Wave Motion

Beats occur when two sound waves of slightly different frequencies interfere. Let the frequencies be and (with ). The combined wave is: The amplitude modulates with frequency , but the perceived beat frequency is the number of amplitude maxima per second, which is .
Proof: The time between consecutive beats is the period of the cosine envelope, but since each cycle of cosine gives two beats (max and min), the beat frequency is .
Answer: Beat frequency = .

Harmonics:
1. Harmonics are integral multiples of the fundamental frequency of a vibrating system (e.g., ).
2. They represent the natural modes of vibration in systems like strings or air columns.
Overtone:
1. Overtones are frequencies higher than the fundamental frequency, numbered sequentially (first overtone is , second is , etc.).
2. They may or may not be integral multiples, depending on the system (e.g., non-harmonic in some cases).
Answer: Harmonics are integral multiples of fundamental frequency; overtones are higher frequencies, numbered sequentially.

Wave equation: .
Standard form: .
Wave speed: \( v = \frac{\omega}{k} = \frac{4}{0.02} = \frac{4 \cdot 100}{2} = 200 \, \text{cm/s}.
Answer: 200 cm/s.