91Ó°ÊÓ

Beat frequency is an interesting phenomenon that occurs when two sound waves of slightly different frequencies are heard simultaneously. It results in a pulsing variation in loudness called beats. These beats can be heard as periodic rises and falls in volume, much like a beating drum.

To understand beat frequency better, consider two sound sources emitting sounds of frequencies close to each other, like in the train whistle example in the exercise. The beat frequency, denoted as \( f_{\text{beat}} \), is calculated using the formula:

• \( f_{\text{beat}} = |f'_A - f'_B| \)

This formula shows that the beat frequency is simply the absolute difference between the two perceived frequencies. In our case, it reflects the variation in sound as detected by a listener between two frequencies altered by the Doppler Effect.
The listener hears these beats as a result of constructive and destructive interference between the two sound waves, which leads to these fluctuations in intensity.

Sound wave frequency is an essential concept in understanding waves and their behaviors. Frequency, denoted by \( f \), is the number of complete wave cycles that pass a point in one second. It is measured in Hertz (Hz).

Higher frequency sound waves correspond to higher-pitched sounds, while lower frequencies are associated with deeper sounds. In the exercise, both train whistles are originally producing sound waves at a frequency of 392 Hz. However, due to the relative movements of the sources and the listener, these frequencies change.
Using the Doppler Effect, we calculate the changes in frequency when either the source or listener, or both, are moving. This effect accounts for the listener hearing one frequency slightly higher from whistle A and another slightly lower from whistle B.

The speed of sound, commonly denoted as \( v \), is the rate at which sound waves travel through a medium. For air, which is a common medium, the speed of sound is approximately 343 m/s at room temperature.

This speed can vary based on factors like temperature, humidity, and air pressure. In our exercise, we use the speed of sound in our Doppler Effect calculations to determine how the motion of the source and listener affects the sound wave frequency as heard by the listener.
Sound travels faster in warmer conditions as the air molecules move more rapidly, aiding in the sound wave's propagation. This means that when solving problems like those in the exercise, it is crucial to use the correct speed of sound based on environmental conditions.

The velocity of the source and listener are key variables in understanding the Doppler Effect. **Velocity** refers to the speed and direction of motion of an object.

• In the exercise, the source whistle B has a velocity of 35 m/s moving away from the stationary whistle A.
• The listener, positioned between these two sources, moves at 15 m/s in the same direction as B.

Understanding their velocities is crucial when using the Doppler Effect formulas. The listener moving towards the source or away from it affects the frequency of the sound perceived.
For our calculations:

• When the listener moves towards the sound source (like whistle A), the frequency heard increases.
• When the listener moves in the same direction as a retreating source (like whistle B), the frequency heard decreases.

Hence, by incorporating their velocities into the Doppler Effect formula, we can account for changes in perceived sound frequency.