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What is the wavelength of the sound?

An observer begins at the center point between the speakers and slowly moves toward one of the speakers. The second very quiet spot encountered (completely destructive interference) is 1.5 m from the center. What is the wavelength of the sound?

Answer:

The wavelength of the sound in this situation is 1.5 m.

In this scenario, we can assume that the sound waves emanating from two speakers are located some distance apart and that the observer is positioned at the midpoint between the two speakers. As the observer moves towards one of the speakers, they will encounter regions of constructive and destructive interference as the sound waves from each speaker combine.

The distance between the two speakers is a critical parameter in this situation, as it determines the wavelengths of the sound waves that are being produced. The wavelength is the distance between successive peaks or troughs in the wave.

If the observer encounters a completely destructive interference (also called a node) at a distance of 1.5 m from the center, this means that the distance between the speakers is equal to an odd multiple of half-wavelengths at that point. Specifically, the distance between the speakers is:

d = (2n + 1)(λ/2)

where n is an integer representing the number of half-wavelengths between the speakers, and λ is the wavelength of the sound.

We know that when the observer is at a distance of 1.5 m from the center, there is a node or completely destructive interference, so we can set n = 1. Then, solving for λ, we get:

λ = 2d/(2n + 1) = 2(1.5 m)/(2(1) + 1) = 1.5 m