I came up with an exercise on Halliday Resnick Krane that asked a question which consfuses me. I premit that I do not look for a solution of the exercise but for suggestions only regarding the highlighted part.

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Consider the acoustic interferometer in the picture. The lenght SAD is fixed, while SBD can be changed. A sound is produced in S and detected in D. We know that for a certain position of B a minimum of intensity of $10\mu W/cm^2$ is detected in D, and if we move B of a quantity $\Delta s=1.65m$ a maximum on intensity of $90\mu W/cm^2$ is detected in D. Find the amplitudes of the waves that arrives in D. How can you explain that these aplitudes are different even if the waves were generated by the same source?

Indeed I find that $\frac{A_1}{A_2}=2$.

Is it correct to say that the differce in amplitude of the two single waves generated by the same source is due to the fact that the wave-fronts are (even if it is not specified in the text) spherical, hence $A\propto \frac{1}{r}$? Therefore a longer path covered implies a smaller amplitude?


It looks like a dubious exercise question, unless the pages preceding the exercise set up a framework of what kind of things you're allowed to assume here. Both the entrance and exit (speaker/ear) represent discontinuities in the acoustic impedance of the tube. The impededance of the speaker (will it absorb reflected acoustic energy or not?) is not described either. These issues make the system much more complicated than just a matter of path lengths.

So, the answer should be: "it has to do with all the impedance mismatches and waves traveling around the loop multiple times". But it could be that the one who wrote down the exercise expects you to answer that the sound wave in the long path undergoes some attenuation (energy loss), so that the interfering amplitudes at the detector are no longer the same.

It could also be that the input sound waves are a mixture of different frequencies - the question does not state anything about the input spectrum.

Wavefronts inside the tubes are not spherical, by the way.


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