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Sound waves are a type of longitudinal wave that travel through a medium, like air, by compressing and rarefying the medium. These waves are characterized by their frequency, wavelength, and speed.

The speed of sound in air is generally around 343 m/s at room temperature, but it was given as 329 m/s in this exercise due to specific conditions. The frequency of a sound wave refers to the number of complete vibrations or cycles per unit of time, usually measured in Hertz (Hz). Higher frequencies correspond to higher pitches, while lower frequencies correspond to lower pitches.

The wavelength is the distance between two identical points in consecutive cycles of a wave, such as crest to crest or compression to compression. Understanding these characteristics of sound waves is essential for examining behaviors like the Doppler Effect, where changes in frequency and wavelength are perceived due to relative movement between the source and the observer.

Calculating frequency is a crucial part of understanding sound waves, especially when dealing with the Doppler Effect. The Doppler Effect describes how the frequency of waves changes for an observer moving relative to the wave source.

In the scenario described, there’s a source A emitting sound at 1200 Hz and a reflecting surface B moving towards it. To find the apparent frequency (\(f'\)) at which B receives these waves, we use the Doppler Effect equation:

• \( f' = \frac{f(v + v_o)}{v - v_s} \)

where \(f\) is the emitted frequency, \(v\) is the speed of sound, \(v_o\) is the observer's speed (toward the source), and \(v_s\) is the source's speed.

Substituting in the values:

• \(f = 1200 \; \text{Hz}\),
• \(v = 329 \; \text{m/s}\),
• \(v_o = 65.8 \; \text{m/s}\),
• \(v_s = 29.9 \; \text{m/s}\)

we find \(f' = 1456.9 \; \text{Hz}\). This indicates a frequency increase due to the relative motion.

The wavelength of sound waves gives us information about how far the waves travel in a single cycle. When calculating wavelengths for waves interacting with moving objects, we can use the formula:

• \( \lambda = \frac{v}{f} \)

where \(\lambda\) represents wavelength, \(v\) is the speed of sound, and \(f\) is the wave frequency.

When the reflecting surface B receives the waves, it interprets them with a frequency of 1456.9 Hz, as derived from the Doppler formula. Thus, the wavelength \(\lambda'\) of these incident waves is:

• \( \lambda' = \frac{329}{1456.9} \approx 0.225 \; \text{m} \)

Once the waves reflect off surface B and travel back to source A, we use the apparent frequency \(f'' = 1600 \; \text{Hz}\) calculated for that specific motion.

Using the same relationship, the wavelength \(\lambda''\) when received again by A is:

• \( \lambda'' = \frac{329}{1600} \approx 0.2056 \; \text{m} \)

Understanding these wavelength changes helps us comprehend the interaction of sound waves with moving entities.