String, normal modes

This is simply the statement that the fluctuations of the normal modes are uncorrelated and that each normal mode has an average energy %kT. (The factor arises because our string is overdamped and has no kinetic energy.)... 

We want to learn how to quantize the radiation field. As a first step, consider a continuous elastic system. Any classical continuous elastic system in one dimension can be treated by a normal-mode analysis. Consider an elastic string of length a [m], tied at both ends to some fixed objects, with density per unit length p [kg m ], and tension, or Hooke s law force constant kH [N m-1]. The transverse displacements of the string along the x axis can be described by a transverse stretch y(x, t) at any point x along the string and at a time t. One can describe the y(x, t) as a Fourier sine series in x ... 

The two terms in the Hamiltonian H are not linked. Therefore one can find product eigenfunctions for the string, with each factor belonging to a different classical normal mode ... 

A membrane is a two dimensional string . There are special frequencies of vibration, normal modes, of the membrane in which all elements vibrate with the same angular frequency, a>. Once all of these special motions are known it is possible to describe any general motion of the membrane. We shall begin the study of the vibrations of membranes by obtaining the equation of motion of the soap film membrane and hence obtain the normal mode frequencies. 

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