I am having lots of difficulty understanding this problem. Any help would be greatly appreciated.
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.
I did not get an image of "the three harmonics shown on the left". I will assume the image would show the first 3 harmonics in a tube with one end closed.
1st harmonic > lambda = 4*L (Only 1/4th of a full wave exists in the pipe. So the length of a full wave would be 4L.)
2nd harmonic > lambda = 4*L/3
3rd harmonic > lambda = 4*L/5
7. The above 3 could be written lambda = 4*L/1, 4*L/3, and 4*L/5. So a general expression that follows that progression would be lambda = 4*L/n where n is any odd integer.
8. A 512 Hz sound has 512 cycles/second.
So the time period of a full cycle of that frequency is
T = 1/f = 1 / (512 cycles/second) = 0.00195 seconds/cycle (See note below)
The speed of sound is 343 m/s. So in 0.00195 seconds, the wave will go
343 m/s * 0.00195 s = 0.67 meters. That is the length of a full wave, or lambda. (That is my attempt to make the formula lambda=velocity/frequency make some sense.)
n = 1 > lambda = 0.67 m = 4*L >> L = 0.167 m
n = 3 > lambda = 0.67 m = 4*L/3 >> L = 0.502 m
n = 5 > lambda = 0.67 m = 4*L/5 >> L = 0.837 m
Note: giving the period the units "seconds/cycle" is the same as giving it the units "seconds" since "cycle" is not a proper unit.
I hope this helps,