Frequency Spectrum of a Continuous Solid

Vibrational Frequency Spectrum of a Continuous Solid.—To find the specific heat, on the quantum theory, we must superpose Einstein specific heat curves for each natural frequency v1y as in Eq. (1.3). Before we can do this, we must find just what frequencies of vibration are allowed. Let us assume that our solid is of rectangular shape, bounded by the surfaces x = 0, x = X, y — 0, y = F, z = 0, z = Z. The frequencies will depend on the shape and size of the solid, but this does not really affect the specific heat, for it is only the low frequencies that art very sensitive to the geometry of the solid. As a first step in investigating the vibrations, let us consider those particular waves that arc propagated along the x axis. 

Rather than using the phase constant a, it is often convenient to use both sine and cosine terms, with independent amplitudes. 4 and B, obtaining 

By using different combinations of functions, we can get standing waves 

Certain boundary conditions must be satisfied at the surfaces of the solid. For instance, the surface may be held rigidly so that it cannot vibrate, or it may be in contact with the air so that it cannot develop a pressure at the surface. The allowed overtones will depend on the particular conditions we assume, but again this is important only for the low overtones and is immaterial for the high frequencies. To be specific, then, let us assume that the surface is held rigidly, so that the displacement is zero on the surface, or when x = 0, x = X. The first condition can be satisfied by using a standing wave containing the factor 

Condition (2.5) can be satisfied in many ways, for we know that the sine of any integer times t is zero. Thus we satisfy our boundary condition if we make 

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