Equations for the Doppler Effect of Sound (DP IB Physics: HL): Revision Note

A police car siren emits a sound wave with a frequency of 450 Hz. The car is travelling away from an observer at a speed of 45 m s⁻¹.

The speed of sound is 340 m s⁻¹.

What frequency of sound does the observer hear?

A. 519 Hz               

B. 483 Hz               

C. 397 Hz               

D. 358 Hz

A bank robbery has occurred and the alarm is sounding at a frequency of 3 kHz. The robber jumps into a car which accelerates and reaches a constant speed.

As he drives away at a constant speed, he hears the frequency of the alarm decrease to 2.85 kHz.

Determine the speed at which the robber must be driving away from the bank.

Speed of sound = 340 m s⁻¹

Answer:

Step 1: List the known quantities

• Source frequency, $f$ = 3 kHz

• Observed frequency, $f^{\text{'}}$ = 2.85 kHz

• Speed of sound, $v$ = 340 m s⁻¹

Step 2: Write down the Doppler shift equation

• The observer is moving away from a stationary source of sound, so the equation to use is

$f^{\text{'}}=f\left(\frac{v-u_o}{v}\right)$

Step 3: Rearrange to find the desired quantity

$\frac{f^{\text{'}}}{f}=\left(\frac{v-u_o}{v}\right)\Rightarrow \frac{v{f}^{\text{'}}}{f}=v-u_o$

$\frac{v{f}^{\text{'}}}{f}+u_o=v\Rightarrow u_o=v-\frac{v{f}^{\text{'}}}{f}$

$u_o=v\left(1-\frac{f^{\text{'}}}{f}\right)$

Step 4: Substitute the values into the equation

$u_o=340\times \left(1-\frac{2.85}{3}\right)=17\mathrm{m}{\mathrm{s}}^{-1}$

• The robber must be driving away at a constant speed of 17 m s⁻¹ based on the change in frequency heard

Pay careful attention as to whether you need to + or – sign in the relevant equation! If it helps, label the 'observer' and 'source' on in your question on the exam paper.