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

132 7. Pipes, Waveguides, and ResonatorsHere p 0 is the maximum amplitude of the pressure wave. The volume flow is givenbyU(x, t) = p 0Sρc ei(ωt−kx)where ω is the angular frequency, k = 2π/λ = ω/c the wave number, S the crosssectional area of the pipe, ρ the density of the fluid inside the pipe, and c thepropagation velocity.7.3 Resonances in a Close-Ended PipeAs shown in Figure 7.1 consider a pipe of length L and cross-sectional area S,filled with a fluid, and sealed off at one end, x = L. Let the fluid inside the pipebe driven by a piston at x = 0. The pipe has a mechanical impedance Z nL. Thepiston vibrates harmonically at a sufficiently low frequency so that only planewaves are considered to exist inside the pipe. The wave inside the pipe can bedescribed byp = Ae i[ωt+k(L−x)] + Be i[ωt−k(L−x)] (7.2)where A and B are established by the boundary conditions at x = 0 and x = L.At x = L the mechanical impedance of the wave must equal the mechanicalimpedance Z nL at the termination so as to sustain the continuities of force andparticles. The force of the fluid acting at the end of the pipe is p(L, t)S, and thecorresponding particle speed u(L, t) derives from the integrated Equation (2.23)∫u =−1ρ δ ( ∂p∂x)dtFigure 7.1. A pipe close-ended at x = L.