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

400 14. Machinery Noise Controlandu R = iω A R e i(ωt+kx) (14.51)We recall that for a plane wave in the direct field, sound pressure is related toparticle velocity byp = ρ cuwhich is used in Equations (14.50) and (14.51) to obtainp I = iρ cω A I e i(ωt−kx) (14.52)p R = iρ cω A R e i(ωt+kx) (14.53)Let x = 0 designate the junction of the inlet pipe and expansion chamber.Pressure must be continuous at this junction, hence(p I 1 + p R1 ) x=0 = (p I 2 + p R2 ) x=0 (14.54)Inserting Equations (14.52) and (14.53) into (14.54) givesA I 1 + A R1 = A I 2 + A R2 (14.55)Subscript 1 refers to the left of the junction and subscript 2 refers to the right.Continuity of flow requires thatA (u I 1 − u R2 ) x=0 = B (u I 2 − u R2 ) x=0Setting B/A = m, where A refers to the flow area of the inlet pipe and the outletpipe and B the flow area of the expansion chamber, we have from continuityA I 1 − A R1 = m(A I 2 − A R2 ) (14.56)In Figure 14.19, C denotes the length of the expansion chamber. At the junctionof the expansion chamber where x = C, pressure and flow continuity necessitatesthatfrom whichandresulting in(p I 2 + p R2 ) x=C = p 3A I 2 e −ikC + A R2 e ikC = A 3 (14.57)B(u I 2 − u R2 ) x=C = Au 3 (14.58)m(A I 2 e −ikC + A R2 e ikC ) = A 3 (14.59)We now have four Equations (14.55)–(14.59) and five unknowns, A I 1 , A R1 , A I 2 ,A R2 , and A 3 . These equations are solved simultaneously to correlate conditions at