenarios. What are some other scenarios and what are the formulas for the obsereved frequency?