Write a definition for each of the following terms.

Doppler effect

Redshift

Ultrasound

The Doppler effect occurs when a source of waves and/or an observer move relative to each other,
resulting in the observer measuring a different frequency of the waves than the frequency at which the
source is emitting.

Redshift is the shift in the position of spectral lines to longer wavelengths due to the relative
motion of a source away from the observer.

Ultrasound or ultrasonic waves are sound waves with a frequency greater than \(\text{20 000}\)
\(\text{Hz}\) (or \(\text{20}\) \(\text{kHz}\)).
The hooter of an approaching taxi has a frequency of \(\text{500}\) \(\text{Hz}\). If the taxi is
travelling at \(\text{30}\) \(\text{m·s$^{1}$}\) and the speed of sound is \(\text{340}\)
\(\text{m·s$^{1}$}\), calculate the frequency of sound that you hear when

the taxi is approaching you.

the taxi passed you and is driving away.
Draw a sketch of the observed frequency as a function of time.

The observer is not moving, so \(v_{L}=0\), and the source is approaching. Therefore:
\begin{align*}
{f}_{L}&=\left(\frac{v}{v{v}_{S}}\right){f}_{S} \\
&=\left(\frac{340}{34030}\right)\left(500\right) \\
&= \text{548,4}\text{ Hz}
\end{align*}

Now the taxi is moving away from the observer, so:
\begin{align*}
{f}_{L}&=\left(\frac{v}{v+{v}_{S}}\right){f}_{S} \\
&=\left(\frac{340}{340+30}\right)\left(500\right) \\
&= \text{459,5}\text{ Hz}
\end{align*}

\(\text{548,4}\) \(\text{Hz}\)

\(\text{459,5}\) \(\text{Hz}\)
A truck approaches you at an unknown speed. The sound of the truck's engine has a frequency of
\(\text{210}\) \(\text{Hz}\), however you hear a frequency of \(\text{220}\) \(\text{Hz}\). The speed of
sound is \(\text{340}\) \(\text{m·s$^{1}$}\).

Calculate the speed of the truck.

How will the sound change as the truck passes you? Explain this phenomenon in terms of the wavelength
and frequency of the sound.

The observer is stationary, so \(v_{L}=0\). Therefore:
\begin{align*}
\frac{{f}_{L}}{f_{S}}&=\frac{v}{v{v}_{S}} \\
\frac{v{v}_{S}}{v} &= \frac{f_{S}}{f_{L}} \\
1  \frac{v_{S}}{v} &= \frac{f_{S}}{f_{L}} \\
\frac{v_{S}}{v} &= 1  \frac{f_{S}}{f_{L}} \\
v_{S} &= 340 \left(1\frac{210}{220} \right) \\
&= \text{15,5}\text{ m·s$^{1}$}
\end{align*}

As the truck goes by, the frequency goes from higher to lower and the wavelength of the sound waves
goes from shorter to longer.

\(\text{15,5}\) \(\text{m·s$^{1}$}\)

As the truck goes by, the frequency goes from higher to lower and the wavelength of the sound waves
goes from shorter to longer.
[Extension question] A police car is driving towards a fleeing suspect at
\(\frac{v}{35} \text{m}·{\text{s}}^{1}\), where v is the speed of sound. The frequency of the
police car's siren is \(\text{400}\) \(\text{Hz}\). The suspect is running away at \(\frac{v}{68}\). What
frequency does the suspect hear?
\({f}_{L}=\left(\frac{v\pm{v}_{L}}{v\pm{v}_{S}}\right){f}_{S}\)
The source is moving towards the listener, so we need to put:
\(v_{S}\)
and the listener is moving away from the source, so we need to put:
\(v_{L}\)
So we use the formula as :
\({f}_{L}=\left(\frac{v{v}_{L}}{v{v}_{S}}\right){f}_{S}\)
We are given that \(v_{L}= \frac{v}{68} \text{m}·{\text{s}}^{1}\) and
\(v_{S}=\frac{v}{35} \text{m}·{\text{s}}^{1}\). Substituting into the formula:
\begin{align*}
{f}_{L}&=\left(\frac{v\frac{v}{68} }{v\frac{v}{35}}\right){f}_{S} \\
&= \left(\frac{\frac{68vv}{68}}{\frac{35vv}{35}} \right){f}_{S} \\
&= \left(\frac{\left(\frac{67}{68}\right)v}{\left(\frac{34}{35}\right)v} \right){f}_{S} \\
&= \left(\frac{\left(\frac{67}{68}\right)}{\left(\frac{34}{35}\right)} \right)400 \\
{f}_{L}&= \text{405,7}\text{ Hz}
\end{align*}
\(\text{405,7}\) \(\text{Hz}\)
Explain how the Doppler effect is used to determine the direction of flow of blood in veins.
An instrument called a Doppler flow meter can be used to transmit ultrasonic waves into a person's body.
The sound waves will be reflected by tissue, bone, blood etc., and measured by the flow meter. If blood flow
is being measured in an artery for example, the moving blood cells will reflect the transmitted wave and due
to the movement of the cells, the reflected sound waves will be Doppler shifted to higher frequency if the
blood is moving towards the flow meter and to lower frequency if the blood is moving away from the flow
meter.
An person in a car travelling at \(\text{22,2}\) \(\text{m·s$^{1}$}\) approaches a source emitting
sound waves with a frequency of \(\text{410}\) \(\text{Hz}\). What is the frequency of the sound waves
observed by the person? Assme the speed of sound in air is \(\text{340}\) \(\text{m·s$^{1}$}\).
The Doppler formula is:
\({f}_{L}=\left(\frac{v\pm{v}_{L}}{v\pm{v}_{S}}\right){f}_{S}\)
The source is stationary so \(v_{S}=0\). Rearranging the formula:
\begin{align*}
{f}_{L}&=\left(\frac{v+{v}_{L}}{v}\right){f}_{S} \\
{f}_{L}& = \left(\frac{340+\text{22,2}}{340}\right)\left(410\right)\\
& = \text{436,8} \text{Hz}
\end{align*}
\(\text{436,8}\) \(\text{Hz}\)