Question

A 2000Hzsiren and a civil defense official are both at rest with respect to the ground. What frequency does the official hear if the wind is blowing at 12m/s.(a) from source to official and (b) from official to source?

Short answer

1. Frequency heard by the official if the wind is blowing at 12m/s from the source to the official is the same as the original that is $2\times {10}^{3}Hz$.
2. Frequency heard by the official if the wind is blowing at 12m/s from the official to the source is the same as the original that is $2\times {10}^{3}Hz$.

Step-by-step solution

The given data

1. The frequency of the siren, f = 2000Hz.
2. The velocity of wind, v = 12m/s.

Understanding the concept of Doppler’s Effect

We can use the concept of the Doppler Effect. We can plug appropriate values in the equation of the Doppler Effect for the motion of the source and listener to get the answer.

Formula:

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

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

a) Calculation of frequency heard by official when wind is blowing from source to official

Speeds of the source and listener are zero. While calculating the frequency, we can consider their speed equal to the speed of air V_l = V_s = V_air . Here, the listener is approaching the source while the source is backing off from the listener.

Frequency heard by the official if the wind is blowing at 12m/s from the source to the official is given using equation (i) as:

$\begin{array}{rcl}f\text{'}& =& \left(\frac{V+V_{air}}{V+V_{air}}\right)f\\ & =& f\\ f\text{'}& =& 2\times {10}^{3}Hz\end{array}$

Since there is no relative motion between the source and listener, there is no change in frequency of the siren. It doesn’t matter that air is blowing from the source to the official. Hence, the frequency heard by the official if the wind is blowing at 12m/s from the source to the official is the same as original.

b) Calculation of the frequency heard by the official if the wind is blowing from the official to the source

Frequency heard by the official if the wind is blowing at 12m/s from the official to the source:

Here, air is blowing from the official to the source, so signs will change. Using equation (i), we get

$\begin{array}{rcl}f\text{'}& =& \left(\frac{V+V_{air}}{V+V_{air}}\right)f\\ & =& f\\ f\text{'}& =& 2\times {10}^{3}Hz\end{array}$

Here also, since there is no relative motion between the source and listener, there is no change in the frequency of the siren. It doesn’t matter that air is blowing from the official to the source. Hence, the frequency value is same as the original.