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

Mechanical impedance Z n , expressed as7.3 Resonances in a Close-Ended Pipe 133Z n = f u(7.3)represents a complex value that is the ratio of the complex driving force f to thecomplex speed u at the point where the force is applied. In the case of the finitepipe, the mechanical impedance at x = L is given byZ nL = ρ 0 cS A + B(7.4)A − BThe value of the input mechanical impedance at x = 0 is expressed asZ n0 = ρ 0 cS AeikL + Be −ikL1 + iZ nLρ 0 cS tan kL (7.5)Eliminating A and B by combining Equations (7.4) and (7.5) yieldsZ n0ρ 0 cS =Z nL+ i tan kLρ 0 cS1 + Z (7.6)nLρ 0 cS tan kLThe term ρ 0 cS is the characteristic mechanical impedance of the fluid. The complexquantity Z n0 can be recast in terms of real and imaginary components, r andψ, respectively,Z nL= r + iψ (7.7)ρ 0 cSThe ratio on the left-hand side of Equation (7.7) constitutes a normalizedimpedance. Inserting Equation (7.7) into Equation (7.6) yieldsZ n0ρ 0 cS=(r + iψ) + i tan kL1 + i(r + iψ) tan kL= r(tan2 kL + 1) − i[ψ tan 2 kL + (r 2 + ψ 2 − 1) tan kL − ψ](ψ 2 + r 2 ) tan 2 (7.8)kL − 2ψ tan kL + 1When r = 0, the input impedance Z n0 vanishes when the reactance vanishes, i.e.,−i[ψ tan 2 kL + (r 2 + ψ 2 − 1) tan kL − ψ](ψ 2 + r 2 ) tan 2 kL − 2ψ tan kL + 1and this results inthat is,= 0 (7.9)ψ tan 2 kL + (ψ 2 − 1) tan kL − ψ = 0 (7.10)ψ =−tan kL (7.11)