Question

Two vehicles, each moving with speed \(u\) on the same horizontal straight road, are approaching each other. Wind blows along the road with velocity \(w\). One of these vehicles blows a whistle of frequency \(f_{1}\). An observer in the other vehicle hears the frequency of the whistle to be \(f_{2}\). The speed of sound in still air is \(V\). The correct statement \((\mathrm{s})\) is (are) (A) If the wind blows from the observer to the source, \(f_{2}>f_{1}\). (B) If the wind blows from the source to the observer, \(f_{2}>f_{1}\). (C) If the wind blows from observer to the source, \(f_{2}

Short answer

The correct statements are (B) If the wind blows from the source to the observer, \(f_{2}>f_{1}\) and (C) If the wind blows from observer to the source, \(f_{2}<f_{1}\).

Step-by-step solution

Understand the Doppler Effect

The Doppler effect is the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. Analyze how the movement of the source and observer, along with the wind's direction, affects the perceived frequency.

Determine Apparent Frequency with Wind Blowing from Observer to Source

When the wind blows from the observer to the source, it carries the sound waves away from the observer, causing the apparent frequency to decrease. Use the Doppler effect formula considering the relative speed of sound in moving air medium. Since the source and observer are moving toward each other, you must add their speeds to the wind's speed when it's assisting the sound waves from the source to the observer.

Determine Apparent Frequency with Wind Blowing from Source to Observer

When the wind blows from the source to the observer, it assists the sound waves in reaching the observer faster, causing the apparent frequency to increase. The speed at which the sound waves travel towards the observer is the sum of the wind speed and the speed of sound.

Apply the Correct Doppler Effect Formulas

Use the formula for the Doppler effect to calculate the frequency heard by the observer, which is: \(f_2 = f_1 \frac{V + v_{observer}}{V + v_{source}}\), where \(v_{observer}\) and \(v_{source}\) are adjusted according to the direction of the wind.

Compare the Frequencies for Each Case

By comparing \(f_2\) and \(f_1\) in each case, you can determine if \(f_2\) is greater than, less than, or equal to \(f_1\). When the observer is approaching the source and the wind aids the motion, \(f_2 > f_1\). Conversely, when the wind opposes the motion, \(f_2 < f_1\).

Identify the Correct Statements

Using the conclusions drawn in previous steps, identify which of the given statements are correct. (B) is correct when the wind assists the sound waves from the source to the observer, making \(f_2\) higher. (C) is correct when the wind carries the waves away from the observer, making \(f_2\) lower.

Key concepts

Sound Wave Frequency

Sound waves are a type of mechanical wave that move through various media, such as air, water, and solids. Sound wave frequency, defined as the number of wave cycles that pass a point per second, is measured in Hertz (Hz). This characteristic is crucial in determining the pitch of the sound we hear.

For instance, a high-frequency sound wave corresponds to a high pitch, such as the sound of a whistle. In the given problem, a whistle with frequency f_{1} is blown. This frequency is the sound's original frequency before any effects of motion are taken into account.

Relative Motion in Wave Propagation

The movement of sound waves is not always through stationary air. Often, either the source of the sound, the observer, or the medium itself -- in this case, air -- is in motion. This influences the speed at which the sound waves reach the observer.

When considering sound wave propagation in relation to relative motion, it is essential to consider the speed and direction of the sound source and observer. In our example, both vehicles (and thus the source and observer of the sound) are moving towards each other with speed u, while the wind is introducing additional motion either aiding or opposing the wave's journey.

Apparent Frequency Change

The Doppler Effect causes an apparent frequency change, which means the frequency observed (f_{2}) may differ from the original frequency emitted by the source (f_{1}). When there is relative motion towards the observer, the apparent frequency increases; conversely, when moving away, the frequency decreases.

Considering our scenario, if the wind blows from the observer to the source, it carries the sound waves in the opposite direction, stretching them out and thus decreasing the frequency that the observer in the second vehicle would perceive. This phenomenon illustrates the Doppler Effect and how relative motion impacts what we hear.

Motion of Sound Source and Observer

The motion of both the sound source and the observer plays a pivotal role in determining the frequency that the observer perceives. If both are moving towards each other, the frequency increases. If they move away from each other, the frequency decreases.

In the given exercise, as the two vehicles approach one another, the observer moves towards the sound waves, compressing them and causing a higher frequency to be heard. However, the wind's effect cannot be ignored—it changes the speed at which the sound travels, thereby affecting the final frequency heard by the observer.