Question

Two trains are traveling toward each other at 30.5m/srelative to the ground. One train is blowing a whistle at 500Hz. (a) What frequency is heard on the other train in still air? (b) What frequency is heard on the other train if the wind is blowing at 30.5m/stoward the whistle and away from the listener? (c) What frequency is heard if the wind direction is reversed?

Short answer

1. The frequency heard on the other train in still air is 598Hz.
2. The frequency heard on the other train if wind is blowing at 30.5m/s toward the whistle and away from the listener is 608Hz.
3. The frequency heard if the wind direction is reversed is 589Hz.

Step-by-step solution

The given data

1. Speed of each train is V_t = 30.5m/s
2. Frequency of whistle is f = 500Hz
3. Speed of the sound is v = 343m/s.

Understanding the concept of Doppler’s Effect

We can use the concept of the Doppler Effect. We can use the equation of the Doppler Effect for motion of source and listener. When there is air moving between trains, we find their relative speed, and then we can find the frequency heard by the listener.

Formula:

The frequency received by the observer according to Doppler’s Effect,

$\mathit{f}\mathbf{\text{'}}\mathbf{=}\left(\frac{V\pm V_l}{V\pm V_s}\right)\mathit{f}$ …(i)

a) Calculation of the frequency heard on the other train in still air

Here boththesource and the listener are moving towards each other, so using equation (i), we can write the frequency as:

$\begin{array}{rcl}f\text{'}& =& 500Hz\times \left(\frac{343m/s+30.5m/s}{343m/s-30.5m/s}\right)\\ & =& 598Hz\end{array}$

Hence, the frequency heard at the other train is 598Hz.

b) Calculation of the frequency heard on the other train if wind is blowing at  toward the whistle and away from the listener

In this frame of reference, the air seems still because both listener and air are moving with the same speed, so their relative speed will be zero.

At the source side, the speed of the train and air is opposite, so their relative speed will be 2(30.5m/s) = 61m/s We can write the frequency using equation (i) as:

$\begin{array}{rcl}f\text{'}& =& 500Hz\times \left(\frac{343m/s+0}{343m/s-61m/s}\right)\\ & =& 608Hz\end{array}$

Hence, the required frequency when wind is blowing at 30.5m/s is 608Hz.

c) Calculation of the frequency heard if the wind direction is reversed

Here, the relative speed of the source and air will be zero while the relative speed of the listener and air will be 61m/s. Hence, the frequency can be given using equation (i) as:

$\begin{array}{rcl}f\text{'}& =& 500Hz\times \left(\frac{343m/s+61m/s}{343m/s-0}\right)\\ & =& 589Hz\end{array}$

Hence, the required frequency is 589Hz.