By Leo L. Beranek, Tim Mellow
Acoustics: Sound Fields and Transducers is a completely up to date model of Leo Beranek's vintage 1954 ebook that keeps and expands at the original's specific acoustical basics whereas including useful formulation and simulation equipment.
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Additional info for Acoustics: Sound Fields and Transducers
15b) Take the divergence of each side of Eq. 13b). 17) where V2 is the operator called the Laplacian. 19) We shall see later that c is the speed of propagation of the sound wave in the medium. Also, the quantity gP0 is the bulk modulus of the fluid medium. 22b) where V2 q ¼ VðV$qÞwhen there is no rota- tion in the medium. 22) apply to sound waves of “small” magnitude propagating in a sourcefree, homogeneous, isotropic, frictionless gas at rest. 5 The wave equation in cylindrical coordinates The one-dimensional wave equations derived above are for plane-wave propagation along one dimension of a rectangular coordinate system.
3 Solutions to the steady-state one-dimensional wave equation. Forward and reverse traveling waves. ) 34 CHAPTER 2 The wave equation and solutions the real part of which is also shown in Fig. 3. 46) for pressure except for one important difference, which is the minus sign preceding p~À . The reason for this is fairly simple. During a positive pressure half-cycle, the resulting velocity is always in the direction of travel. Therefore, in the case of the wave with amplitude p~þ traveling in the positive x direction, positive pressure produces positive velocity because it is in the positive x direction, as shown by the dashed arrows in Fig.
The “2” in parentheses denotes an outgoing cylindrical wave and the “1” denotes an incoming one. FIG. 17 Bessel functions of the first kind. 58 CHAPTER 2 The wave equation and solutions FIG. 18 Bessel functions of the second kind. The azimuthal equation in f. 135) It can be seen that the integer n denotes the nth harmonic of the azimuthal modes of vibration where f ¼ 2p represents a full rotation about the z axis. The values of An and Bn depend on where the nodes and antinodes lie on the circumference.
Acoustics: Sound Fields and Transducers by Leo L. Beranek, Tim Mellow