1、外文部分 Chapter2 Plane waves 2.1 Introduction In this chapter we present the foundations of Fourier acoustics-plane wave expansions.This material is presented in depth to provide a firm foundation for the rest of the book ,introducing concepts like wavenumber space and the extrapolation of wavefields f
2、rom one surface to another .Fouries acoustics is used to derive some famous tools for the radiation from planar sources； the Rayleigh integrals ,the Ewald sphere construction of farfield radiation, the first product theorem for arrays, vibrating plate radiation, and radiation classification theory.
3、Finally,a new tool called supersonic intensity is introduced which is useful in locating acoustic sources on vibrating structures.We begin the chapter with a review of some fundamentals； the wave equation, Eulers equation, and the concept of acoustic intensity. 2.2 The Wave Equation and Eulers Equat
4、ion Let p(x,y,z,t) be an infinitesimal variation of acoustic pressure from its equilibrium value which satisfies the acoustic wave equation 22221 0pp ct (2.1) for a homogeneous fluid with no viscosity .c is a constant and refers to the speed of sound in the medium .At 020C c=343 m/s in air and c=148
5、1 m/s in water. The right hand side of Eq.(2.1) indicates that there are no sources in the volume in which the equation is valid. In Cartesian coordinates 2 2 222 2 2x y z A second equation which will be used throughout this book is called Eulers equation, 0 v pt (2.2) Where v (Greek letter upsilon)
6、 represents the velocity vector with components u ,v ,w ； v ui vj wk (2.3) where i j and k are the unit vectors in the the x, y, and z directions, respectively, and the gradient in terms of the unit vectors as i j kx y z (2.4) We use the convention of a dot over a displacements quantity to indicate
7、velocity as is done in Junger and Feit. The displacements in the three coordinate directions are given by u, v, and w . The derivation of Eq.(2.2) is useful in developing some understanding of the physical meaning of p and v . Let us proceed in this direction. Figure2.1 : Infinitesimal volume elemen
8、t to illustrate Eulers equation Figure 2.1 shows an infinitesimal volume element of fluid x y z, with the x axis as shown .All six faces experience forces due to the pressure p in the fluid. It is important to realize that pressure is a scalar quantity. There is no direction associated with it .It h
9、as units of force per unit area , 2/Nm or Pascals.The following is the convention for pressure, P 0 Compression P 0 Rarefaction At a specific point in a fluid .a positive pressure indicates that an infinitesimal volume surrounding the point is under compression ,and forces are exerted outward from t
10、his volume. It follows that if the pressure at the left face of the cube in Fig. 2.1 is positive, then a force will be exerted in the positive x direction of magnitude p(x,y,z) y z. The pressure at the opposite face p(x+ x,y,z)is exerted in the negative x direction. We expand p(x+ x,y,z)in a Taylor
11、series to first order, as shown in the figure .Note that the force arrows indicate the direction of force for positive pressure .Given the directions of force shown,the total force exerted on the volume in the x direction is ( , , ) ( , , ) pp x y z p x x y z y z x y zx Now we invoke Newtons equatio
12、n ,f =ma =m ut,where f is the force, 0m x y z and 0 is the fluid density, yielding 0 uptx Carrying out the same analysis in the y and z directions yields the following two equations: 0 upty and 0 uptz We combine the above three equations into one using vectors yielding Eq(2.2) above, Eulers Equation
13、. 2.3 Instantaneous Acoustic Intensity It is critical in the study of acoustics to understand certain energy relationships. Most important is the acoustic intensity vector. In the time domain it is called the instantaneous acoustic and is defined as ( ) ( ) ( )I t p t v t , (2.5) with units of energ
14、y per unit time (power) per unit area, measured as (joules/s)/ 2m or watts/ 2m . The acoustic intensity is related to the energy density e through its divergence, e It , (2.6) where the divergence is yx zI II Ix y z (2.7) The energy density is given by 2211022| ( ) | ( )e v t p t (2.8) where is the
15、fluid compressibility, 201c (2.9) Equation (2.6) expresses the fact that an increase in the energy density at some point in the fluid is indicated by a negative divergence of the acoustic intensity vector； the intensity vectors are pointing into the region of increase in energy density. Figure 2.2 should make this clear. If we reverse the arrows in Fig. 2.2, a positive divergence results and the energy density at the center must decrease, that is, et 0. This case represents an apparent source of energy at the center.
