Two loudspeakers, 4.0 m apart and facing each other, play...

Two loudspeakers, $4.0 \mathrm{m}$ apart and facing each other, play identical sounds of the same frequency. You stand halfway between them, where there is a maximum of sound intensity. Moving from this point toward one of the speakers, you encounter a minimum of sound intensity when you have moved $0.25 \mathrm{m}$. a. What is the frequency of the sound? b. If the frequency is then increased while you remain $0.25 \mathrm{m}$ from the center, what is the first frequency for which that location will be a maximum of sound intensity?

Two loudspeakers, $4.0 \mathrm{m}$ apart and facing each other, play identical sounds of the same frequency. You stand halfway between them, where there is a maximum of sound intensity. Moving from this point toward one of the speakers, you encounter a minimum of sound intensity when you have moved $0.25 \mathrm{m}$.
a. What is the frequency of the sound?
b. If the frequency is then increased while you remain $0.25 \mathrm{m}$ from the center, what is the first frequency for which that location will be a maximum of sound intensity?

Key Concepts

Interference

Interference refers to the phenomenon that occurs when two or more waves overlap in space, resulting in a new wave pattern. In acoustics, when sounds from different sources meet, they can either amplify each other (constructive interference) or cancel each other out (destructive interference), leading to regions of varying sound intensity.

Path Difference

Path difference is the difference in the distances that two waves travel from their respective sources to a given observation point. This difference in distance translates into a phase shift between the waves, which is the primary factor in determining whether they will interfere constructively or destructively.

Constructive and Destructive Interference

Constructive interference happens when the phase difference between the waves is an integral multiple of 2?, leading to an increase in amplitude or intensity. Destructive interference occurs when the phase difference is an odd multiple of ?, resulting in a reduction or cancellation of the wave amplitude. These principles are crucial for predicting the locations of maxima and minima in wave interference patterns.

Wavelength and Frequency Relationship

The relationship between wavelength and frequency is given by the equation v = f?, where v is the speed of sound, f is the frequency, and ? is the wavelength. This inverse relationship is essential for calculating one parameter when the others are known, and it plays a key role in the analysis of interference and resonance phenomena in sound waves.

Standing Waves

Standing waves are formed when two waves of identical frequency and amplitude traveling in opposite directions interfere with each other. This results in a stationary pattern characterized by nodes, where the amplitude is consistently zero, and antinodes, where the amplitude reaches a maximum. Standing wave patterns are fundamental to understanding resonance and the distribution of sound intensity in interference setups.

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