Chapter 12, Problem 2TC

A fire engine sounds its siren as it approaches you. Rank the Doppler effect you hear from highest to lowest frequency.

$\left(\text{a}\right)$ The speed of the fire engine is $20\text{km}/\text{h}$ and you are at rest.

$\left(\text{b}\right)$ The speed of the fire engine is $30\text{km}/\text{h}$ and you are at rest.

$\left(\text{c}\right)$ The speed of the fire engine is $30\text{km}/\text{h}$ while you drive away from it at $20\text{km}/\text{h}$.

$\left(\text{d}\right)$ The speed of the fire engine is $30\text{km}/\text{h}$ while you drive toward it at $20\text{km}/\text{h}$.

To determine

The ranking of the Doppler effect for greatest to least frequency.

Answer to Problem 2TC

The Doppler effect of the case (d) is greatest, followed by Doppler effect of case (b) and case (a) and the Doppler effect of the case (c) is least.

Explanation of Solution

The variation in the frequency of the sound due to the source's motion, receiver, or both is called the Doppler effect. The variation in the frequency depends on the relative separation of the source and receiver. If the source and receiver move away from each other, then the Doppler effect decrease and if the source and receiver move toward each other, then the Doppler effect increase.

The speed of the fire engine in case (a) is $20\text{km}/\text{h}$ and source is at rest, so the fire engine approaches you with speed $20\text{km}/\text{h}$.

The speed of the fire engine in case (b) is $30\text{km}/\text{h}$ and source is at rest, so the fire engine approaches you with speed $30\text{km}/\text{h}$.

The speed of the fire engine in case (c) is $30\text{km}/\text{h}$ and you approaches away from the fire engine with speed $20\text{km}/\text{h}$, so the fire engine approaches you with relative speed of $10\text{km}/\text{h}$.

The speed of the fire engine in case (d) is $30\text{km}/\text{h}$ and you approaches toward the fire engine with speed $20\text{km}/\text{h}$, so the fire engine approaches you with relative speed of $50\text{km}/\text{h}$.

Doppler effect varies with separation between the source and receiver. The separation between source and receiver decrease with increase in speed of the source toward receiver. The faster is the separation reduces higher is the Doppler effect.

The fire engine in the case (d) is moving toward you with highest speed, so the separation between you and siren of fire engine decreases with highest speed. The Doppler effect is greatest when the engine moves toward you with speed $30\text{km}/\text{h}$ and you moves toward engine with $20\text{km}/\text{h}$.

The Doppler effect is least for the case (c) because the separation between the siren and you decrease with least speed of $10\text{km}/\text{h}$.

The Doppler effect of the case (b) follows the greatest Doppler effect as the fire engine moves toward you with speed .$30\text{km}/\text{h}$.

The Doppler effect of the case (a) follows the case (b) as the separation between you and fire engine decreases with speed $20\text{km}/\text{h}$.

Conclusion:

Therefore, the Doppler effect of the case (d) is greatest, followed by Doppler effect of case (b) and case (a) and the Doppler effect of the case (c) is least.