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

5.6 General Boundary Conditions for a Freely Vibrating Bar 99applying Equation (5.3) to supplant the compressive forces and by expressing theparticle velocity as u = ∂ξ/∂t:( ) ∂ξ= Z ( )mio ∂ξ(5.21)∂x x=0 ρ L c 2 ∂t x=0( ) ∂ξ= Z ( )miL ∂ξ(5.22)∂x x=L ρ L c 2 ∂t x=Lwhere ρ L = ρ Â is the linear density (kg/m) of the bar.If the loads Z mio and Z miL are purely reactive, there is no transient or spatialdamping, and hence no loss of acoustical energy occurs. Equation (5.16) constitutesa proper solution. And because no loss of acoustical energy occurs, a wave travelingin the +x-direction must equal the energy of a wave moving in the oppositedirection. The absolute magnitudes of the complex wave amplitudes must thereforebe equal, i.e., |A| =|B|. The boundary conditions (5.21) and (5.22) establish thephase angles of the complex amplitudes.But if the mechanical impedances contain some measure of resistive components,a solution more general than that of Equation (5.16) needs to be applied. Asin the case of a freely vibrating string terminated by a resistive support, transient(or temporal) damping has to occur in the presence of resistance. The transientbehavior of the bar is characterized by a complex angular frequency ω = ω + iβ.The real portion of this frequency is the angular frequency ω; the imaginary partrepresents the transient absorption coefficient β. But no internal losses occur inthe bar, so wave equation (5.7) still applies, and we infer the solutionξ(x, t) = (Ae −ikx + Be ikx )e iωt (5.23)where ω 2 = c 2 k 2 . If the losses are quite small we can use the approximationω ≈ ck to simplify the solution. Applying boundary conditions (5.21) for x = 0and (5.22) for x = L to (5.23) and making use of the approximation we obtain thefollowing pair of equationsA −ikLeA − B =− Z mi0(A + B)ρ L c− B ikLe= Z miL (A−ikLeρ L c+ BeikL )Solution of these preceding two equations by elimination of A and B results in thetranscendental equationtan kL = iZ mi0ρ L c + Z miLρ L c1 + Z mi0ρ L cZ miLρ L cThe characteristics of the vibration are determined from the complex impedancesZ mi0 and Z miL . The solution of the preceding transcendental equation is rendered