Question

Two train whistles, A and B, each have a frequency of 393Hz. A is stationary and B is moving toward the right (away from A) at a speed of 35m/s. A listener is between the two whistles and is moving toward the right with a speed of 15m/s. No wind is blowing. (a) What is the frequency from A as heard by the listener? (b) What is the frequency from B as heard by the listener? (c) What is the beat frequency detected by the listener?

Short answer

a) 375Hz

b) 370.75Hz

c) 4 Hz

Step-by-step solution

Doppler’s effect

The Doppler’s effect is given by the formula, ${\mathit{f}}_{\mathbf{L}}\mathbf{=}\frac{\mathbf{v}\mathbf{+}{\mathbf{v}}_{\mathbf{L}}}{\mathbf{v}{\mathbf{+}}_{\mathbf{s}}}{\mathit{f}}_{\mathbf{s}}$, where

• ${\mathit{f}}_{\mathbf{L}}$is the frequency observed by the listener.
• vis the speed of sound.
• ${\mathit{v}}_{\mathbf{L}}$is the speed of the listener.
• ${\mathit{v}}_{\mathbf{s}}$is the speed of the source of sound.
• ${\mathit{v}}_{\mathbf{s}}$is the frequency of the source of sound.

Calculate frequency when the listener is moving away from the source

$$\begin{gathered} f_L=\frac{v+v_L}{v+v_S}{f}_S \\ {f}_L=\frac{344-15}{344+0}\times 392 \\ {f}_L=375Hz \end{gathered}$$

Calculate frequency when listener is moving towards the source and source is moving away from the listener

$$\begin{gathered} f_L=\frac{v+v_L}{v+v_S}{f}_S \\ {f}_L=\frac{344+15}{344+35}\times 392 \\ {f}_L=370.75Hz \end{gathered}$$

Calculate beat

$$\begin{gathered} beat=|f_1-f_2| \\ beat=|375-370.75| \\ beat=4Hz \end{gathered}$$