String and Soundboard

Our model of blackbody radiation has an acoustic analog in the form of a string interacting with a soundboard, with the following key components (as total or for each frequency in a spectral decomposition):

  • internal string energy
  • string amplitude
  • soundboard amplitude
  • force balance between string and soundboard = wave equation
  • small damping in the interaction expressed by the wave equation.

The small damping is an important feature of the model since it creates an “optimal” interaction between the string and the soundboard in the sense that the tone generated by the soundboard will be both loud and have long sustain. With zero damping the interaction will small and the tone weak, and with too big damping the sustain will be short. In radiation the damping is small.

The effect of the small damping is that the string velocity is out-of-phase with the forcing, which is the key to optimal interaction between string and soundboard.

Radiative equilibrium corresponds to acoustic equilibrium, which can be interpreted in two different ways:

  1. The vibration of the soundboard is sustained by the vibration of the string = outgoing acoustic waves.
  2. The vibration of the string is sustained by the vibration of the soundboard = incoming acoustic waves.

The same equilibrium can thus be interpreted as incoming or outgoing, which can be expressed as absorptivity = emissivity, which thus is built into the model.

This argument applies to frequencies below cut-off with only radiative damping being active. For frequencies above cut-off the interaction between string and soundboard is different with internal heating of the string, and emissivity in general different from absorptivity.

There is also non-equilibrium dynamics below cut-off with e.g. the string energy being transferred to the soundboard as the string amplitude decreases.