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1、Harmonic Plane Wave Traveling in Harmonic Plane Wave Traveling in One MediumOne Medium 祈聚 舆泊 享峭 妇符 消曹 贞磨 剿陌 镜统 讶碑 檄蒂 傀得 括呐 忱艰 括躁 季蝗 遏炎 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 If all the acoustic variables are functions I
2、f all the acoustic variables are functions of only one spatial coordinate, the phase of only one spatial coordinate, the phase of any variable is a constant on any plane of any variable is a constant on any plane , such a wave is called a , such a wave is called a plane waveplane wave. . The simples
3、t solutions to the wave The simplest solutions to the wave equation (3-4) are those that depend on equation (3-4) are those that depend on only one of the three spatial coordinates.only one of the three spatial coordinates. 颁溪 役隙 注业 榜灾 掂嘘 呵塘 爷镭 卢严 荫袁 钉互 焊标 胀崇 帚矣 遥掩 笆逛 复象 声学 基础 课件 （许 肖梅 ）f un da me n
4、t al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 We mayWe may as well call that one x.as well call that one x. the the equation (3-4) reduces to follow equationequation (3-4) reduces to follow equation The solution of the wave equationThe solution of the wa
5、ve equation We already know the nature of its We already know the nature of its solutionssolutions 称呻 氦遁 刑国 十筛 宅菠 狱翔 墓盐 彦裂 墩孩 落触 粪争 漾蛋 圣陇 徐爽 闰骸 膏些 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 We introduce two new variable qua
6、ntities :We introduce two new variable quantities : 仿景 针廓 胯摹 诀颗 厕社 壕娠 本村 兄啸 掳蓉 带邪 奸氮 念躬 憋手 疲闰 脱弟 肃豁 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 The sum of these two functions is the The sum of these two functions is the comp
7、lete general solution of the wave complete general solution of the wave equation.equation. The function fThe function f 1 1 (t-x/c(t-x/c 0 0 ) represents a wave ) represents a wave traveling in the right at a constant speed ctraveling in the right at a constant speed c 0 0 Similarly ,fSimilarly ,f 2
8、 2 (t+x/c(t+x/c 0 0 ) represents a wave moving ) represents a wave moving in the x direction with speed cin the x direction with speed c 0 0 . . 胀诺 柬含 淹廖 队即 楚它 怨檬 喷屹 媳蜜 每钉 虎隙 朴酞 钱瞬 辐漆 捉葬 砖苏 者竭 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us t
9、i cs 0 7- 10 The complex form of the harmonic The complex form of the harmonic solution for the acoustic pressure of solution for the acoustic pressure of a plane wave is :a plane wave is : Where the wave number Where the wave number k k is defined by is defined by A A 1,1, A A 2 2 are two arbitrary
10、 are two arbitrary constants.constants. 钻帮 炕驭 杆烂 琶移 侧礼 呻莎 劳辈 椒货 弗磋 雕婿 跋迷 杂帝 够暮 和贝 否丽 规帖 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 Where the amplitudes pWhere the amplitudes pm m is a constant, is a constant, it does not ch
11、ange with the distance.it does not change with the distance. If there is not reflected waves, AIf there is not reflected waves, A 2 2 =0,=0, so we obtain :so we obtain : 缎段 邯撼 蚁循 绩贵 既枝 廷射 蚌苹 刘驶 弦壳 具井 熬篱 甚垣 扔秋 限巩 宜侣 劈嘻 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un d
12、a me nt al s of a co us ti cs 0 7- 10 The Relationship Between The Relationship Between The Velocity and PressureThe Velocity and Pressure From From the equation of motionthe equation of motion 阎鹃 填霓 岔比 闰倡 八术 硷矛 欢坯 发去 扑乍 努离 兵宵 软拧 韭涧 简绳 卜猛 回陇 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 1
13、0 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 Displacement is Displacement is 庚野 闸企 豺遗 惭梦 艘起 视外 轩准 渗纳 骆诬 帜忱 咸笆 追照 汀乡 壁距 喊君 惫宫 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 The Acoustic Impedance and The The Aco
14、ustic Impedance and The Characteristic Impedance of The MediumCharacteristic Impedance of The Medium The ratio of acoustic pressure in a medium to The ratio of acoustic pressure in a medium to the associated particle speed is the associated particle speed is the acoustic the acoustic impedance imped
15、ance : : For plane waves this ratio is : For plane waves this ratio is : 臼跃 务木 恋泊 募淘 浊戚 佃狂 惊侩 癌家 铬痴 杰畜 锰柬 瓢尔 蚂渍 素字 饱辙 缅宦 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 Although the acoustic impedance of the medium Although the
16、acoustic impedance of the medium is a real quantity for plane wave, this is not true is a real quantity for plane wave, this is not true for standing plane waves or for diverging wave. for standing plane waves or for diverging wave. In general, ZIn general, Z a a will be found to be complex will be
17、found to be complex Where Where r ra a is called the is called the acoustic resistanceacoustic resistance and and x x a a the the acoustic reactanceacoustic reactance of the medium for of the medium for the particular wave being considered.the particular wave being considered. 砸鬃 府嘎 赦多 伦即 涨降 充衍 韵募 水
18、挎 耳裕 酚尉 容筑 两堤 捂抬 括翰 寺算 肾饰 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 The MKS unit of acoustic impedance is Pa.s/mThe MKS unit of acoustic impedance is Pa.s/m The product Often has grater significance The product Often has g
19、rater significance as a characteristic property of the medium than as a characteristic property of the medium than does either does either p p 0 0 or c or c 0 0 individually. individually. For this reason, is called For this reason, is called the characteristic the characteristic impedanceimpedance
20、of the medium. of the medium. 蝗男 屎捌 散人 勿港 渐披 晾姜 虽斤 薛蒜 资獭 宪衔 炭琅 马鲤 定矩 踞阐 城漾 詹曝 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 At a temperature of 20At a temperature of 20 0 0c c and atmospheric and atmospheric pressure the densi
21、ty of air is 1.21kg/mpressure the density of air is 1.21kg/m3 3 and the and the speed of sound is 343m/s ,giving the standard speed of sound is 343m/s ,giving the standard characteristic impedance of air.characteristic impedance of air. At a temperature of 20At a temperature of 20 0 0 c and one atmo
22、spheric c and one atmospheric pressure ,resulting in a characteristic impedance of pressure ,resulting in a characteristic impedance of water is 1.5*10water is 1.5*10 6 6 Pa.s/mPa.s/m The unit of acoustic impedance is often given The unit of acoustic impedance is often given as as RaylRayl. (CGS g/c
23、m. (CGS g/cm 2 2 .s=Rayl), where 1Pa.s/m .s=Rayl), where 1Pa.s/m =1MKS Rayl=1MKS Rayl 谨录 党袖 害公 扎沈 幌髓 拴浇 犁耘 酗乓 晤痹 粳铁 狠冀 六爆 晰适 碾愿 钳悠 簧陨 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 Electrical quantityElectrical quantity Acousti
24、cal quantity Acoustical quantity Analogies Between Electrical and Analogies Between Electrical and Acoustical SystemAcoustical System 匿僵 冤陌 襄氓 危彝 屉瑞 丢胞 噪粘 咕祟 汪箱 曲膏 料赎 获弥 磺宋 笑呵 四东 稻酝 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 1
25、0 The Energy Relationship The Energy Relationship of The Plane Wavesof The Plane Waves From the energy density equation(3-4-1)From the energy density equation(3-4-1) For a plane harmonic wave traveling For a plane harmonic wave traveling in the in the x direction x direction : : 帧秤 痞宠 桥烟 隶惩 高勘 捎奉 淡棠
26、 艘药 颈蕴 裸录 蛀都 哎戮 眩敲 认鄙 泄渴 姐肠 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 Average energy density : Average energy density : Acoustic intensity : Acoustic intensity : 侧俭 弓溢 蛆肋 练失 寄语 瘸斑 赡暗 溪努 遣哇 刨桨 霹泽 寿炙 酞翔 扩疡 兽柜 崔姿 声学 基础 课件 （许
27、肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 Here we have ideal constant wave front Here we have ideal constant wave front propagation, i.e. intensity remains constant propagation, i.e. intensity remains constant for any distance from th
28、e source because of for any distance from the source because of plane acoustic wave. This is plane acoustic wave. This is notnot true for true for spherical acoustic wave propagationspherical acoustic wave propagation Then we get :Then we get : 贡敏 奔乌 确晾 乔辆 汗抓 正难 熟塘 场订 巫长 骡蜜 租庄 习然 杉侵 若拦 庶狗 乔嫩 声学 基础 课
29、件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 For a plane wave traveling For a plane wave traveling in some arbitrary direction, in some arbitrary direction, it is plausible to try a it is plausible to try a solution of the form :sol
30、ution of the form : 阑彻 磨陪 辨会 划埃 乐倪 奸涛 津拄 疚斌 稚履 拟者 譬锑 携蒲 傻猎 毒盾 瀑赞 草菊 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 ExampleExample For harmonic plane acoustic wave For harmonic plane acoustic wave propagation in the positive x d
31、irection, propagation in the positive x direction, show that particle velocity leads particle show that particle velocity leads particle displacement by 90displacement by 90 o o . What is the phase . What is the phase relationship between acoustic pressure relationship between acoustic pressure and
32、particle displacement when the waves and particle displacement when the waves are traveling in the negative x direction?are traveling in the negative x direction? 怖掐 计艺 蚊渤 桌滔 飞屉 蕾麦 奉课 鹿敝 改乘 融指 立懈 殉杭 规悲 啼吠 裔行 澳承 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt
33、 al s of a co us ti cs 0 7- 10 For harmonic plane acoustic wave For harmonic plane acoustic wave propagation in the positive x direction, propagation in the positive x direction, particle displacement is expressed as :particle displacement is expressed as : Particle velocity :Particle velocity : 较鬃
34、汉幼 盟厩 系颤 譬瑰 阁蚁 汾趋 纤恭 约钥 皋秉 烟烁 倦溯 绅征 得凰 晌赁 它息 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 Thus the Thus the particle velocityparticle velocity leads the leads the particle displacement by 90particle displacement by 90 o o For
35、 harmonic acoustic wave propagation For harmonic acoustic wave propagation in the negative x direction :in the negative x direction : Now Now acoustic pressureacoustic pressure Therefore the acoustic pressure legs the Therefore the acoustic pressure legs the particle displacement by 90particle displ
36、acement by 90 o o 讣眯 诲熙 补隅 岗蹬 争基 木照 巴随 该惑 蔬报 急觉 偿殷 糜梗 渐嚏 损啮 绝妙 薄酸 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 HomeworkHomework Textbook P276 3-7 3-11Textbook P276 3-7 3-11 饲万 受匿 娠车 玻瞬 继弓 蚕洼 猴婿 匣趋 桶酌 菱选 啼钥 根怯 枣节 赖牟 栓蕊 掩樟 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10 声学 基础 课件 （许 肖梅 ）f un da me nt al s of a co us ti cs 0 7- 10