By Karl F. Herzfeld

ISBN-10: 1483230570

ISBN-13: 9781483230573

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5-4) [5] I. THE STOKES-NAVIER EQUATIONS OF HYDRODYNAMICS 33 It follows from Eqs. (5—2) and (5—4) that in a purely progressive nondissipative wave, s, u, and p — p0 are in phase, as far as space and time are concerned. If, on the other hand, one has a pure standing wave s = s0 cos œt cos — x, (5—5) Eq. (5—4) stands but, from Eq. (5—2) u . œ — = s0 sin œt sin — x. (5-6) In the progressive wave, the energy contained in 1 cc (if all dimensions are large compared to the wave length) is j / v 2 + i(^:) s 2 = jPoi* + ®2s2}· (5~7) The two terms are equal at any time and place, but vary with time and place.

Equation (4—6) can be rewritten as \dp)s TV-ß-. (4-7) This is the equation connecting temperature and pressure change in an adiabatic reversible compression for any fluid. 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.

Accordingly, we now assume all the quantities involved, s, u, p — p0, and T — T0, to be proportional to exp { - a* + io>\t - ! j j = exp + χω |f - ~(l - -^-jj (7-4) where a is called the absorption coefficient for the amplitude, and has the dimension cm - 1 . ) decreases for each cm of travel by e~a, or decreases by the factor e~x for the distance a - 1 . The absorption coefficient for the intensity is 2a. 40 A. GENERAL THEORY OF RELAXATION IN FLUIDS [7] Occasionally it is useful to define an absorption coefficient per wave length a* = αλ = In — a.

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