The Doppler effect describes the change in frequency of a sound as perceived by an observer moving relative to the sound source. This phenomenon occurs when there is relative motion between the listener and the sound source, leading to a shift in the frequency of the sound waves. The frequency heard by the listener is denoted as \( f_L \) (frequency of the listener), while the frequency emitted by the sound source is \( f_S \) (frequency of the sound source). When the source and listener are stationary relative to each other, the frequency heard is equal to the frequency emitted, meaning \( f_L = f_S \).

When either the listener or the source is in motion, the frequency perceived changes. If the listener moves towards the sound source or the source moves towards the listener, the listener perceives a higher frequency than that emitted by the source. Conversely, if the listener moves away from the source or the source moves away from the listener, the perceived frequency decreases. This results in a higher pitch sound as the source approaches and a lower pitch sound as it recedes.

The mathematical representation of the Doppler effect can be expressed with the equation:

\( f_L = f_S \cdot \frac{v + v_L}{v - v_S} \)

In this equation, \( v \) represents the speed of sound in air, which is approximately 343 meters per second. The variable \( v_L \) is the velocity of the listener, and \( v_S \) is the velocity of the sound source. It is important to note that the signs of \( v_L \) and \( v_S \) depend on their direction relative to the listener. If the listener is moving towards the source, \( v_L \) is positive; if moving away, it is negative. Similarly, if the source is moving towards the listener, \( v_S \) is positive; if moving away, it is negative.

To illustrate this concept, consider a scenario where a car alarm at rest produces a sound frequency of 550 Hz. If a listener on a motorcycle travels directly towards the car, they perceive a frequency of 600 Hz. To find the listener's speed, we can rearrange the Doppler effect equation:

\( 600 = 550 \cdot \frac{343 + v_L}{343} \)

By solving for \( v_L \), we can determine the speed of the listener. After performing the calculations, we find that the listener must be traveling at approximately 31.2 meters per second towards the sound source.

Understanding the Doppler effect is crucial in various fields, including astronomy, radar technology, and medical imaging, as it helps in interpreting the behavior of waves in motion.

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