The three lowest normal frequency modes are fundamental frequency mode, first overtone, and the second overtone.

The first overtone is at the second harmonic frequency and the second overtone is at the third harmonic frequency.

In the case of the fundamental frequency mode, the length of the pipe can be given by:

L = λ/2

λ = 2L

The normal modes of this system are products of trigonometric functions. (For linear systems, the time dependance of a normal mode is always sinusoidal, but the spatial dependence need not be.) Specifically, for this system a normal mode is described by

yi(x,t)=Ai sin(2p*x/?i)sin(2pfi*t)

Find the three lowest normal mode frequencies f1, f2, and f3.

Express the frequencies in terms of L, v, and any constants. List them in increasing order, separated by commas.

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