Two loudspeakers in a 20^∘ C room emit 686 Hz sound waves along the x -axis. What is the smalles

step 1 Hi everyone this is the problem based on concept of interference of sound. Here it is given two

Two loudspeakers in a 20^∘ C room emit 686 Hz sound waves...

Two loudspeakers in a $20^{\circ} \mathrm{C}$ room emit $686 \mathrm{Hz}$ sound waves along the $x$ -axis. What is the smallest distance between the speakers for which the interference of the sound waves for a point on the axis is destructive?

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Key Concepts

Phase Difference and Destructive Interference

Destructive interference occurs when the phase difference between two waves results in their amplitudes canceling out. This typically happens when the path difference between the waves corresponds to an odd multiple of half wavelengths, producing a phase difference of ? (or 180 degrees) between them.

Wavelength Determination

Wavelength is calculated using the relationship between the speed of a wave, its frequency, and wavelength (? = v/f). In acoustics, the speed of sound varies with environmental conditions such as temperature, and knowing the wavelength is essential for predicting interference patterns.

Wave Interference

Wave interference refers to the phenomenon where two or more waves superpose to form a resultant wave of greater, lower, or the same amplitude. In contexts involving sound, the interference can be constructive (amplitudes add) or destructive (amplitudes cancel out), depending on the phase relationship between the contributing waves.

Path Difference in Wave Propagation

The path difference is the difference in distances that waves from two sources travel to reach a common point. This difference governs the phase difference at that point and thus determines whether the interference will be constructive or destructive.

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