理想媒質(zhì)中的聲波方程.ppt

1、THREE BASIC EQUATIONS,理想媒質(zhì)中的三個(gè)基本方程,1. The equation of motion,1. the equation of motion ( Eulers equation) First, we write the relation between sound pressure and velocity, Consider a fluid element,When the sound waves pass, the pressure is,So the force on area ABCD will be,is the force of per unit a

2、rea,The force on area EFGH will be,The net force experienced by the volume dV in the x direction is,According to Newtons second law F=ma, the acceleration of small volume in x direction will be,For small amplitude , we can neglect the second order variable terms,When,For small amplitude,Similarly ,i

3、n the direction of y and z, we can obtain,Now let the motion be three dimensional, so write,is gradient operator,Since P0 is a constant, and obtain,This is the linear inviscid equation of motion, valid for acoustic processes of small amplitude,2. The equation of continuityrestatement of the law of t

4、he conservation of matter,To relate the motion of the fluid to its compression or dilatation, we need a functional relationship between the particle velocity u and the instantaneous density p .,Consider a small rectangular-parallelepiped volume element dV=dxdydz which is fixed in space and through w

5、hich elements of the fluid travel. The net rate with which mass flows into the volume through its surface must equal the rate with the mass within the volume increases.,That the net influx of mass into this spatially fixed volume,resulting from flow in the x direction , is,Similar expressions give t

6、he net influx for the y and z directions,So that the total influx must be,We obtain the equation of continuity,Note that the equation is nonlinear; the right term involves the product of particle velocity and instantaneous density, both of which are acoustic variables. Consider a small amplitude sou

7、nd wave, if we write p=p0(1+s) .Use the fact that p0 is a constant in both space and time, and assume that s is very small,We obtain,Similar expressions gibe the net influx for the y and z directions,Where,is the divergence operator,3. The equation of state,We need one more relation in order to dete

8、rmine the three functions P , and u. It is provided by the condition that we have an adiabatic(絕熱的) process, ( there is insignificant exchange of thermal energy from one particle of fluid to another). Under these conditions, it is conveniently expressed by saying that the pressure p is uniquely dete

9、rmined as a function of the density ( rather than a depending separately on both and T),Generally the adiabatic equation of state is complicated. In these cases it is preferable to determine experimentally the isentropic (等熵)relationship between pressure and density fluctuations.,We write a Taylors

10、expansion,Where S is adiabatic process, the partial derivatives are constants determined for adiabatic compression and expansion of the fluid about its equilibrium density.,If the fluctuations are small, only the lowest order term in,Need be retained .This gives a linear relationship between the pre

11、ssure fluctuation and the change in density,We suppose,In the case of gases at sufficiently low density, their behavior will be well approximated by the ideal gas law. An adiabatic process in an ideal gas is governed by,Here r is the ratio of specific heat at constant pressure to that at constant vo

12、lume. Air , for instance ,has r =1.4 at normal conditions,For idea gas,In the sound field of small amplitude,Speed of sound in fluids,This is the equation of state, gives the relationship between the pressure fluctuation and the change in density.,We get a thermodynamic expression for the speed of s

13、ound,Where the partial derivative is evaluated at equilibrium conditions of pressure and density. For a sound wave propagates through a perfect gas, the speed of sound is :,For air, at 00C and standard pressure P0=1atm =1.013*105Pa.Substitution of the appropriate values for air gives,This is in exce

14、llent agreement with measured values and thereby supports our earlier assumption that acoustic processes in a fluid are adiabatic. Theoretical prediction of the speed of sound for liquids is considerably more difficult than for gases. A convenient expression for the speed of sound in liquids is,Bs i

15、s adiabatic compression constant,The wave equation,From the requirement of conservation of matter we have obtained the equation of continuity, relating the change in density to the velocity ; form the thermodynamic laws we have obtained the equation of state, relating the change in pressure to the c

16、hange in density,By using one more equation (the equation of motion), that relating the change in velocity to pressure. We shall have enough equation to solve for all three quantities.,The three equations must be combined to yield a single differential equation with on dependent variable.,In small a

17、mplitude sound field , we can neglect the second order small quantity ,so that,We obtain,Form,Equation (3-4) is the linearized , lossless wave equation for the propagation of sound in fluids. c is the speed for acoustic waves in fluids. Acoustic pressure p (x, y, z, t) is a function of x, y, z, and

18、time t.,Where,is the three dimensional Laplacian operator.,In different coordinates the operator takes on different forms,Rectangular coordinates,-Spherical coordinates,Cylindrical coordinates,The velocity potential of sound,From the equation (3-1) ,we get,Where rot is rotation operator,So the velocity must be irrotational（無(wú)旋的）. This means that it can be expressed as the gradient of a scalar（標(biāo)量） function,where,is defined as the velocity potential of sound .,The physical meaning of this important result