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

170 8. Acoustic Analogs, Ducts, and Filters(a) Find the resonance frequency of the room considered as a Helmholtzresonator.(b) Find the acoustic compliance of the room.(c) Find the inertance of the door opening.(d) What is the acoustic impedance presented to the sound source at 30 Hz insidethe room? Consider only the compliance of the room and the inertanceof the door opening.4. Pipe 1 having cross-section A 1 connects to pipe 2 with cross section A 2 .(a) Obtain an expression for the ratio of intensity of the waves transmitted intothe second pipe to that of the incident waves.(b) Under what conditions will the transmitted energy exceed the incidentenergy?(c) Develop a general expression for the standing wave ratio (SWR) producedin pipe 1 in terms of relative areas A 1 and A 2 .5. Consider two pipes that are connected to each other but separated by a thinrubber diaphragm. Pipe 1 is filled with a fluid with characteristic impedanceρ 1 c 1 and pipe 2 contains a fluid with characteristic impedance ρ 2 c 2 . A planewave travels in pipe 1 in the positive x direction toward pipe 2.(a) Obtain an expression for the power-transmission ratio from pipe 1 intopipe 2.(b) Under what conditions does 100% power transmission occur?6. An infinitely long pipe of cross-sectional area A has a side branch, also infinitelylong, of cross-sectional area A b . The main pipe is transmitting planewaves with frequencies such that their wavelengths are much larger than thediameter of either pipe.(a) Develop an equation for the transmission coefficient in the main pipe.(b) Do the same thing for the branch pipe.(c) Let the area of the main pipe be twice as much as that of the branch pipe.Obtain numerical values for the transmission coefficient into each pipe. Dothe sum of the two coefficients equal unity? If not, where did the remainingpower go?7. A ventilating duct in a basement has a square cross-sectional area, 35 cm tothe side. In order to quiet the duct in part, a Helmholtz resonator-type filter isconstructed about the duct by drilling a hole of 9 cm radius in one wall of theduct, leading into a surrounding closed chamber of volume V .(a) What is the volume V necessary to most effectively filter sounds at afrequency of 30 Hz?(b) What will be the sound power transmission coefficient of the filter at 45Hz and 60 Hz?8. Demonstrate that the radius r of the hole drilled into a pipe of radius r 0 as toresult in a 50% sound power transmission coefficient at a frequency f is givenbyr = 64 r0 2 f3 c