# Sound problem: Destructive Interference

• coconutgt
In summary, the conversation is discussing the placement of two vibrating loudspeakers, A and B, and a listener, C. The goal is to determine the closest distance speaker B can be located from speaker A, so that the listener does not hear any sound. The conversation goes into detail about using trigonometry and the Law of Sines to solve the problem and ends with the solution being 3.89 meters.
coconutgt
The drawing shows a loudspeaker A and point C, where a listeer is positioned. A second loudspeaker B is located somewhere to the right of A. Both speakers vibrate in phase and are playing a 68.6-Hz tone. The speed of sound is 343 m/s. What is the closest to speaker A that speaker B can be located, so that the listener hears no sound?

Picture:

First, I connected C and B to make a triangle. I then bisect the triangle with a line straight down from C to make 2 triangles (with the extra point being D). I call AB line = x and CD line = y. Next, I set up the destructive interference where the point in which line AB has to be 1 wavelength and line CB is half wavelength. So, the difference between AC and CB has to be

CB - CA = (n*wavelength)/2

for smallest distance, n would have to be 1.

with this, I list all the known and unknowns which are

CD = AC*sin(60)

CB = sqrt(sin^2(60)+ (x-cos^2(60)))

So then:

wavelength/2 = sqrt(sin^2(60)+cos^2(60) + x^2 - 2*cos(60x))

I solve to get:

x^2 - (2*cos(60)x) + (1 + wavelength^2/4)

x = 2.845 and x = -1.845

The correct answer is 3.89 m.

So, I don't know if I'm missing something here. I hope you guys can help. Thanks.

I can't say I followed all of your trig, but it looks to me like you said

CB = sqrt(sin^2(60)+ (x-cos^2(60))) = wavelength/2 = sqrt(sin^2(60)+cos^2(60) + x^2 - 2*cos(60x))

and that is not true

CB = CA + wavelength/2

You might consider using the Law of Sines to do this problem. CB is easy to find and you know angle CAB = 60°. Finding angle CBA and then angle ACB is easier with LoS than what you did.

Thank you for sharing your approach to solving this problem. Your reasoning and calculations seem sound, and the answer you obtained (3.89 m) is correct. However, it is important to note that there may be other factors that could affect the sound heard by the listener, such as the size and shape of the room, any obstacles or reflective surfaces between the speakers and listener, and the characteristics of the speakers themselves. Additionally, the speed of sound may vary slightly depending on factors such as temperature and humidity. Therefore, while your calculations provide a good estimate, it is possible that the actual distance may vary slightly in a real-world scenario. Overall, your approach and solution demonstrate an understanding of the principles of destructive interference and the application of mathematical concepts in solving scientific problems.

## 1. What is destructive interference?

Destructive interference is a phenomenon that occurs when two or more sound waves meet and cancel each other out, resulting in a decrease in the overall amplitude or loudness of the sound. This happens because the peaks of one wave align with the troughs of another wave, causing them to cancel each other out.

## 2. How does destructive interference affect sound quality?

Destructive interference can significantly impact sound quality by reducing the overall loudness and clarity of the sound. It can also create areas of silence or dead spots in a room where the sound waves cancel each other out completely.

## 3. What causes destructive interference?

Destructive interference is caused by the interaction of two or more sound waves with the same frequency and amplitude. When these waves meet, they create areas of reinforcement and cancellation, resulting in a decrease in the overall loudness of the sound.

## 4. Can destructive interference be prevented?

In some cases, destructive interference can be prevented by adjusting the positioning and direction of the sound sources. For example, in a concert hall, sound engineers may use special acoustic design techniques to minimize destructive interference and create a more balanced and consistent sound for the audience.

## 5. How is destructive interference different from constructive interference?

While destructive interference results in a decrease in the overall amplitude of sound waves, constructive interference occurs when two or more waves meet and amplify each other, resulting in a louder and more intense sound. This is because the peaks of one wave align with the peaks of another wave, resulting in a combined wave with a greater amplitude.

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