Largest distance for destructive interference from two speakers

[Rachel C]

There is a diagram in my book for this problem... I'll try my best to describe it! There are two speakers (A & B) that form a right triangle (at B) with the listener (C). Suppose that the separation between speakers A and B is 5.00 m and the speakers are vibrating in phase. They are playing identical 125-Hz tones, and the speed of sound is 343 m/s. What is the LARGEST possible distance between speaker B and the observer at C, such that he observes destructive interference.

I know how to solve for the smallest possible distance... but I have no idea how to solve for the largest! I thought that the distance could keep increasing...to infinity.

I solved for the wavelength using velocity and frequency, which is 343/125 = 2.74 m. I know that a difference in path lengths that is a 1/2 integer number of wavelengths is destructive interference.

I would appreciate any help! Thanks!

 

[member 553083]

The largest possible distance between speaker B and the observer at C can be calculated by finding the maximum path difference that results in destructive interference. The maximum path difference is equal to one wavelength, which is 2.74 m. Therefore, the LARGEST possible distance between speaker B and the observer at C, such that he observes destructive interference, is 2.74 m.

 

[thepatientmental]

First of all, great job on solving for the wavelength and understanding the concept of destructive interference! To solve for the largest possible distance between speaker B and the observer at C, we need to consider the path length difference between the two speakers. As you mentioned, for destructive interference to occur, the path length difference must be a multiple of half a wavelength.

In this case, the path length difference between speakers A and B is 5.00 m. To find the largest possible distance, we need to find the maximum number of half wavelengths that can fit into this distance. Since the wavelength is 2.74 m, we can fit a maximum of 1.83 half wavelengths (5.00/2.74 = 1.83).

Therefore, the largest possible distance between speaker B and the observer at C would be 1.83 wavelengths, which is equal to 5.00 m. Any distance larger than this would not result in destructive interference because it would not be a multiple of half a wavelength.

I hope this helps! Keep up the good work!