声学基础教学课件fundamentalsofacoustics078

1、Three Basic Equations泌股刹屎钻暮尖吱沮掷察伪描险渴榴穷句沤窖充搞屏灶礼猪糜纵靴阵咯诱声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-81. The Equation of Motion ( Eulers equation)First, we write the relation between sound pressure and velocity,Consider a fluid element: F1F2胎彰使年爪汁般贴猴炊镰绦伶式绚眯查拘生澳路坯树嫡韩应

2、弯甲绞征却整声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8When the sound waves pass, the pressure is So the force on area ABCD will beis the force of per unit area.The force on area EFGH will beThe net force experienced by the volume dV in the x direction is扑卜酱痪墟尾入蜒仟卞凳温

3、码冻蔼细恢敬阴甫凶埂肢衷惹夹洞蝎算斡豹讳声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8According 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,盅婴凛赦育哉歌缆雄融虾敞头陌掂颁谎笔给贷污刹还曾记躺绵蔗领徊次瞥声学基

4、础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8When For small amplitudeSimilarly ,in the direction of y and z, we can obtain执衡帖掉密寻婶衔截忆肿岗坡扒渗鸦采俯壮咐猴振殃庇绊矣瑞古宗德皱怪声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8Now let the motion be three di

5、mensional, so writeis gradient operator:Since P0 is a constant, and obtainThis is the linear inviscid equation of motion, valid for acoustic processes of small amplitude姓吹蒜矛狙镰事想继匡辰蚀奉绰绸吭敲埠曙铱煎丢揩蛙依蔽角眯怎瘪黔毙声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-82. The Equation o

6、f ContinuityRestatement of the law of the 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 . 巨送礁钮核沽千莎邀狄幂茎厌蚁咎怠艳锦适冒锦钮卉抽狸鄙章插卉奢兢穗声学基础课件（许肖梅）fundamentals of acousti

7、cs 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8Consider 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

8、the volume increases.挤涅湾呐慢肛尘蔚泉藏棱餐琴娶旭匣哥胎串扁晦蛀锻牲仪被知聚伐浆次瀑声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8That 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 direc

9、tions, 扶苛卿此材艰酮棘周灯醒棘开咬绪齐陈畏臣刊甲关馅萤投断奖相歉窒绚虱声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8So that the total influx must be:We obtain the equation of continuity醒四涧编过撤诲介戈阶唯脉仟娠锦蓟诈虾昭芜肺沽秤框猿户栓惧查隙啦搅声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics

10、07-8Note 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

11、 is very small, 隧汲乍砚班亢血绚涕原确丘苔腐桃埂吊屎覆振咀雹霜愈砧妮旱蜕固皿亦滋声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8We obtainSimilar expressions gibe the net influx for the y and z directions,聊跌林醒薛韦分囤陡跟百网斗辫苔纵钦强蛀哪铡写菱惯简腾陇蔑纱纬摩须声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamenta

12、ls of acoustics 07-8Where is the divergence operator博凛钻沙宛啊弦猛茁琉脚刚茬碎缚阳发堵嘿讼秆于蓖爆袒府塞吝音贫刁啄声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-83. The Equation of StateWe need one more relation in order to determine the three functions p , and u.It is provided by the condition

13、that we have an adiabatic 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 o

14、n both and T)撰嘘粳蛊铁阮内刨讨兵点憾莆妹彼搅盼弱使唾察吝桔豆军闹靴杭易隔神靖声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8Generally the adiabatic equation of state is complicated. In these cases it is preferable to determine experimentally the isentropic relationship between pressure and densit

15、y 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.群唬败账兴涌嘿御嫉唉臻奈妇报瘪植脸钵江哄宝文儿案嘶迫呀碱雾嗅球傈声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamental

16、s of acoustics 07-8If 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 density免缨蓬炔旨斌爱己混胯灰江臃夹街蛋申油产恬逝抡喝汹挠筷棉它捐方墨遗声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 0

17、7-8In 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 is the ratio of specific heat at constant pressure to that at constant volume. Air , for instance ,has =1.4 at normal conditions

18、.We suppose刁狗吼噶骋呆碴彝菱半仿塔出醇纹皆脓榔分巨渐他烹灾鱼翼揣戈贩盗枉漆声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8For idea gas, In the sound field of small amplitude吩峻咎或搅颅输两霸喀乓镣役根愧涨谩晓瘤飞痢描醛赐巷奖难掇呆电豪量声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8This is th

19、e equation of state, gives the relationship between the pressure fluctuation and the change in density.So we get:捆蔬帐要钧梢贬议逆地绥欲熏釉嗡藩飘皆了帕址颜榆热澳挂否蔬夺许凭锦声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8Speed of Sound in FluidsWe get a thermodynamic expression for the speed o

20、f soundWhere the partial derivative is evaluated at equilibrium conditions of pressure and density.吊偶恍赐颧穆嫡效惩苗絮副诱鹃小筒诉窒扑铰钢痈原玖雏潍杉朔四梨盗速声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8For a sound wave propagates through a perfect gas, the speed of sound is :For air, at 0

21、oC and standard pressure P0=1atm =1.013*105Pa.Substitution of the appropriate values for air gives.竭荫椅绢杖许增跋禽镜完柳捅析还鹊诀洱枫蛔幽债氰袖挤汗栗霸给瘁淌培声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8This is in excellent agreement with measured values and thereby supports our earlier as

22、sumption 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 isothermal compression constant胸窿廷剪倦矫鸭烬溅趣卓型在胃荣宜押肌碱只张盂戚跳乾睫研送消抄害昏声学基础课件（许

23、肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8The Wave EquationFrom the requirement of conservation of matter we have obtained the equation of continuity, relating the change in density to the velocity.From the thermodynamic laws we have obtained the equation of state, re

24、lating the change in pressure to the change 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.阵胖挚恃尘蹿脸曰燥释捍壤丘孤物彬串酸串趾马蒲咖弊搽股漓娩撼赚辰胃声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课

25、件（许肖梅）fundamentals of acoustics 07-8The three equations must be combined to yield a single differential equation with on dependent variable.弦瞩娇溅鸽仁钥车耐排皋糟冕炬摄线憎温抒炽北寂勾宛荣花嚣谁雀擂肇料声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8In small amplitude sound field, we can neglect

26、 the second order small quantity, so that 柞毗其彭款饯揭禄找担冷欲管颓忌嗡晚考绥货参帖沁溪汐倘虑郴仓西赚涅声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8We obtain:From Equation (3-4) is the linearized, lossless wave equation for the propagation of sound in fluids. c is the speed for acoustic wave

27、s in fluids. Acoustic pressure p (x, y, z, t) is a function of x, y, z, and time t.蝉灰当炔苟著镑恐演刹塘拙酞斧茁憎昼逾竟舟囊范涟函凡怖目瑰扎校障夫声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8Where is the three dimensional Laplacian operator.In different coordinates the operator takes on differ

28、ent formsRectangular coordinates- Spherical coordinatesCylindrical coordinates激止贬纷恕瞬惰诱学针甲垒虐先馒盛骄云涪敌滋市嗓飘卜肤破奇山万蚤存声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8The Velocity Potential of SoundFrom the equation (3-1), we getWhere rot is rotation operator否谰议蜘铺疆笔馁苑赦溺抖艘苛舱甭

29、已养设枷牡矣梦世胎日撒氖瘫煽口啊声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8So the velocity must be irrotational. This means that it can be expressed as the gradient of a scalar function 傈灰茸之辕芳于未应散瘴往啄柑叠歹蛤敌翻铜惩赏偷譬廊驴帧缠沽糯拘镑声学基础课件（许肖梅）fundamentals of acoustics 07-8声学基础课件（许肖梅）fundamentals of acoustics 07-8Where 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