Opening a Pipe
Classical Mechanics
Level
1
Compared to a half-open pipe, how much longer would a pipe open on both ends need to be in order to have the same fundamental frequency for longitudinal pressure waves?
The same length
Eight times as long
Four times as long
Twice as long
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1 solution
In a half-open pipe, the fundamental frequency is:
f 1 = 4 L v
as per applying boundary conditions to standing waves . The fundamental in a pipe open at both ends satisfies:
f 1 = 2 L ′ v
where L ′ is the length of the pipe open at both ends. This is because the pressure displacement at either end of the double-open pipe is maximal, whereas the pressure displacement at one end of the half-open pipe is zero. So the pressure displacement must vary from maximum to maximum over half a wavelength in the double-open pipe (try drawing a picture to see that this is true!). If the two formulas above are equal, then L ′ = 2 L as claimed.