12.3 The Doppler Effect

Doppler Effect

Fire engine doppler effect video

Car doppler effect video

Doppler Effect

The pitch (frequency) of the horn of a passing car changes from high to low. This is due to the Doppler effect, named after Christian Doppler (1803-1853) who first studied it scientifically (with regard to trains).

The Doppler effect occurs when the velocity of the sound source---or of the observer---is a significant fraction of the velocity of the waves themselves.

The Doppler Effect: Description

“When a sound source moves toward an observer, the frequency heard is higher than the frequency produced by the source.

When a sound source moves away from an observer, the frequency heard is lower than the frequency produced by the source.”

The Doppler Effect

𝑡 = 0: Sound source emitting 1st wave crest:

𝑡 = 𝑇0: Sound source emitting 2nd wave crest:

λ𝑛𝑒𝑤 = 𝑣𝑤𝑇0 − 𝑣𝑠𝑇0

𝑣𝑤𝑓𝑛𝑒𝑤

=𝑣𝑤− 𝑣𝑠𝑓0

𝑓𝑛𝑒𝑤 =𝑓0

1 − 𝑣𝑠 𝑣𝑤

wavelength decreasesfrequency increases

λ𝑛𝑒𝑤 = 𝑣𝑤𝑇0 + 𝑣𝑠𝑇0

𝑣𝑤𝑓𝑛𝑒𝑤

=𝑣𝑤+ 𝑣𝑠𝑓0

𝑓𝑛𝑒𝑤 =𝑓0

1 + 𝑣𝑠 𝑣𝑤

wavelength increasesfrequency decreases

𝑣𝑠

The Doppler Effect

So the frequency heard depends on the motion of the sound source relative to the observer.

Moving Observer

A stationary sound source with a moving observer will also cause a Doppler effect. Here the wavelength is the original but the apparent velocity is not. For an observer moving toward a source:

𝑓𝑛𝑒𝑤 =𝑣𝑛𝑒𝑤λ

=𝑣𝑠 + 𝑣𝑜𝑏𝑠

λ=

𝑣𝑠 + 𝑣𝑜𝑏𝑠 𝑓0𝑣𝑠

𝑓𝑛𝑒𝑤 = 1 + 𝑣𝑜𝑏𝑠 𝑣𝑠 𝑓0

Doppler Effect: General Form

If both the sound source and the observer are moving:

𝑓𝑜𝑏𝑠 =1 ± 𝑣𝑜𝑏𝑠 𝑣𝑤𝑎𝑣𝑒

1 ± 𝑣𝑠𝑜𝑢𝑟𝑐𝑒 𝑣𝑤𝑎𝑣𝑒𝑓0

What signs to use?

If the observer is moving toward the source, or the source toward the observer, the pitch will be higher: use +|𝑣𝑜𝑏𝑠| or −|𝑣𝑠𝑜𝑢𝑟𝑐𝑒| .

If the observer is moving away from the source, or the source away from the observer, the pitch will be lower: use −|𝑣𝑜𝑏𝑠| or +|𝑣𝑠𝑜𝑢𝑟𝑐𝑒| .

Example 1

An express train sounds its whistle as it approaches a station. If the whistle’s pitch is 655 Hz and the train’s speed is 21.2 m/s, what is the frequency heard by an observer at the station?

𝑓𝑜𝑏𝑠 =1 ± 𝑣𝑜𝑏𝑠 𝑣𝑤𝑎𝑣𝑒

1 ± 𝑣𝑠𝑜𝑢𝑟𝑐𝑒 𝑣𝑤𝑎𝑣𝑒𝑓0

𝑓𝑜𝑏𝑠 =1 + 0

1 − 21.2 m s−1 343 m s−1655 Hz

𝑓𝑜𝑏𝑠 = 698 Hz

Example 2

A street musician sounds the A string on his violin at 440.0 Hz. What frequency does a bicyclist hear as she moves away at 11.0 m/s?

𝑓𝑜𝑏𝑠 =1 ± 𝑣𝑜𝑏𝑠 𝑣𝑤𝑎𝑣𝑒

1 ± 𝑣𝑠𝑜𝑢𝑟𝑐𝑒 𝑣𝑤𝑎𝑣𝑒𝑓0

𝑓𝑜𝑏𝑠 =1 − 11.0 m s−1 343 m s−1

1 + 0440 Hz

𝑓𝑜𝑏𝑠 = 426 Hz

The Doppler Effect: Light

The Doppler effect also occurs for electromagnetic waves. The frequency of light waves coming from distant galaxies is slightly higher (“redder”) than that typical for stars, indicating that they are moving away from us at a significant fraction of the speed of light.

In fact, the velocity of galaxies and stars can be estimated from their amount of “red-shift” (or “blue-shift”, if moving toward us).

Doppler Effect: Applications

The Doppler effect is used with radio waves to estimate the speed of cars, baseballs and winds based on the frequency of the reflected waves.

The amplitude of the reflected radar wave tells the number of rain drops; the Doppler shift tells their horizontal velocity.

Shock Wave / Sonic Boom

For the leading edge of a moving sound source:

𝑓𝑜𝑏𝑠 =1

1 − 𝑣𝑠𝑜𝑢𝑟𝑐𝑒 𝑣𝑤𝑎𝑣𝑒𝑓0

What happens when 𝑣𝑠𝑜𝑢𝑟𝑐𝑒 ≥ 𝑣𝑤𝑎𝑣𝑒 = 343 m/s?

No sound is heard on approach, then a sonic boomoccurs: a sound of large amplitude due to the constructive interference of multiple wave crests.

Shock wave physics video

Sonic boom video

Supersonic Speeds

An object, such as an airplane, moving at supersonic speed---faster than the speed of sound---is given a Mach number:

Mach number =𝑣𝑜𝑏𝑗𝑒𝑐𝑡

𝑣𝑠𝑜𝑢𝑛𝑑

A conical shock wave is produced, which is heard by an observer as a sonic boom, which may contain enough energy to break windows and cause other damage.

Shock Waves

Shock waves are analogous to the bow waves produced by a boat going faster than the speed of the water waves it produces.

A Boat’s wakeA bit more complicated…though the wake does have an amplified leading crest.

Shock Wave Angle

The shock wave is a cone with its apex at the moving object. The angle of this cone is given by

sin 𝜃 =𝑣𝑤𝑎𝑣𝑒𝑣𝑜𝑏𝑗𝑒𝑐𝑡

A Boat’s wakeThe angle and amplitude of the wake are also related to the speed of the boat.

Ultrasonic and Infrasonic

Sound waves can have any frequency; the human ear can hear sounds between about 20 Hz and 20,000 Hz.

Sounds with frequencies greater than 20,000 Hz are called ultrasonic; sounds with frequencies less than 20 Hz are called infrasonic.

Elephants and whales communicate, in part, by infrasonic waves. Ultrasonic waves are used in medical applications.

Applications: Sonar, Ultrasound, and Medical Imaging

The reflection of sound is used in many applications to determine the distance and the spatial structure of objects, by measuring the time it takes a sound pulse to reflect back to the receiver. Changes in amplitude indicate changes in density.

Ultrasonic waves are often used, as the shorter wavelengths have less diffraction relative to objects and thus allow the sensing of smaller objects.

EcholocationBats, whales and dolphins use echolocation to find prey.

Sonar

Sonar (SOund NAvigation Ranging) is used to locate objects underwater, and the depth an features of the sea floor.

Similar techniques can be used to learn about the internal structure of the Earth, including the location of oil deposits.

Ultrasound and Medical Imaging

Ultrasound sonograms are used in medicine, such as to detect tumors or pockets of fluid, or to monitor the health of a fetus.