Modal energy distribution in closed pipe resonance

[quasar44]

When one causes the air column in a closed pipe to vibrate in its well-known modes (harmonics) or a plucked string to vibrate similarly, how is the exciting energy distributed amongst the modes?

 

[Ackbach]

This sounds like a Fourier Analysis problem. You'd have to take your actual wave function, and perform a Fourier series expansion on it to determine the relative strengths of each harmonic.

The exact makeup of each harmonic influences what in music we call timbre. An alto recorder, for example, has very few overtones and has a "purer" sound than a transverse flute.

 

[marcusl]

The system geometry (size and length of pipe, mass density, tension and length of string) determine the possible modes that could be excited. The "excitation" or initial condition determines which ones actually are, and how much energy goes into each. For a string, you can change the modal spectrum by plucking it at different positions from the end. Pluck at the center and and you excite the lowest mode plus odd harmonics. Pluck elsewhere and the spectrum (timbre) changes. Pull at two points simultaneously with different displacements (amplitudes) and you can get still different spectra.