Closed Pipe Harmonics

Closed Cylindrical Pipe

A pipe closed at one end (clarinet, stopped organ pipe) has a pressure node at the closed end and a pressure antinode at the open end. This boundary condition permits only odd harmonics:

f_n = rac{(2n-1)v}{4L}, quad n = 1, 2, 3, ldots

• $n=1$ gives the fundamental: f_1 = rac{v}{4L}
• $n=2$ gives the 3rd harmonic: f_2 = rac{3v}{4L}
• $n=3$ gives the 5th harmonic: f_3 = rac{5v}{4L}

Why Only Odd Harmonics?

The closed end forces a node; the open end forces an antinode. Only standing waves with an odd number of quarter-wavelengths between the ends satisfy both conditions.

Closed vs Open (same length)

A closed pipe's fundamental is one octave lower than an open pipe of the same length — the same reason a stopped organ pipe sounds an octave below its open counterpart.

Your Task

Implement closedPipeMode(n, L) returning the nth resonant frequency ($v = 343$ m/s).

Run the code to hear the hollow, woody tone of odd harmonics only.