How to Minimize Sound Wave Interference Using a Tube with Variable Radius

AI Thread Summary
To minimize sound wave interference in a tube with a variable radius, it's essential to understand the principles of constructive and destructive interference. A sound wave with a 40cm wavelength needs to be analyzed to determine the smallest radius that produces a minimum at the detector end. The discussion highlights the importance of the difference in travel distances between two paths—straight through the tube and around an obstruction—resulting in destructive interference when the path length difference is an odd multiple of half a wavelength. The user seeks guidance on modeling the path around the bump, questioning if the arc-length of the semi-circle should be considered. Understanding these concepts is crucial for solving similar physics problems effectively.
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Homework Statement


A sound wave with 40cm wavelength enters a tube at the source end. What must be the smallest radius r such that a minimum is heard at the detector end (picture attached)?


Homework Equations



I am really lost in this problem, but I BELIEVE constructive/destructive wave interference:
f = (2n+1)V/(2dX)

where n is an integer, V is the wave speed, dX is the difference in travel distances between the two waves, and f is the frequency at which destructive interference occurs at a given point.

The Attempt at a Solution



I have honestly been stumped by this question. I have a physics exam very soon and so I need to know how to solve these sorts of problems. I am not looking for anyone to help me find the answer; I was just wondering if someone could help me get started off?

Thus far, can conceptually see that if r is the same size as the bump, then the wave will flow through as if it were an open ended pipe, but beyond that I do not know what is going on. Could someone please throw me on the right track?
 

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Sound can take two paths from source to detector: either straight through the tube, or around the "bump". If the length difference between the two paths is an odd multiple of a half-wavelength, the two waves would interfere destructively and cancel.
 
Correct, but I am failing in my attempt to model the path that the wave could take around the "bump". Would this be the arc-length of the semi-circle?
 

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