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What is Doppler shift? Obtain an expression for the apparent frequency of sound heard when the observer is in motion with respect a source at rest?
Description:
Doppler shift refers to the change in frequency or wavelength of a wave as perceived by an observer due to relative motion between the observer and the source of the wave. It is named after the Austrian physicist Christian Doppler, who first described this phenomenon in 1842.
When an observer is in motion relative to a stationary sound source, the apparent frequency of the sound heard by the observer will be different from the actual frequency emitted by the source. This change in frequency occurs because the motion of the observer affects the relative speed at which the sound waves reach the observer.
To derive an expression for the apparent frequency of sound heard by an observer in motion, we consider two scenarios: one where the observer is moving towards the source (approaching), and the other where the observer is moving away from the source (receding).
Let’s assume the actual frequency emitted by the source is denoted as f, the velocity of sound in the medium as v, the velocity of the observer as v_observer, and the velocity of the source as v_source.
Observer Approaching:
In this case, when the observer is moving towards the source, the apparent frequency (f’) heard by the observer is given by:
f’ = (v + v_observer) / (v + v_source) * f
Observer Receding:
When the observer is moving away from the source, the apparent frequency (f’) heard by the observer is given by:
f’ = (v – v_observer) / (v – v_source) * f
In both expressions, the actual frequency f is multiplied by a ratio of the relative velocities of the observer and the source to the velocity of sound v. The numerator represents the net speed of the wavefronts reaching the observer, while the denominator represents the net speed of the wavefronts as emitted by the source.
It’s important to note that these expressions assume that the speed of sound remains constant and that the relative velocities of the observer and the source are small compared to the speed of sound. Additionally, the expressions hold true for both sound waves and other types of waves, such as electromagnetic waves, as long as the medium through which the waves propagate remains unchanged.
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