• Sound propagates through air as a longitudinal wave. In a longitudinal wave, air particles vibrate back and forth in the same direction as the wave’s motion, creating alternating regions of compression (high pressure) and rarefaction (low pressure).
• These periodic pressure variations travel through the air and cause small oscillations in objects they encounter, like the candle flame.
• The flame responds to the changing air pressure by moving rhythmically back and forth. It is not being pushed by a stream of air or heated; rather, it is reacting to oscillating pressure from the sound wave generated by the vibrating tuning fork.

\(\Rightarrow B\)

a.    Sound wave propagation:

• Reflection of sound occurs when sound waves bounce off hard surfaces, like walls or cliffs. This allows echoes to be heard in large halls or open spaces.
• Diffraction allows sound to bend around obstacles or spread out after passing through narrow openings. This is why we can still hear someone speaking around a corner, even without direct line of sight.

b.   Musical instruments sound production:

• Resonance occurs when the frequency of the sound wave matches the natural frequency of the air column in the instrument, causing amplification of the sound.
• Superposition of the incident and reflected waves inside the tube creates standing waves, with nodes and antinodes, reinforcing specific frequencies and allowing clear, sustained musical notes to be produced.

a.    \(f_{\text{beat}} = \abs{300-f_1} = 6\).

• The absolute difference between the two frequencies is equal to \(6\). 
•   \(f_1 = 300 \pm 6\ \ \Rightarrow \ \ f_1 = 306\ \text{or } 294\).

b.    Description of “beats”:

• Beats are caused by the superposition of two sound waves with slightly different frequencies.
• As the two waves interfere, they produce constructive interference (louder sound) when their compressions and rarefactions align, and destructive interference (softer sound) when they are out of phase.
• This interference results in a periodic fluctuation in volume, called a beat, with a frequency equal to the difference between the two original frequencies.
• This demonstrates the wave nature of sound, where sound behaves as a longitudinal wave capable of interference and superposition.

\(f_{\text{beat}} = \abs{f_2-f_1} = \abs{260-256} = 4\ \text{Hz}\)

\(\Rightarrow B\)

• Using \(\lambda = \dfrac{v}{f}\) to find the wavelength of each wave.
•    830 Hz wave:  \(\lambda = \dfrac{340}{830} = 0.41\ \text{m}\)
•    4000 Hz wave:  \(\lambda = \dfrac{340}{4000} = 0.085\ \text{m}\)
• The width of the gap \(=\dfrac{40}{100} = 0.4\ \text{m}\).
• The 850 Hz wave has a wavelength equal to the gap size, so it undergoes significant diffraction, spreading out widely after passing through the gap.
• The 4000 Hz wave has a much shorter wavelength than the gap, so it undergoes minimal diffraction and continues in a narrow beam-like path.

• The speed of sound in air is 340 ms\(^{-1}\).
•    \(d = v \times t = 340 \times 4 = 1360\ \text{m}\)
• As the sound travels 1360 m to and from the tunnel way, the distance from the student to the tunnel wall will be half of that distance, i.e. 680 m.

\(\Rightarrow B\)

• Beats occur when two sound waves of slightly different frequencies interfere with each other, creating a pulsing effect in loudness. The frequency of the beats is equal to the absolute difference between the two frequencies:
•    \(f_{\text{beat}} = \abs{f_1-f_2}\)
• As the two notes get closer in pitch, the difference between their frequencies decreases. This causes the beat frequency to get lower, so the beats sound slower.
• When the two frequencies become exactly the same, the beat frequency becomes zero. This means no more interference pattern occurs, and the beats disappear completely. At this point, the two instruments are perfectly in tune, and you hear a single, steady tone.

\(\Rightarrow A\)