A sound source producing 1.00-{\rm kHz} waves moves toward a stationary listener atone-half the speed of sound.
The frequency that the listener will hear is2000 Hz.
when the source is stationary and the listener moves towardthe source at one-half the speed of sound, the frequency that thelistener hear became 1500 Hz.
Explain on physical grounds why thetwo frequencies differ.
I am not sure what is meant by "Explain
on physical grounds", but it can be shown mathematically why the difference
exist.
The difference in the frequencies heard by the stationary listener in the two scenarios can be explained using the concept of the Doppler effect. The Doppler effect is the perceived change in frequency of a wave when there is relative motion between the source of the wave and the observer.
In the first scenario, where the sound source is moving towards the stationary listener, the frequency of the sound waves appears higher to the listener. This is because as the sound source moves towards the listener, it compresses the sound waves, effectively decreasing the wavelength of the waves.
The formula that describes the relationship between the observed frequency and the actual frequency of the sound source is:
observed frequency (f') = actual frequency (f) * (speed of sound + velocity of observer) / (speed of sound + velocity of source)
Here, since the source is moving towards the listener at one-half the speed of sound, the velocity of the source is -(1/2) * (speed of sound). The velocity of the observer is 0, as the listener is stationary. Substituting these values into the formula:
f' = f * (speed of sound + 0) / (speed of sound - (1/2) * (speed of sound))
Simplifying the expression:
f' = f * (2 * speed of sound) / (2 * speed of sound - speed of sound) = 2 * f
As a result, the frequency heard by the stationary listener is 2 times the frequency of the sound source, which is 2000 Hz.
In the second scenario, where the listener moves towards the stationary sound source, the frequency of the sound waves heard by the listener appears lower. This is because as the listener moves towards the sound source, it encounters the compressed sound waves sooner, effectively reducing the wavelength of the waves.
Using the same formula as before:
f' = f * (speed of sound + velocity of observer) / (speed of sound + velocity of source)
Here, the velocity of the observer is -(1/2) * (speed of sound) as the listener moves towards the source at one-half the speed of sound. The velocity of the source is 0, as it is stationary. Substituting these values into the formula:
f' = f * (speed of sound + (-(1/2) * (speed of sound))) / (speed of sound + 0)
Simplifying the expression:
f' = f * (2 * speed of sound) / (2 * speed of sound + speed of sound) = (2/3) * f
As a result, the frequency heard by the moving listener is (2/3) times the frequency of the sound source, which is 1500 Hz.
In conclusion, the difference in frequencies heard by the stationary listener in the two scenarios is due to the Doppler effect, which depends on the relative motion between the sound source and the listener. When the source moves towards the listener, the observed frequency appears higher, and when the listener moves towards the source, the observed frequency appears lower.
The two frequencies differ due to the Doppler Effect. The Doppler Effect is the change in frequency of a wave due to the relative motion between the source of the wave and the observer.
When the sound source is moving towards the listener, the sound waves are compressed in front of the source, resulting in a higher frequency. This is because the wave crests are closer together when the source is moving towards the listener. As a result, the listener hears a higher frequency than the actual frequency of the source.
On the other hand, when the listener is moving towards the stationary source, the sound waves are stretched out in front of the listener, resulting in a lower frequency. This is because the wave crests are farther apart when the listener is moving towards the source. As a result, the listener hears a lower frequency than the actual frequency of the source.
In the given scenario, the sound source producing 1.00 kHz moves towards the stationary listener at one-half the speed of sound. The listener hears a frequency of 2000 Hz, which is higher than the actual frequency of the source. This is because the source is moving towards the listener, causing the sound waves to be compressed, thus increasing the frequency.
When the source is stationary and the listener moves towards the source at one-half the speed of sound, the listener hears a frequency of 1500 Hz, which is lower than the actual frequency of the source. This is because the listener is now moving towards the stationary source, causing the sound waves to be stretched out, thus decreasing the frequency.
In summary, the frequencies differ due to the Doppler Effect, which is caused by the relative motion between the source and the listener.