DOPPLER EFFECT WITH SOUND

Learning Outcomes
Contents for this page	Related topics
Introduction Moving source, stationary listener General case Blood flow rates measurements Additional questions	Doppler effect with light Light and colour 2-D and 3-D wavefronts Wave nature of matter	Data Glossary
After studying this section, you will be familiar with the Doppler effect with sound and be aware of some of it practical applications.
Helpful background knowledge
Introduction to vectors	Addition of vectors	Equations of motion

Please note: Some links will play sound files. They are labelled with "". You will need a multimedia computer to play them.

Introduction:

When a sound source moves relative to a listener, the listener hears hears sound of variable frequency, depending on the relative speed of the source with respect to the listener. For example, if you stand alond a road and an ambulance approaches sounding its siren , you will hear sound of increasing frequency as the ambulance approaches, and decreasing frequency as the ambulance recedes. This is known as the DOPPLER EFFECT after its discoverer, Christian Johann Doppler.

Stationary listener, moving source:

If the source is moving, the wave front is bunched up in the direction of motion and spread out in the opposite direction:

If f_S is the frequency of the sound emitted by the source, then the period, T of the wave will be given by T = 1/f_S.

During this time, the source will have moved a distance Tv_S = cv_S/f_S.

The wavelength λ is the RELATIVE DISPLACEMENT of the wave with respect to the source:

λ = c/f_S - v_S/f_S = (c - v_S)/f_S

Which shows why the wavelength in the direction of motion is shorter than for the stationary source, and longer in the opposite direction (v_S then being negative).

The stationary listener hears a frequency f_o such that f_o = c/λ, thus

If a sound source, such as the ambulance described above, passes by a stationary observer, the pitch (frequency) of the sound will increase according to f_o = f_s(c/c - v_s) as the sound source approaches the observer, and then decrease according to f_o = f_s(c/c + v_s) as the sound source recedes from the observer.

Reflection of sound emitted by a source moving towards a stationary object:

The stationary object has a speed v_o = 0, and will receive a frequency f = (c/c - v_s). The sound at that frequency is now reflected back towards the original source, which acts as a moving observer, while the stationary object from which the sound is relected acts as a stationary source. The received frequency is f_r = f(c + v/c) = f_s(c + v_s)/(c - v_s).

Reflection of sound emitted by a stationary source towards a receding object:

You can try and show that the received frequency after reflection is f_r = f_s(c - v_o)/(c + v_o). See below for a medical application in determining the rate of flow of blood in blood vessels.

General case:

We will use the following symbols:
f_s: frequency of the sound source
f_o: frequency of sound as measured by the observer
c: velocity of the sound wave
v_s: velocity of the sound source
v_o: velocity of the observer.

The general equation that governs f_o is:

where the upper signs (shown in red) are for cases where the source and observer APPROACH each other, and the lower signs (shown in blue) are for cases where the source and observer RECEDE from one another. The table below shows how f_o is affected in different situations.

Condition	General equation reduces to	Effect on frequency detected by observer
Observer and source are both at rest: v_s = v_s = 0	f_o = f_s	No change in detected frequency
Stationary observer (v_o = 0), source approaching observer	f_o = f_sc/(c - v_s)	f_o increases
Stationary observer (v_o = 0), source receding from observer	f_o = f_sc/(c + v_s)	f_o decreases
Stationary source (v_s = 0), observer moving towards source	f_o = f_s(c + v_o)/c	f_o increases
Stationary source (v_s = 0), observer receding from the source	f_o = f_s(c - v_o)/c	f_o decreases
Source approaches the observer at velocity v_s = c	f_o → ∞	Condition for shock wave
Source recedes from the observer at velocity v_s = c	f_o = f_s/2	Sound now has half the source frequency
Observer moves towards stationary source at a velocity v_o = c	f_o = 2f_s	Sound now has double the source frequency
Observer moves away from stationary source at a velocity v_o = v_s	f_o = 0	No sound is detected

Blood flow rates measurements:

The Doppler effect is used for determining the rate of flow of blood in blood vessels. In the diagram below, blood is flowing from left to right within an artery, at a velocity, v. A crystal ultrasound transmitter sends out a signal at an angle q to the flow, where it is reflected back by blood corpuscles, to be picked up by a crystal receiver.

For low values (< 5º) of θ, cosθ, the component of the sound velocity parallel to the blood flow, approximates to 1. We have here a case of the "listener" (the blood corpuscles) moving away from a stationary source, and thus the frequency, f_corpuscle that will be reflected by the corpuscles will be

For the reflected waves, reflection, the corpuscles are the source and the "listener" is the receiver. The frequency at the receiver will be

from which we get

The flow rate, v, is then readily obtained from a knowledge of the difference between the transmitted and received frequencies, and c, the velocity of sound wave in blood.