A Quincke tube is a way to produce total destructive interference with sound by

Question

A Quincke tube is a way to produce total destructive interference with sound by splitting the sound (of the same frequency) and then recombining it. Suppose the shorter tube is length 1.66 m.
- If the speaker produces a sound of frequency 313 Hz, what is the minimum length of the longer tube for which you get total destructive interference?
length = m
- If the frequency produced by the speaker is increasing at a steady rate of df/dt = 80.9 Hz/second. To keep the total destructive interference, How fast must the longer tube be increasing in length when the frequency is 680 Hz?
NOTE: You must give the correct sign: positive if L is increasing, negative if L is decreasing.
dL/dt = cm/s

Explanation / Answer

Wavelength of the sound = v/frequency = 340/313 Hz = 1.086 m

Longer tube can be odd multiple of wavelenght/2 higher than the shorter tube. Since destrcutive interferernce requires a wave with phase diff of pi radians which is achieved every half wavelegth.

Therefore minimum legth of longer tube = 1.66 + 1.086/2 = 2.20 m

The frequecy is increasing df/dt = 80.9 Hz/sec

d (wavelgth)/dt = d(v/f)/ dt = dv/dt* 1/f - v df/dt * 1/f^2

As dv/dt = 0

d(wavelgth)/dt = - 340 * 80.9/sec * 1/680^2 = - 0.059 m/sec

Therefore longer tube should increase at a rate equal to rate of change in wavelgth = - 0.059m/sec = - 5.9 cm/sec

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