thus showing that the preceding simple analysis has reproduced the 

 essential features of the simplified solutions for the transmission and 

 reflection coefficients. 



b. Explicit Determination of the Linearized Friction Factor . The 

 general graphical solution for R and T (Figs. 2 and 3) and the simpli- 

 fied solutions (eqs. 35 and 36) require knowledge of the linearized 

 friction factor, f, to be of use. This friction factor which was 

 formally introduced by equation (6) may be determined by invoking 

 Lorentz' principle of equivalent work. This principle, whose use and 

 application are discussed in detail in Appendix A, is particularly 

 appropriate for use in the present context, since the flow resistance 

 within the structure contributes to the problem as a dissipator of 

 energy. Hence, invoking Lorentz' principle, which states that the 

 average rate of energy dissipation should be identical whether evaluated 

 using the true nonlinear resistance law or its linearized equivalent, 

 yields : 



r ,-T 



^ 1 f ,,, 4 f . 1^ ,,2 



E, =J)^d^[^J^ pf^U dtj 



1 „, rl r . ,,2 „|,,1„2, 



„ I d^ [^ \ p(aU" + B|U|r)dtj (47) 



in which V- is the volume per unit length occupied by the porous structure, 

 T is the wave period, and E^ is the spatial and temporal average rate of 

 energy dissipation per unit volume. 



The value of U to be used in equation (47) should correspond to the 

 general solution given by equation (21). However, keeping in mind the 

 approximate nature of Lorentz' principle as well as the uncertainties 

 involved in assessing the values of a and B, the simple solution, valid 

 only for nk^Jl < 0.2 (eq, 37) is used. Equation (37) shows |u| and hence 

 U, as given by equation (8) to be independent of location within the 

 porous structure. Since U is necessarily periodic, with period T, the 

 averaging process indicated by equation (47) is readily performed and 

 leads to the following relationship: 



f ^ = a + B |- |u| . (48) 



n 3tt ' ' ^ -^ 



With |u| given by equation (37) this is seen to be a quadratic 

 equation in f 



28 



