91Ó°ÊÓ

The concept of frequency change is essential when discussing the Doppler Effect. Whether it's a train approaching or receding, the change in the sound's frequency is noticeably different when compared to a stationary observer. This difference in frequencies is a direct outcome of motion between the source and the observer.

• Approaching Source: The frequency increases because the sound waves are compressed.
• Receding Source: The frequency decreases as the sound waves are stretched out.

This change in frequency is described by the expressions for the detected frequencies:
- When moving toward the source, the frequency is given by: \[ f_{\text{app}}' = f \left( \frac{v + v_D}{v} \right) \]- When moving away from the source, the frequency is given by: \[ f_{\text{rec}}' = f \left( \frac{v - v_D}{v} \right) \]These formulas capture how the relative velocity influences the frequency observed. By understanding these changes, one can predict how sound behaves when there's movement involved.

The speed of the detector, often denoted as \(v_D\), is crucial in the Doppler effect calculations. This speed refers to how fast the listening device (or an observer) is moving relative to the sound source. When the detector moves:

• Toward the source: The detected frequency increases because the waves are encountered more frequently.
• Away from the source: The frequency decreases as the waves are encountered less frequently.

In the context of the given problem, the detector's speed forms part of the equation that solves for the change in frequency:\[ \frac{f_{\text{app}}' - f_{\text{rec}}'}{f} = 0.500 \]By substituting the Doppler effect equations, we eventually find that the ratio of the detector's speed to the speed of sound, \( \frac{v_D}{v} \), is 0.250. Meaning, the detector moves at a quarter of the speed of sound for the given frequency change proportion.

The speed of sound, denoted as \(v\), significantly affects how sound waves travel in a medium. Typically, in air at 20°C, sound travels at approximately 343 meters per second.This speed determines how quickly the sound waves reach the observer or detector. In Doppler effect scenarios, such as the one described:

• The formula for approaching frequencies depends heavily on this speed:
 \[ f_{\text{app}}' = f \left( \frac{v + v_D}{v} \right) \]
• When receding, the speed is just as important:
 \[ f_{\text{rec}}' = f \left( \frac{v - v_D}{v} \right) \]

Sound speed in a medium like air doesn't change rapidly with small temperature variations, making our calculations reliable for everyday scenarios. This constancy allows us to make precise determinations about other variables, such as the detector speed compared to sound, as shown by finding the ratio \( \frac{v_D}{v} = 0.250 \).
Hence, knowing the speed of sound enables practical applications like measuring speed using Doppler radar.