声波的辐射课件.ppt

1、1第6章 声波的辐射6.1 脉动球的辐射单极子声源6.2 反向小球源的远场辐射偶极子声源6.3 同相小球源的远场辐射和声柱6.4 点声源和活塞辐射6.5 镜象原理21、辐射场的各种规律：声源与辐射场的关系；声压随距离的变化关系；声源的指向性2、辐射场反过来对声源振动状态的影响：由于辐射声波而附加于声源的辐射阻抗物体振动产生声波声 源辐 射 场反作用36.1脉动小球的辐射单极子声源小球表面振动速度0exp()r auuitka为了方便基本方程球坐标中的声波方程22220pcpt最基本的声源4球坐标)0 ,20 ,(0 r cossinsincossinrzryrx2222010ppct22222

2、222111sinsinsinrrrrrr 5xyzOr(r,)eeer6球对称(,)(,)p rtp r t 222220110pprctrrr时间简谐解(,)()exp()p r tp ri t2221()0ddprk p rr drdrk=/c0令()()rp rr7得到222()()0drkrdr()exp()exp()rAikrBikr(,)exp()exp()ABp r titkritkrrr前一项：向外辐射解；后一项：向原点会聚波。(,)exp()Ap r titkrr8速度场0pt v01rpvir 0 0111exp()rAvitkrc rikr边界条件球面上媒质的速度等于球

3、面振动速度！00 0111Auc aikarr ar avu9200 000 02()|exp()1()1uc auc kaAikaikaAiikaka210 0021|;tan()1c kaAukaka声场和速度场0 0|(,)exp()|111exp()rAp r titkrrAvitkrc rikr10结论：1、低频ka120 000 002|()1Hc kaAuc auka|1|LHAkaA3、频率一定，球面越大，辐射声压越大；球面大小一定，频率越高，辐射声压越大。11脉动球源，辐射与方向无关，单极子辐射音箱中扬声器，低频辐射，可看作单极声源！12声场对脉动球源的反作用辐射阻抗0rr

4、aFS p 球面在外力F作用下作振动，如果忽略向周围媒质中的声辐射，振动方程22mmmddMRKFdtdt球面径向位移dudtudt13考虑向周围媒质中的声辐射，振动球受到声压 p|r=a 声场的反作用力0rr aFS p 新的振动方程22mmmrddMRKFFdtdt200 02(,)exp()()()1Ap r titkrruc kaAikaka140000 020 002()exp()()1()()()1rr aFS pS uc kaikaitkakac kaika S u tka 令220 00 00022;()1()1rrc k ac kaRSXSkaka()()rrrrrdFRiX

5、Z u tZdt 15()mmrmduMRZ uKudtFdt特解时间简谐解(/)mmmrmrFFuRi MKZZZ新的力阻抗(/)()(/)()mrmmmrmrrmmmrmrmZZRi MKZRRi XMKKXRRiM161.Rr声辐射产生振动阻尼力学系统的能量耗损转变成声能；2.Xr声辐射产生质量抗，增加的等效质量为002()1rraXMSka好象有质量为Mr的媒质粘附在球上一切振动！171、低频ka120 002000300()0;(4)4333rrrRc kaSXMaSaaaM0 00 000;0rrrcXRc SMSka辐射很小，同振质量很大，同振质量很小，辐射极大。1819辐射声场

6、的特性|(,)exp()Ap r titkrrq等相位面tkr常数在时刻t，等相位面方程krt常数球面方程球面波！20q振幅随距离的变化|aApr2|aAdpdrr aadpdrpr 远场：0aaprpr 远场的声压变化很小，近似平面波！近场：小的距离变化能引起大的声压变化！1、r=1m处，r=1m 100%aaprpr2111|SPL20logarrApAp22|SPL20log22arrrAApp 12|SPLSPL20log20log220log26dBrrAApp221、r=10m处，r=1m 110|SPL20log1010arrAApp211|SPL20log1111arrrAAp

7、p 12|SPLSPL20log20log101120log1020log110.8dBrrAApp 110%10aaprpr23声强和辐射功率01Re()Re()TpdtTIv0 0|(,)exp()|111exp()rAp r titkrrAvitkrc rikrv,v=0,声强只有径向分量，即声能只沿径向传播242200 0220 01|cos()sin()cos()|2TrAItkrTc rtkrtkrdtkrAc r2220 00 0|2erpAIc rc声强通过任意球面的能量即声功率为220 02|4rAWr Ic与r无关！01|22epApr256.2 反向小球源的远场辐射偶极子

8、声源声偶极子是由两个相距很近，并有相同的振幅而相位相反（即相差180度）的小脉动球源所组成的声源。26q声压两个相同振动小球（反相）在r处产生的声压(,)exp()exp()AAp r titkritkrrr两个特殊方向：1、=/2，即两个振动球的连线的垂直线上的点rr(,)0p r t 272、=0，即两个振动球的连线上的点，并且rl;22llrrrr(,)exp()exp(/2)exp(/2)2exp()sin2Ap r titkrikliklriAklitkrr 分母上：111111;/2/2rrlrrrlr指数里：;22klklkrkrkrkr不能够忽略！28一般cos;cos22ll

9、rrrr2cos(,)exp()sin2iAklp rtitkrr 因为两小球相距很近，当频率不很高，kl1(,)exp()cosklAp rtiitkrr 定义指向性(,)()|cos|(,0,)p rtDp rt29q速度场q声强q辐射声功率22 22200 01|Re()Re()cos2TrrAk lIpv dtTc r0 0111cosexp()rkAlviitkrc rikr222 20 02|sin3rrSSAWI dSI rd dk lc 30单极子与偶极子辐射功率比较222 20 00 02|2|3AAWWk lcc单极偶极子；kll12sin;sin22llrrrr(,)ex

10、p()2cos()Ap r titkrkrsin2l 声程差的一半！34sin(2)(,)exp()sin()Akp r titkrrkq指向性 指向特性：声程差与波长的比值有关 sin(2)()2sin()kDk指向性与波长有关！351、当sin,(0,1,2,.)kmlmm()1D声程差是波长的整数倍！主极大：m=0,=0出现极大的角度arcsin ml副极大：m0,0如果l,不出现副极大！注意：主极大和副极大的幅值大小相同362、当2sin,(1,3,5,.)2kmlmm()0D声程差是半波长的奇数倍！出现极小的角度arcsin2ml主声束的角宽度：=0与第一次出现极小的角度的2倍2ar

11、csin2ll增加，声束变窄！37抑制副极大：要求l；减小主声束的宽度：要求l增加矛盾！l60ol增加：主声束的宽度较小，但出现副极大。3、当kl1,k12(,)exp()Ap r titkrr两小球靠的很近！重要结论3839q自辐射阻抗和互辐射阻抗 每一个小球源的振动状态受到合成声场的影响,它不仅受到自己产生的声场的反作用，还会受到另一个小球源产生的声场的影响小球源 I 11112FFF相当于在它的振动系统上附加了一项辐射阻抗 1111121FZZZu 4011111FZu 小球源 I 自身的辐射阻抗，为自辐射阻抗。111111111FZRiXu 22110 01010110 01010;R

12、c k r SXc kr S12121FZu 小球源 II 在小球 I 上产生的辐射阻抗，它反映了声源之间的相互作用，称为互辐射阻抗。41Z12的计算lIII假定：球源线度都很小，对声波的散射作用很微弱，声源 I 放在声源 II 产生的声场中时，对声场的干扰可以忽略不计，球表面所受的声压近似地认为和该点自由声场声压相等。小球 II 的辐射声场在小球 I 表面处的声压 20 0020212(,)exp()c kr upl tiitkll小球 II 的辐射声场作用在小球 I 表面的力 20 00202121211exp()c kr uFp SiSitkll 42互阻抗 Z12 为 20 00202

13、12101sin()cos()c kruZSklikllu两个小球完全一样 121212ZRiX20 001220 0012()sin();()cos()c krRSklklc krXSklkl43小球源 I 的总辐射阻抗 111ZRiX1111110sincos1;1klklRRXXkrklkl小球源 I 的平均辐射声功率 221101102220 000011sin1221sin12klWRuRuklklc k r Sukl44小球源I的低频辐射功率kl122210 000012Wc k r S u11WW单一小球122WWW单一小球当两个小球间距离较远或者频率较高时，两个小球间的影响已经

14、小得可以忽略。这时组合声源辐射功率等于两个小球单独存在时的辐射功率之和。46q声柱电声技术中广泛应用：由许多小扬声器单元按直线或曲线排列而成的声柱辐射的指向特性。设 n 个体积速度相等、相位相同的小脉动球源均匀分布在一直线上，小球源间距 l，声柱总长度(1)Lnl改变声辐射的指向特性：抑制副极大和减小主声束的宽度是互相矛盾的。47远场声压（rL）1(,)exp()i niiiAp rtitkrr远场21311sin;2 sin;.;2(1)sinnrrlrrlrrnl122(1)(,)exp()1.i ki k nAp rtitkrree4821211(,)exp()1sin()exp(1)s

15、in()sin()exp()sin()i nki kAep rtitkrreAknitkrk nrkAknitkrrk1(1)sin(1)2nrrlrn49远场指向性（rL）sin(2)()sin()knDnkn=4501、当sin,(0,1,2,.)kmlmm()1D声程差是波长的整数倍！主极大：m=0,=0出现极大的角度arcsin ml副极大：m0,0如果l,不出现副极大！注意：主极大和副极大的幅值大小相同512、当sinmnkmln()0Dm为除了n的整数倍以外的整数！出现极小的角度arcsinmn l主声束的角宽度：=0与第一次出现极小的角度的2倍2arcsinnln增加，声束变窄！

16、523、当(21)(21)sin221,2,.mnkmlnm max()1D次极大！第一个次极大位置：3arcsin2n l第一个次极大与主极大的比1112;limsin(3/2)3nDDnn53重要结论1、L增加，主声束宽度下降，但必须增加n，以保证l而不出现副极大；2、L一定，增加n，主声束宽度下降，次极大的幅值下降；54n=4,不同长度声阵的辐射方向图55声柱的能量关系0 0(,)exp()()(,)exp()()rAp rtnitkr DrAv rtnitkr Dc r远场声压和速度场声强22220 0|()()2rAInDc r主方向=02220 0|(0)2rAInc r56如果n

17、个小球随机分布，总声强是n个小球声强之和220 0|(0)2rAInc r主要结论1、由于声波的干涉，声柱使主方向的声强增加n倍！在听堂、剧院的扩声系统有重大应用。2、对很低的频率，kl1,D()=1,由于声波的干涉，声柱使所有方向的声强增加n倍！声柱使低频辐射功率大大增加。2220 0|()2rAInc r57例：不相干小球源的线阵高速公路上汽车噪声模型 Pr0rnnb每个声源的相位是无关的，因此，P点的总平方平均声压等于每个声源的平方平均声压。设每辆汽车的噪声辐射功率为W220 02|4rAWr Ic58|(,)exp()Ap r titkrr222|2Apr220 02|4rAWr Ic

18、20 024Wcpr20 022014()nWcprnbP点的总平方平均声压等于每个声源的平方平均声压5920 022020 022200020 0222000 0000 0014()124(/)124coth44nnnWcprnbWcbrr bnWcbrxnWcrbrbWcbr0/xrb001coth1rbrb6020 0012eWcppbrP点的有效声压随距离01/r衰减！因此，高速公路上汽车噪声能传播很远防治困难！相干声源的叠加2|()2eAppnDr随距离1/r衰减！而且有方向性！616.4 点声源和活塞辐射点源的组合来处理较复杂声源(例如活塞)的辐射!q点声源在空间所辐射的声压 20

19、0 02(,)exp()()()1Ap r titkrruc kaAikakakaacossincoshrr()sincos0002()sincos00000(,)ee2ee2it kriSait kric kup rtid drc kuiddr 利用2cos00011()e2()()ixJxdxJx dxxJ x682()0012(sin)(,)e2sinit kru aJ kap rtirka00 01111rpvpircikr 声强0222210 0021Re()Re()2(sin)1()8sinTrrIpv dtTJ kaac ukarka6970远场的指向性利用11()2J xx12

20、(sin)(,)()(,0,)sinJ kap rtDp rtka低频kazfsin()sin222ffzzkRzzz远场，与距离反比！zf 远近场的分界线远近场临界距离！79近场特性应用于扬声器低频特性测量近场声压起伏很大扬声器测量时应该避免在近场！利用近场特性可以测量扬声器低频特性。低频特性测量的困难：消声室自由场截止频率(75Hz）测量原理0 00(0,)2esinexp22i tkaaptic uik扬声器中心附近(z=0)80低频kazg)因此，扬声器中心附近的声压幅值为0 00napc u ka0 00()(,)2eexp22gi tzRzp z tic uikz因此，扬声器轴线且

21、远场的声压幅值为8120 000 0012gfazpc uc u kazz两者之比;(1)22fanapaazpz测得扬声器中心附近的声压，就可以计算出1m处的声压，前者容易测量而无需很低自由场截止频率的消声室！82活塞的辐射阻抗尽管活塞面振动的速度一致，但活塞面上的声压各点不同！面元dS在面元dS处产生的声压为()000e2it khc kudpidSh活塞面S在面元dS处产生的声压为()0001e2it khSc kupidSh83面元dS受到声场的反作用力rdFpdS 活塞表面受声场的总反作用力0 001ee2i tikhrSSSc kuFpdSidSdSh 最后结果2110 0022(

22、2)2(2)1e2(2)i trJkaKkaFca uikaka 令111122()2()()1;()J xK xR xXxxx 8411();()R xXx活塞辐射的阻函数活塞辐射的抗函数85200011(2)(2)ei trFca uRkaiXka 活塞辐射阻抗20011(2)(2)rrFZcaRkaiXkau 22001001(2);(2)rrRca RkaXca Kka辐射阻和辐射抗低频辐射阻和辐射抗(ka5)3200022;()rraRcaXka低频辐射功率2222200001()24Lrc kWR uau与频率的平方成正比！87高频辐射功率222000011()22LrWR uca

23、 u与频率无关！声源的高频辐射比低频辐射容易！886.5 镜象原理刚性平面边界前的点源辐射问题平面波：遇到硬边界，声压同相；速度反相。球面波：在平面界面的反射很复杂！89P点的声波=直达波+发射波；反射波可等效成由刚性壁后的一个虚源发出的同相位声波P数学上：在边界条件（硬边界，法向速度为0）下解非齐次波动方程Green函数。90P点声压1212(,)exp()exp()AAp ritkritkrrr空间声压增加一倍！低频辐射功率为不存在硬边界时的4倍！能量从哪里来？声场对声源的反作用提高了声源的辐射能力！在硬边界附近更容易辐射声波房间声学中，声源放在墙角！水空气声源91柔软平面边界前的点源辐射

24、问题P平面波：遇到软边界，声压反相；速度同相。P点的声波=直达波+发射波；反射波可等效成由软性壁后的一个虚源发出的反相位声波92数学上：在边界条件（软边界，声压为0）下解非齐次波动方程Green函数。P点声压1212(,)exp()exp()AAp ritkritkrrr低频辐射功率很低！水空气声源93Radiation from a Baffled Piston(model for a loudspeaker)A circular piston in a large baffle is a good starting approximation for investigating the r

25、adiation of sound from a boxed loudspeaker.The far-field pressure radiated by a baffled piston depends on the radius of the piston a,the frequency(through the wavenumber )and the direction (with =0o being directly in front of the piston)according to()0102(sin)(,)e2sinit krkQJ kap rticrka障板上的简单点源指向性9

26、4Low frequency(ka1)As the frequency gets higher,but assuming the speaker diameter does not change,the value of ka increases and the speaker becomes directional.That is,the sound energy produced by the speaker becomes channeled into a preferred direction and very little energy is radiated at other di

27、rections.In the animation below the radiated sound is pretty much contained within a cone of 55o from the center axis.Also,from the darkness of the contour shading(darker means higher pressure)you can see that the radiated sound field is strongest right in front of the speaker and weakens as you mov

28、e to either side.98Animation of sound field99Directivity pattern 3-D directivity pattern100High frequency(ka1)As the frequency becomes even higher(and ka becomes much bigger than 1)the sound field radiated by the speaker becomes even narrower and side lobes appear.Now the main lobe of radiated sound

29、 is limited to about 20o on either side of the central axis,and the pressure amplitude falls off rapidly as you move away from the central axis.Notice that the side lobes are much lower in amplitude than the main lobe(the darker the contour the higher the pressure-louder the sound).Also notice that

30、the sound waves in the side lobes have the opposite phase as the sound wave in the main lobe.If you were using the same speaker(a large woofer)to produce both low and high frequencies,you would definitely notice a severe drop-off in the loudness of the higher frequencies as you step away from in fro

31、nt of the speaker.Fortunately,well designed loudspeaker systems dont attempt to send all frequencies through the same speaker so you probably wont observe this problem.101Animation of sound field102Directivity pattern 3-D directivity pattern103Measured Data for a Real Loudspeaker The animation shows

32、 actual directivity data measured for a 4-inch boxed loudspeaker.At low frequencies(250 Hz)the speaker radiates sound equally well in all directions.At higher frequencies(10 kHz)the speaker radiates all of its sound in front-the sound level behind the speaker is almost 25 dB lower than the level in

33、front,indicating that much more sound energy is being radiated directly in front and very little behind.104105This tendency for a loudspeaker to become directional at higher frequencies is one of the main reasons for using a cross-over network when designing a multi-speaker system.The goal is to hav

34、e each speaker in the system radiate sound with approximately the same spatial distribution over its own frequency range,so that the entire frequency output of the speaker(20-20,000 Hz)is radiated evenly into the listening space.106The cross-over filter sends low frequencies(really small k)to the la

35、rger speaker(medium sized a)so that it is fairly omnidirectional(ka5 is a typical upper limit).Likewise,mid-range frequencies(larger k)are sent to the mid-size speaker(smaller a)so that it has the same upper limit of directivity(ka5).Finally,the highest frequencies(large k)are sent to the smallest s

36、peaker(really small a)so that it also has approximately the same spatial directivity(tiny speaker size ensures that ka is still 5).The more omni-directional you want each speaker to be,the more different sized speakers you will need in order to cover the entire frequency range.2107The table below compares the speaker sizes and frequency ranges.(Kenwood JL-840W 4-way speakers)Speaker Diameter(cm)Frequencies(Hz)Cutoff kaWoofer 30 20-2,000 5.5 mid-range 12 2,000-5,000 5.5 Tweeter 6 5,000-10,000 5.5 Super-tweeter 3 10,000-20,000 5.5