THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

The effective specific heat can be established from( (dE =(C v ) eff dT tr = ¯C v dT tr + C ′ dT ′ = ¯C v)vand16.2 Relaxation Processes 449+ C′dT′dT tr)dT tr(C v ) eff = ¯C v +1 + iωτ = C v − C′ iωτ1 + iωtThe acoustic propagation constant k can be written in the formk 2 ( 1ω = 2 c − iα ) 2= ρ 0κ TωC ′γ eff(16.9)where c represents the acoustic velocity, α is attenuation coefficient, ρ 0 is theequilibrium density, κ T is the compressibility of the gas, and γ eff is given byγ eff ≡ (C v) eff + R(C v ) effwhere R is the gas constant. In the case of this simple single relaxation, for α/ω≪ 1andwhereλa = π( c0c( cc 0) 2εωτ1 + (ωτ s ) 2 (16.10)) 2= 1 −ε(ωτ s ) 21 + (ωτ s ) 2 (16.11)ε = c2 ∞ − c2 0c∞2c 0 = speed of sound for ωτs ≪ 1c ∞ = speed of sound at frequencies ≫ relaxation frequencyλ = wavelengthThe adiabatic relaxation time τ s is related to the isothermal relaxation time τ asfollows:τ s = C v + RCv∞ + R τThe frequency at which the maximum absorption per wavelength occurs is calledthe relaxation frequency, symbolized by f r . It is related to the adiabatic absorptiontime τ s as follows:f r = 1 c ∞2πτ s c 0We take as an example the gas Fl 2 at 102 ◦ C in Figure 16.1 that shows curves for absorptionper wavelength and velocity dispersion due to a single-relaxation process(Shields, 1962). Measured values are also plotted for the purpose of comparingwith theory.