理想媒质中的声波方程.ppt

THREE BASIC EQUATIONS理想媒质中的三个基本方程1 1.1.The equation of motion1.the equation of motion(Eulers equation)First,we write the relation between sound pressure and velocity,Consider a fluid element F1F22 2.F1F2When the sound waves pass,the pressure is So the force on area ABCD will beis the force of per unit areaThe force on area EFGH will beThe net force experienced by the volume dV in the x direction is3 3.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,4 4.When For small amplitudeSimilarly,in the direction of y and z,we can obtain5 5.Now let the motion be three dimensional,so writeis gradient operatorSince P0 is a constant,and obtainThis is the linear inviscid equation of motion,valid for acoustic processes of small amplitude6 6.2.The equation of continuityrestatement of the law of the conservation of matterTo 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 .7 7.Consider a small rectangular-parallelepiped volume element dV=dxdydz which is fixed in space and through which 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.8 8.That the net influx of mass into this spatially fixed volume,resulting from flow in the x direction,is Similar expressions give the net influx for the y and z directions,9 9.So that the total influx must beWe obtain the equation of continuity1010.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 sound 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,1111.We obtainSimilar expressions gibe the net influx for the y and z directions,1212.Where is the divergence operator1313.3.The equation of stateWe need one more relation in order to determine the three functions P,and u.It is provided by the condition that we have an adiabatic(绝热的)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 determined as a function of the density (rather than a depending separately on both and T)1414.Generally the adiabatic equation of state is complicated.In these cases it is preferable to determine experimentally the isentropic(等熵等熵)relationship between pressure and density fluctuations.We write a Taylors expansionWhere S is adiabatic process,the partial derivatives are constants determined for adiabatic compression and expansion of the fluid about its equilibrium density.1515.If the fluctuations are small,only the lowest order term in Need be retained.This gives a linear relationship between the pressure fluctuation and the change in densityWe suppose1616.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 byHere r is the ratio of specific heat at constant pressure to that at constant volume.Air,for instance,has r=1.4 at normal conditions1717.For idea gas,In the sound field of small amplitude1818.Speed of sound in fluidsThis 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 sound1919.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 gives2020.This is in excellent 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 is adiabaticadiabatic compression constant2121.The wave equationFrom 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 change in density2222.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.2323.The three equations must be combined to yield a single differential equation with on dependent variable.2424.In small amplitude sound field,we can neglect the second order small quantity,so that 2525.We obtainForm 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 time t.2626.Whereis the three dimensional Laplacian operator.In different coordinates the operator takes on different formsRectangular coordinates-Spherical coordinatesCylindrical coordinates2727.The velocity potential of soundFrom the equation(3-1),we getFrom the equation(3-1),we getWhere rot is rotation operator2828.So the velocity must be irrotational（无旋的）.This means that it can be expressed as the gradient of a scalar（标量）function 2929.whereis defined as the velocity potential of sound.The physical meaning of this important result is that the acoustical excitation of an inviscid fluid involves no rotational flow;there are no effects such as boundary layers,shear waves,or turbulence.3030.In different coordinates it takes on different formsRectangular coordinatesSpherical coordinatesCylindrical coordinates3131.Differentiating the equation(3-5)expression with respect to t,and eliminating Substitution of this equation in(3-4)and integrating with respect to time will show that the velocity potential of sound also satisfies the wave equation3232.HomeworkCalculate the speed of sound in air at 200C and standard atmospheric pressure.(0=1.29kg/m3)3333.