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

138 7. Pipes, Waveguides, and ResonatorsMechanical impedances at terminations occur as complicated functions of thefrequencies, so it may be necessary to conduct measurements over the range offrequencies under consideration. Smith nomographs (Beranek, 1949) are usefultools to expedite computations for r and ψ, the real and imaginary components ofthe impedances, from the measurements of the standing wave ratio and the positionof the node most adjacent to the end.The impedance tube is used to measure the reflective and absorptive propertiesof small sections of materials, such as acoustic tiles and sound control absorbers,mounted at the end of the tube.7.7 The Rectangular CavityIn Figure 7.2 a rectangular cavity is shown having dimensions L x , L y , L z in thex-, y-, and z-directions, respectively. This parallelepiped can represent a simpleauditorium or any other rectangular space that contains rigid walls, few windows˘and other openings. We assume the walls of the cavity to be perfectly rigid thatn · ⃗u = 0 at all of the boundaries (i.e., the walls will not move in the directions oftheir normals). This also means that n ·∇p = 0, i.e.,( ) ( ) ⎫∂p ∂p== 0∂x x=0 ∂x x=L x( ) ( )∂p ∂p⎪⎬=(7.29)∂y)( ∂p∂zy=0z=0˘== 0∂y y=L y)= 0⎪⎭z=L z( ∂p∂zFigure 7.2. The rectangular cavity.