By Karl F. Herzfeld
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Additional resources for Absorption and Dispersion of Ultrasonic Waves
Truesdell4 has questioned the consistency of introducing a ''constant" into the hydrodynamical equations — which are supposed to hold for any time dependence — and then making this "constant" frequency dependent, which assumes simple harmonic motion. From the standpoint of consistency, he is quite right, and the only consistent procedure is the one applied in the following sections. However, the other procedure has long been followed in other fields of physics, the most conspicuous example being the dielectric "constant" introduced in Maxwell's equations, which also has to be taken as frequency dependent.
The Stokes-Navier Equations of Hydrodynamics 1. The State of the Fluid1 (a) Variation in Space and Time The condition and motion of a fluid — which name shall include both gases and liquids — filling a certain volume of space may be characterized by a number of scalars and vectors, and by one tensor. Among the first are the density p and the temperature T; among the second is the flow velocity w, with the components uv u2, u3. In general these quantities will, at a given moment of time, be different at different places in space.
Inserting Eq. (4—7) in Eq. (4—2), one gets However Tv£- KT = Cp-C9 = A. (4-9) Therefore, Eq. r-KT = — * r . tp y (4—10) Equations (4—7) and (4—10) are applicable to the isentropic compression of any fluid. The physical reason for the adiabatic compressibility being smaller than the isothermic one is simple. In an adiabatic compression not only does the density increase, but the temperature does also ; both processes increase the pressure. One also finds from a combination of Eqs. (4—7), (4—8), (4—9), and (4-10) that 32  A.