理想媒质中的声波方程PPT课件

1、理想媒质中的三个基本方程zyxvabcdefghyxzzxyf1f2abcdefghyxzzxyf1f2when the sound waves pass, the pressure is 0( , , , )pp x y z t10()fppy z 0pp20()fpppy z 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 is12x

2、fffp y z tudtduxxdtdzzudtdyyudtdxxutudtduxxxxxfor small amplitude , we can neglect the second order variable terms,zyxvmxup y zx y zt xupxt0 xxpxpx0limtuxpx00for small amplitudesimilarly ,in the direction of y and z, we can obtaintuypy0tuzpz00uptpp 0dupdt ijkxyz ppppijkxyz is gradient operatorsince

3、p0 is a constant, and obtainthis is the linear inviscid equation of motion, valid for acoustic processes of small amplitudetzyxuxuxx)(that the net influx of mass into this spatially fixed volume,resulting from flow in the x direction , is tzyxuxx)(similar expressions give the net influx for the y an

4、d z directions, tzyxuyy)(tzyxuzz)()()()xyzmx y zuuux y z txyz )()()(zyxuzuyuxt0tttt0lim)()()(zyxuzuyuxtwe obtain the equation of continuity0 xuuxuxxxx00)()(yuuyyy0)(zuuzzz0)()(0zuyuxutzyx0ut similar expressions gibe the net influx for the y and z directions,()yxzaaaaxyzwhere is the divergence operat

5、or0 ut( )pp0220s,00d1()()()()()d2!pd pppd,20,()sdpdpdd00s,0d() ()()dpp p0need be retained .this gives a linear relationship between the pressure fluctuation and the change in densitywe suppose02,()sdpcd00vppvhere r is the ratio of specific heat at constant pressure to that at constant volume. air ,

6、for instance ,has r =1.4 at normal conditions0 0vv00vppv00000000dpppppdin the sound field of small amplitude01d2000ppdpppdc d0ppp02dpdcdtdtthis 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

7、of sound0,sddpc002pc for air, at 00c and standard pressure p0=1atm =1.013*105pa.substitution of the appropriate values for air gives51.402 1.103 10331.6/1.293cm s01sc bs is compression constantfrom the requirement of conservation of matter we have obtained the equation of continuity, relating the ch

8、ange 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 density0(3 1)upt 0(32)ut 2(33)dpdcdtdt202(32)()0uformtt22()0pt ()xyzddxdydzdttx dty dtz dtuuutxyz()dudtttdtddppdtt2pctt22()0pt 2pctt2222(34)pcptequat

9、ion (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.2222222xyz 22222222111()(sin)sinsinrrrrrr 2is the three dimensional laplacian operator.in di

10、fferent coordinates the operator takes on different formsrectangular coordinates2222221)(1zrrrrr-spherical coordinatescylindrical coordinates0(3 1)upt 01(35)updt 000yxxzyzuuyxuuzxuuzyrot0u where rot is rotation operatorrot()()()0yyxxzzuuuuuuuijkyzzxxyu is 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.,xyzuuuxyz 11,sinruuurrr 1,rzuuurrz spherical coordinatescylindrical coordinates0pt22