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Question
I am confused about this whole question. Thanks for any help.

Two loudspeakers emit out-of-phase sound waves along the x-axis. The sound has maximum loudness
when the speakers are 30 cm apart. The sound becomes quieter as the distance between the speakers is
increased, reaching zero at a separation of 60 cm.
a) What is the frequency of the sound?
b) If you keep increasing the separation between the speakers, at what separation will you hear the
loudest sound again?

Answer
Hello Nick,

These speakers produce maximum loudness when the waveforms from both speakers add with constructive interference. This is very similar to the phenomenon we see with organ pipes. If the  length of the pipe is such that the reflection of the end of the pipe reinforces the new waveform being produced, there is resonance. So when these speakers are separated by 30 cm, the peaks occur at the same point and reinforce. We get destructive interference when the peak of one waveform and the trough of the other occur at the same location. In the waveform, the distance between the peak and the trough is 1/2 of the wavelength. So when one of the speakers was shifted by 30 cm, that 30 cm must have been 1/2 of the wavelength and the full wavelength must be 60 cm.

a. From our formula v = lambda*f,
the frequency is given by f = v/lambda.
Convert the wavelength to meters, plug in a good number for the speed of sound, and calculate the frequency.

b. If the separation is increased by another 1/2 wavelength, so the 2 speakers are 1 1/2 wavelengths apart, you will get constructive interference again. So at 1.5*lambda, you again hear the loudest sound.

I hope this helps,
Steve

Physics

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Steve Johnson

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I would be delighted to help with questions up through the first year of college Physics. Particularly Electricity, Electronics and Newtonian Mechanics (motion, acceleration etc.). I decline questions on relativity and Atomic Physics. I also could discuss the Space Shuttle and space flight in general.

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I have a BS in Physics and an MS in Electrical Engineering. I am retired now. My professional career was in Electrical Engineering with considerable time spent working with accelerometers, gyroscopes and flight dynamics (Physics related topics) while working on the Space Shuttle. I gave formal classroom lessons to technical co-workers periodically over a several year period.

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BS Physics, North Dakota State University
MS Electrical Engineering, North Dakota State University

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