How far do i have to sit to hear the loudest at a concert

[yjk91]

Homework Statement

A college student is at a concert and really wants to hear the music, so she sits between two in-phase loudspeakers, which point toward each other and are 46.4 m apart. The speakers emit sound at a frequency of 535 Hz. At the midpoint between the speakers, there will be constructive interference, and the music will be at its loudest. At what distance closest to the midpoint could she also sit to experience the loudest sound?

The Attempt at a Solution

v = f * lambda
so lambda = v/f = 343 m/s / 535 = 0.641 m
the answer is 0.641m but when i got the answer key from my teacher the answer seems to be λ/2 but the answer was still right on blackboard auto grading and when i entered λ/2 as my answer it was wrong maybe it depends on the "full wavelength" the answer key talks about.
can some body explain the bottom paragraph is the answer key's solution with
frequency as 490 and 49.7 m apart i think since λ is .7 and it says exactly 71 full wavelength
i think something depends on anti node and nodes also number of wavelength

thank you.


The wavelength of the sound is λ=v/f=(343m/s)/(490.Hz)=0.700m. Since the speakers are in phase and are facing each other, and since they are exactly 71 full wavelengths apart, their interference will yield a standing wave with an anti-node at the center between them. If she sits a half wavelength away from the center, then she will be at another anti-node. Therefore, the minimum distance away from the center that she can move on the straight line connecting the two speakers and again hear the loudest sound is: d=λ/2=0.350m. However, she can also move away form the midpoint in a direction perpendicular to the straight line connecting the two speakers. The next instance for which she encounters a maximum proceeding in this way is when the path from each speaker is 12λ longer. This gives the condition of a right triangle with side lengths d, 35.5λ, and 36λ, which we have to solve for the d, the distance she has to move: d=(36λ)2−(35.5λ)2=5.979λ. This is much greater than the 0.5 wavelength she has to move along the line connecting the speakers, and thus d=λ/2=0.350m is our desired solution.

 

[anathema]

Dear college student,

Thank you for your question. The answer to your question depends on the specific situation and the properties of the sound waves being emitted by the speakers. In this case, the frequency of the sound is 535 Hz, which means that the wavelength of the sound is 0.641 m. When two waves are in phase and interfere constructively, they create a standing wave pattern with nodes and antinodes. In this case, the speakers are facing each other and are exactly 71 full wavelengths apart. This means that there will be an antinode at the midpoint between the speakers, where the sound will be the loudest.

If the student were to sit exactly halfway between the speakers, she would also be at an antinode and experience the loudest sound. However, she can also move away from the midpoint in a direction perpendicular to the line connecting the two speakers. In this case, the next instance of constructive interference will occur when the path from each speaker is 12λ longer. This creates a right triangle with side lengths d, 35.5λ, and 36λ. Solving for d, we get d = 0.350 m, which is half of the wavelength.

In conclusion, the closest distance the student can sit from the midpoint to experience the loudest sound is half of the wavelength, which is 0.350 m. However, this distance may vary depending on the specific properties of the sound waves and the positioning of the speakers.

I hope this helps clarify the answer for you. Keep being curious and asking questions!