1.4 Doppler effect (Page 2/3)

Siyavula textbooks: grade 12 Page 2 / 3

This change in frequency can be calculated by using:

f_{L} = \frac{v + v_{L}}{v + v_{S}} f_{S}

where $f_{L}$ is the frequency perceived by the listener,

$f_{S}$ is the frequency of the source,

$v$ is the velocity of the waves,

$v_{L}$ the velocity of the listener and

$v_{S}$ the velocity of the source.

Note: Velocity is a vector and has magnitude and direction. It is very important to get the signs of the velocities correct here:

Source moves towards listener	$v_{S}$ : negative
Source moves away from listener	$v_{S}$ : positive

Listener moves towards source	$v_{L}$ : positive
Listener moves away from source	$v_{L}$ : negative

Khan academy video on the doppler effect

The siren of an ambulance has a frequency of 700 Hz. You are standing on the pavement. If the ambulance drives past you at a speed of 20 m $\cdot$ s $^{- 1}$ , what frequency will you hear, when

the ambulance is approaching you
the ambulance is driving away from you

Take the speed of sound to be 340 m $\cdot$ s $^{- 1}$ .

Determine how to appoach the problem based on what is given
$f_{L} = \frac{v + v_{L}}{v + v_{S}} f_{S}$

$\begin{matrix} f_{s} & = & 700 Hz \\ v & = & 340 m \cdot s^{- 1} \\ v_{L} & = & 0 \\ v_{S} & = & - 20 m \cdot s^{- 1} for (a) and \\ v_{S} & = & + 20 m \cdot s^{- 1} for (b) \end{matrix}$
Determine $f_{L}$ when ambulance is appoaching
$\begin{matrix} f_{L} & = & \frac{340 + 0}{340 - 20} (700) \\ = & 743, 75 Hz \end{matrix}$
Determine $f_{L}$ when ambulance has passed
$\begin{matrix} f_{L} & = & \frac{340 + 0}{340 + 20} (700) \\ = & 661, 11 Hz \end{matrix}$

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What is the frequency heard by a person driving at 15 m $\cdot$ s $^{- 1}$ toward a factory whistle that is blowing at a frequency of 800 Hz. Assume that the speed of sound is 340 m $\cdot$ s $^{- 1}$ .

Determine how to approach the problem based on what is given
We can use

$f_{L} = \frac{v + v_{L}}{v + v_{S}} f_{S}$

with:

$\begin{matrix} v & = & 340 m \cdot s^{- 1} \\ v_{L} & = & + 15 m \cdot s^{- 1} \\ v_{S} & = & 0 m \cdot s^{- 1} \\ f_{S} & = & 800 Hz \\ f_{L} & = & ? \end{matrix}$

The listener is moving towards the source, so $v_{L}$ is positive.
Calculate the frequency
$\begin{matrix} f_{L} & = & \frac{v + v_{L}}{v + v_{S}} f_{S} \\ = & \frac{340, 6 m \cdot s^{- 1} + 15 m \cdot s^{- 1}}{340, 6 m \cdot s^{- 1} + 0 m \cdot s^{- 1}} (800 Hz) \\ = & 835 Hz \end{matrix}$
Write the final answer
The driver hears a frequency of 835 Hz.

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Interesting fact

Radar-based speed-traps use the Doppler Effect. The radar gun emits radio waves of a specific frequency. When the car is standing still, the waves reflected waves are the same frequency as the waves emitted by the radar gun. When the car is moving the Doppler frequency shift can be used to determine the speed of the car.

Ultrasound and the doppler effect

Ultrasonic waves (ultrasound) are sound waves with a frequency greater than 20 000 Hz (the upper limit of human hearing). These waves can be used in medicine to determine the direction of blood flow. The device, called a Doppler flow meter, sends out sound waves. The sound waves can travle through skin and tissue and will be reflected by moving objects in the body (like blood). The reflected waves return to the flow meter where its frequency (received frequency) is compared to the transmitted frequency. Because of the Doppler effect, blood that is moving towards the flow meter will change the sound to a higher frequency and blood that is moving away from the flow meter will cause a lower frequency.

Ultrasound can be used to determine whether blood is flowing in the right direction in the circulation system of unborn babies, or identify areas in the body where blood flow is restricted due to narrow veins. The use of ultrasound equipment in medicine is called sonography or ultrasonography.

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Source: OpenStax, Siyavula textbooks: grade 12 physical science. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11244/1.2

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