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

[xavior6]

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?

 

[ideasrule]

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.

 

[xavior6]

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?