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Question
Write down an expression for the length of the air column L as a function the wavelength, lambda, for each of the three harmonics shown on the left.
7. From the expressions above, find the expression for the length of the air column of the n-th harmonic as a function of lambda(where is an odd number).
8. A tuning fork emits a 512 Hz sound at the open end of a tube (one end is open, the other one is closed). Find the length of tube required to have the first three resonances (occurrences of standing waves). Assume that sound travels at 343 m/s.
Resonance
Harmonic, n

Answer
If a tube is closed at one end resonance will occur at odd multiple of 1/4 wavelength such as 1/4L, 3/4L, 5/4L, etc. When you take the inner diameter of the tube into account the wavelength of the first resonance will be:
Lambda=4(L+0.4d)
While the second resonance will be:
Lambda=4/3(L+0.4d)
And for the third resonance,will be:
Lambda=4/5(L+0.4d)
Given that the wave speed is s=343m/s and that the frequency is f=512Hz the corresponding wavelength will be:
Lambda=s/f=343/512=0.67m
The first resonance will occur at:
Lambda=4(L+0.4d)  becomes  0.67=4(L+0.4d)  assuming that the diameter is small|
L1=0.67/4=0.168m
L2=0.67*3/4=0.503m
L2=0.67*5/4=0.838m

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I am teaching or have taught AP physics B and C [calculus based mechanics & electricity and magnetism] as well as Lab Physics for college bound students. I have a BS in Physics from the University of Pittsburgh and a Master of Arts in Teaching from same. I have been teaching physics for 34 years. I am constantly updating my skills and have a particular interest in modern physics topics.

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