How long is the open-closed pipe.. physics

Question

An open-open organ pipe is 87.8 cm long. An open-closed pipe has a fundamental frequency equal to the third harmonic of the open-open pipe.

How long is the open-closed pipe?

Answer 1

The length of the open-open pipe is 1/2 the wavelength of the fundamental.

This is because the pressure of an open end has to be zero (relative to atmospheric). The magnitude of the pressure is the greatest at the center of this pipe. A full wavelength would go from zero pressure to a max positive then back to zero then to a max negative then to zero. The open-open pipe is half of this

The length of the open-closed is 1/4 the wavelength of the fundamental. Again, on the open end the pressure is always zero. On the closed end it is a maximum.

For the open-open (call it #1)

wavelength(1) = 2 * length(1)

For the open-closed (call it #2)

wavelength(2) = 4 * length(2)

Third harmonic just means 3 x the fundamental frequency, or 1/3 the wavelength

wavelength(2) = 1/3 wavelength(1)

Solving:

length(2) = length(1)/6

length(2) = 14.6 cm