Sound waves travel horizontally from a source to a receiver....

Sound waves travel horizontally from a source to a receiver. Assume that the source has a speed $u$, that the receiver has a speed 0 (in the same direction) and that a wind of speed $w$ is blowing from the source toward the receiver. Show that, if the source emits sound of frequency $v_{0}$, and if the speed of sound in still air is $V$, the frequency recorded by the receiver is given by $$ \nu=\nu_{0} \frac{V-v+w}{V-u+w} $$ Note that if the velocities of source and receiver are equal, the existence of the wind makes no difference to the observed frequency of the received signal.

Sound waves travel horizontally from a source to a receiver. Assume that the source has a speed $u$, that the receiver has a speed 0 (in the same direction) and that a wind of speed $w$ is blowing from the source toward the receiver. Show that, if the source emits sound of frequency $v_{0}$, and if the speed of sound in still air is $V$, the frequency recorded by the receiver is given by
$$
\nu=\nu_{0} \frac{V-v+w}{V-u+w}
$$
Note that if the velocities of source and receiver are equal, the existence of the wind makes no difference to the observed frequency of the received signal.

Key Concepts

Reference Frames and Relative Motion

When analyzing the Doppler shift for sound, it is essential to consider the reference frame in which the speeds are measured. The speeds of the source and receiver are compared to the speed of sound in the medium rather than to each other directly. This use of relative motion in a defined frame helps in correctly determining the observed frequency by accounting for the additional factors such as wind or any other movement of the medium.

Doppler Effect

The Doppler effect is a change in observed wave frequency (or wavelength) when there is relative motion between the source of the waves and the observer. In acoustics, when either the source, the observer, or both are moving relative to the medium (such as air), the frequency perceived by the observer differs from the frequency emitted by the source. This concept explains why a passing vehicle’s siren sounds higher in pitch as it approaches and lower as it recedes.

Role of the Medium in Wave Propagation

In wave propagation, especially for sound in air, the medium through which the wave travels plays a crucial role. The speed of sound is measured relative to the medium, and any movement of the medium (like wind) will affect the effective speed at which the sound waves travel. This concept is critical because it means that the velocities of the source and the observer must be considered relative to the medium, not just to each other.

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