# The Big Bang Theory of the Doppler Effect

Season 01, Episode 06: “The Middle-Earth Paradigm”

Sheldon Cooper dresses as the Doppler Effect on the Season 1, Episode 6 “The Middle-Earth Paradigm” of the Big Bang Theory

Yes. It’s the apparent change in the frequency of a wave caused by relative motion between the source of the wave and the observer.

-Sheldon Cooper

In the “Middle-Earth Paradigm” episode, Sheldon Cooper dresses as the “Doppler Effect” for Penny’s Halloween party. The Doppler Effect (or Doppler Shift) describes the change in pitch or frequency that results as a source of sound moves relative to an observer; moving relative can mean either the source is moving while the observer is stationary or vice versa. It is commonly heard when a siren approaches and recedes from an observer. As the siren approaches, the pitch sounds higher and lowers as it moves away. This effect was first proposed by Austrian physicist Christian Doppler in 1842 to explain the color of binary stars.

In 1845, Christophorus Henricus Diedericus (C. H. D.) Buys-Ballot, a Dutch chemist and meteorologist conducted the famous experiment to prove this effect. He assembled a group of horn players on an open cart attached to a locomotive. Ballot then instructed the engineer to rush past him as fast as he could while the musicians played and held a constant note. As the train approached and receded, Ballot noted that the pitch changed as he stood and listened on the stationary platform.

# Physics of the Doppler Effect

A stationary sound source has sound waves radiating outward and can be viewed as concentric circles.

As a stationary sound source produces sound waves, its wave-fronts propagate away from the source at a constant speed, the speed of sound. This can be seen as concentric circles moving away from the center. All observers will hear the same frequency, the frequency of the source of the sound.

When either the source or the observer moves relative to each other, the frequency of the sound that the source emits does not change but rather the observer hears a change in pitch. We can think of the following way. If a pitcher throws balls to someone across a field at a constant rate of one ball a second, the person will catch those balls at the same rate (one ball a second). Now if the pitcher runs towards the catcher, the catcher will catch the balls faster than one ball per second. This happens because as the catcher moves forward, he closes in the distance between himself and the catcher. When the pitcher tosses the next ball it has to travel a shorter distance and thus travels a shorter time. The opposite is true if the pitcher was to move away from the catcher.

If instead of the pitcher moving towards the catcher, the pitcher stayed stationary and the catcher ran forward. As the catcher runs forward, he closes in the distance between him and the pitcher so the time it takes from the ball to leave the pitcher’s hand to the catcher’s mitt is decreased. In this case, it also means that the catcher will catch the balls at a faster rate than the pitcher tosses them.

## Sub Sonic Speeds

The source radiates sound waves outward. As it moves, the center of each new wavefront is slightly displaced to the right and the wavefronts bunch up on the right side (front) and spread out further apart on the left side (behind) the source.

We can apply the same idea of the pitcher and catcher to a moving source of sound and an observer. As the source moves, it emits sounds waves which spread out radially around the source. As it moves forward, the wave-fronts in front of the source bunch up and the observer hears an increase in pitch. Behind the source, the wave-fronts spread apart and the observer standing behind hears a decrease in pitch.

### The Doppler Equation

When the speeds of source and the receiver relative to the medium (air) are lower than the velocity of sound in the medium, i.e. moves at sub-sonic speeds, we can define a relationship between the observed frequency, $$f$$, and the frequency emitted by the source, $$f_0$$.
$f = f_{0}\left(\frac{v + v_{o}}{v + v_{s}}\right)$
where $$v$$ is the speed of sound, $$v_{o}$$ is the velocity of the observer (this is positive if the observer is moving towards the source of sound) and $$v_{s}$$ is the velocity of the source (this is positive if the source is moving away from the observer).

#### Source Moving, Observer Stationary

We can now use the above equation to determine how the pitch changes as the source of sound moves towards the observer. i.e. $$v_{o} = 0$$.
$f = f_{0}\left(\frac{v}{v – v_{s}}\right)$
$$v_{s}$$ is negative because it is moving towards the observer and $$v – v_{s} < v$$. This makes $$v/(v - v_{s})$$ larger than 1 which means the pitch increases.

#### Source Stationary, Observer Moving

Now if the source of sound is still and the observer moves towards the sound, we get:
$f = f_{0}\left( \frac{v + v_{o}}{v} \right)$
$$v_{o}$$ is positive as it moves towards the source. The numerator is larger than the denominator which means that $$v + v_{o}/v$$ is greater than 1. The pitch increases.

## Speed of Sound

As the source of sound moves at the speed of sound the wave fronts in front of the source all bunch up at the same point.

As the source of sound moves at the speed of sound, the wave fronts in front become bunched up at the same point. The observer in front won’t hear anything until the source arrives. When the source arrives, the pressure front will be very intense and won’t be heard as a change in pitch but as a large “thump”.

The observer behind will hear a lower pitch as the source passes by.
$f = f_{0}\left( \frac{v – 0}{v + v} \right) = 0.5 f_{0}$

Early jet pilots flying at the speed of sound (Mach 1) reported a noticeable “wall” or “barrier” had to be penetrated before achieving supersonic speeds. This “wall” is due to the intense pressure front, and flying within this pressure front produced a very turbulent and bouncy ride. Chuck Yeager was the first person to break the sound barrier when he flew faster than the speed of sound in the Bell X-1 rocket-powered aircraft on October 14, 1947.

Bell X-1 rocket plane of the United States Air Force (NASA photography)

As the science of super-sonic flight became better understood, engineers made a number changes to aircraft design that led the the disappearance of the “sound barrier”. Aircraft wings were swept back and engine performance increased. By the 1950s combat aircraft could routinely break the sound barrier.

## Super-Sonic

As the source moves faster than the speed of sound, i.e. faster than the sound waves it creates, it leads its own advancing wavefront. The sound source will pass by the stationary observer before the observer hears the sound.

As the sound source breaks and moves past the “sound barrier”, the source now moves faster than the sound waves it creates and leads the advancing wavefront. The source will pass the observer before the observer hears the sound it creates. As the source moves forward, it creates a Mach cone. The intense preseure front on the Mach cone creates a shock wave known as a “sonic boom”.

### Twice the Speed of Sound

Something interesting happens when the source moves towards the observer at twice the speed of sound — the tone becomes time reversed. If music was being played, the observer will hear the piece with the correct tone but played backwards. This was first predicted by Lord Rayleigh in 1896 .

We can see this by using the Doppler Equation.
$f = f_{0}\left(\frac{v}{v-2v}\right)$
This reduces to
$f=-f_{0}$
which is negative because the sound is time reversed or is heard backwards.

# Applications

U.S. Army soldier uses a radar speed gun to catch speeding violators at Tallil Air Base, Iraq.

The Doppler Effect is used in radar guns to measure the speed of motorists. A radar beam is fired at a moving target as it approaches or recedes from the radar source. The moving target then reflects the Doppler-shifted radar wave back to the detector and the frequency shift measured and the motorist’s speed calculated.

We can combine both cases of the Doppler equation to give us the relationship between the reflected frequency ($$f_{r}$$) and the source frequency ($$f$$):
$f_{r} = f \left(\frac{c+v}{c-v}\right)$
where $$c$$ is the speed of light and $$v$$ is the speed of the moving vehicle. The difference between the reflected frequency and the source frequency is too small to be measured accurately so the radar gun uses a special trick that is familiar to musicians – interference beats.

To tune a piano, the pitch can be adjusted by changing the tension on the strings. By using a tuning instrument (such as a tuning fork) which can produce a sustained tone over time, a beat frequency can be heard when placed next to the vibrating piano wire. The beat frequency is an interference between two sounds with slightly different frequencies and can be herd as a periodic change in volume over time. This frequency tells us how far off the piano strings are compared to the reference (tuning fork).

To detect this change in a radar gun does something similar. The returning wave is “mixed” with the transmitted signal to create a beat note. This beat signal or “heterodyne” is then measured and the speed of the vehicle calculated. The change in frequency or the difference between $$f_{r}$$ and $$f$$ or $$\Delta f$$ is
$f_{r} – f = f\frac{2v}{c-v}$
as the difference between the speed of light, $$c$$, and the speed of the vehicle, $$v$$, is small, we can approximate this to
$\Delta f \approx f\frac{2v}{c}$
By measuring this frequency shift or beat frequency, the radar gun can calculate and display a vehicle’s speed.

# “I am the Doppler Effect”

The Doppler Effect is an important principle in physics and is used in astronomy to measure the speeds at which galaxies and stars are approaching or receding from us. It is also used in plasma physics to estimate the temperature of plasmas. Plasmas are one of the four fundamental states of matter (the others being solid, liquid, and gas) and is made up of very hot, ionized gases. Their composition can be determined by the spectral lines they emit. As each particle jostles about, the light emitted by each particle is Doppler shifted and is seen as a broadened spectral line. This line shape is called a Doppler profile and the width of the line is proportional to the square root of the temperature of plasma gas. By measuring the width, scientists can infer the gas’ temperature.

We can now understand Sheldon’s fascination with the Doppler Effect as he aptly explains and demonstrates its effects. As an emergency vehicle approaches an observer, its siren will start out with a higher pitch and slide down as as it passes and moves away from the observer. This can be heard as the (confusing) sound he demonstrates to Penny’s confused guests.