外文翻譯--平面波

1、<p><b>　　外文部分</b></p><p><b>　　Chapter2</b></p><p>　　Plane waves</p><p>　　2.1 Introduction</p><p>　　In this chapter we present the foundati

2、ons of Fourier acoustics-plane wave expansions.This material is presented in depth to provide a firm foundation for the rest of the book ,introducing concepts like wavenumber space and the extrapolation of </p>&l

3、t;p>　　wavefields from one surface to another .Fouries acoustics is used to derive some famous tools for the radiation from planar sources; the Rayleigh integrals ,the Ewald sphere construction of farfield radiation, t

4、he first product theorem for arrays, vibrating plate radiation, and radiation classification theory. Finally,a new tool called supersonic intensity is introduced which is useful in locating acoustic sources on vibrating

5、structures.We begin the chapter with a review of some fundamentals; t</p><p>　　2.2 The Wave Equation and Euler’s Equation</p><p>　　Let p(x,y,z,t) be an infinitesimal variation of acoustic pressu

6、re from its equilibrium value which satisfies the acoustic wave equation</p><p><b>　　(2.1)</b></p><p>　　for a homogeneous fluid with no viscosity .c is a constant and refers to the s

7、peed of sound in the medium .At c=343 m/s in air and c=1481 m/s in water. The right hand side of Eq.(2.1) indicates that there are no sources in the volume in which the equation is valid. In Cartesian coordinates</p&

8、gt;<p>　　A second equation which will be used throughout this book is called Euler’s equation,</p><p><b>　　(2.2)</b></p><p>　　Where (Greek letter upsilon) represents the veloc

9、ity vector with components ,,;</p><p><b>　　(2.3)</b></p><p>　　where i j and k are the unit vectors in the the x, y, and z directions, respectively, and the gradient in terms of the u

10、nit vectors as </p><p><b>　　(2.4)</b></p><p>　　We use the convention of a dot over a displacements quantity to indicate velocity as is done in Junger and Feit. The displacements in t

11、he three coordinate directions are given by u, v, and w .</p><p>　　The derivation of Eq.(2.2) is useful in developing some understanding of the physical meaning of p and . Let us proceed in this direction.&l

12、t;/p><p>　　Figure2.1 : Infinitesimal volume element to illustrate Euler’s equation</p><p>　　Figure 2.1 shows an infinitesimal volume element of fluidxyz, with the x axis as shown .All six faces exp

13、erience forces due to the pressure p in the fluid. It is important to realize that pressure is a scalar quantity. There is no direction associated with it .It has units of force per unit area , or Pascals.The following i

14、s the convention for pressure,</p><p>　　P﹥0 → Compression</p><p>　　P﹤0 → Rarefaction</p><p>　　At a specific point in a fluid .a positive pressure indicates that an infinitesimal vol

15、ume surrounding the point is under compression ,and forces are exerted outward from this volume. It follows that if the pressure at the left face of the cube in Fig. 2.1 is positive, then a force will be exerted in the p

16、ositive x direction of magnitude p(x,y,z)yz.</p><p>　　The pressure at the opposite face p(x+x,y,z)is exerted in the negative x direction. We expand p(x+x,y,z)in a Taylor series to first order, as shown in th

17、e figure .Note that the force arrows indicate the direction of force for positive pressure .Given the directions of force shown,the total force exerted on the volume in the x direction is </p><p>　　Now we in

18、voke Newton’s equation ,f =ma =m,where f is the force, and is the fluid density, yielding</p><p>　　Carrying out the same analysis in the y and z directions yields the following two equations:</p><

19、;p><b>　　and</b></p><p>　　We combine the above three equations into one using vectors yielding Eq(2.2) above, Euler’s Equation.</p><p>　　2.3 Instantaneous Acoustic Intensity </

20、p><p>　　It is critical in the study of acoustics to understand certain energy relationships. Most important is the acoustic intensity vector. In the time domain it is called the instantaneous acoustic and is de

21、fined as </p><p>　　, (2.5)</p><p>　　with units of energy per unit time (power) per unit area, measured as (joules/s)/ or watts/. </p><p>　　The acoustic in

22、tensity is related to the energy density e through its divergence, </p><p>　　, (2.6)</p><p>　　where the divergence is </p><p><b>　　(2.7)</b>&

23、lt;/p><p>　　The energy density is given by</p><p><b>　　(2.8)</b></p><p>　　whereis the fluid compressibility,</p><p><b>　　(2.9)</b></p><p

24、>　　Equation (2.6) expresses the fact that an increase in the energy density at some point in the fluid is indicated by a negative divergence of the acoustic intensity vector; the intensity vectors are pointing into th

25、e region of increase in energy density. Figure 2.2 should make this clear.</p><p>　　If we reverse the arrows in Fig. 2.2, a positive divergence results and the energy density at the center must decrease, tha

26、t is,﹤0. This case represents an apparent source of energy at the center.</p><p>　　Figure2.2: Illustration of negative divergence of acoustic intensity.</p><p>　　The region at the center has an

27、increasing energy density with time ,that is, an apparent sink of energy.</p><p>　　2.4 Steady State</p><p>　　To consider phenomena in the frequency domain ,we obtain the steady the steady state

28、solution through transforms </p><p><b>　　(2.10)</b></p><p>　　leading to the steady state solution </p><p><b>　　(2.11)</b></p><p>　　Equation (2.1

29、0) can be differentiated with respect to time to yield the important relationship</p><p><b>　　so that </b></p><p><b>　　(2.12)</b></p><p>　　where the calligra

30、phic letter f represents the Fourier transform of the time domain wave equation,Eq,(2.1), yielding the Helmholtz equation</p><p><b>　　(2.13)</b></p><p>　　where the acoustic wavenumbe

31、r is k=w/c,the frequency is given by ,</p><p>　　and is the function (x,y,z,).For simplicity of notation we drop the bar above the variable. It will be clear from the context of the discussion if the quantit

32、y is in the frequency or in the time domain. The Fourier transform of Euler’s equation, Eq .(2.2), becomes, in the frequency domain</p><p><b>　　(2.14) </b></p><p>　　where Eq.(2.12) h

33、as been used again for the time derivative.</p><p>　　2.5 Time Averaged Acoustic Intensity</p><p>　　Now consider the intensity relationship for steady state fields .This is defined as the averag

34、e of the instantaneous intensity over a period T, where T=1/f and f is the excitation frequency:</p><p><b>　　(2.15)</b></p><p>　　Using complex variable notation this relationship be

35、comes</p><p><b>　　(2.16)</b></p><p>　　where stands for complex complex conjugate and Re for the real part .The one-half results from the time average process . is the average power o

36、ver one period passing through unit area. For example , the x component of this flow represents the power passing through an element of area .</p><p>　　Important in this chapter is the radiation from planar

37、 radiators .Of particular interest is the power flow crossing an infinite plane. For example, consider the total power crossing the corrdinate plane z=0, a quantity expressed is watts or joules persecond .We use the symb

38、ol to represent the total power in watts crossing the boundary:</p><p><b>　　(2.17)</b></p><p>　　If there are no sources in the upper half space, then is the total power radiated by

39、 has the same power passing through it, since is no absorption in the fluid and there are no sources above the boundary.</p><p>　　The equation of continuity,Eq.(2.6), becomes</p><p><b>　　(

40、2.18)</b></p><p>　　By the definition of stesdy state the energy density at time at time T is the same as the density at time 0, so that we have </p><p><b>　　(2.19)</b></p>

41、;<p>　　This means that in a source-free field the divergence of the time averaged acoustic intensity must always be zero. The only way the intensity field can have a non-zero divergence is if there are sources or

42、sinks of energy within the medium, or losses in the medium.</p><p>　　Plane Wave Expansion </p><p>　　We turn now to plane wave solutions of the wave equation in Cartesian coordinates .Th

43、ese solutions will be useful in the study of sources which are planar (or nearly planar) in geometry such as vibrating plates .We note that Eq.(2.1) is very similar to the equation for a vibrating string;</p><

44、p><b>　　(2.20)</b></p><p>　　whereis normal displacement of the string, and is the wave speed, a constant. A solution to this equation is given by </p><p><b>　　(2.21)</b&g

45、t;</p><p>　　where A and B are arbitrary constants. For this solution to satisfy Eq.(2.20) we must have</p><p><b>　　(2.22)</b></p><p>　　We introduce the string solution t

46、o understand the plane wave solutions of Eq.(2.1). In Eq .(2.21) is called the wavenumber in the x direction.</p><p>　　Consider the phase term in Eq. (2.21) given by . We track the crest of a wave traveling

47、down the string by choosing a constant value of phase and then following it as a function of position and time .The position of the crest .Choosing arbitrarily, is given by . Thus , is the velocity of the crest in the

48、positive x direction and is called the phase velocity of the wave. The solution corresponding to the second term in Eq. (2.21) is a wave traveling in the negative x direction. At a fixed time </p><p><b&g

49、t;　　(2.23)</b></p><p>　　is the wavelength in the x direction and is the distance over which the phase of the wave changes by when time is held constant.</p><p><b>　　第二章</b><

50、;/p><p><b>　　平面波</b></p><p><b>　　2.1介紹</b></p><p>　　在這章中我們提出傅立葉平面波擴(kuò)展的基礎(chǔ)。這些觀點(diǎn)的提出以有利于像波速空間和推理這樣的書(shū),引入概念提供一個(gè)堅(jiān)固的基礎(chǔ).從一個(gè)面到另一個(gè)面.傅立葉聲學(xué)的勵(lì)磁波被用來(lái)作為平面的來(lái)源.從輻射引出一些著名的理論； 整流場(chǎng)輻射的

51、微量部分,the Ewald半球建設(shè)，陣列的第一定理，振動(dòng)盤(pán)輻射和輻射分類(lèi)理論。 最后，在結(jié)構(gòu)振動(dòng)時(shí)定位聲音的來(lái)源時(shí)也是有用的,從而引出一個(gè)新的概念被稱(chēng)為超音速?gòu)?qiáng)度。下面我們從一些基本概念引出平面波方程，Euler’s方程，和聲學(xué)的強(qiáng)度的概念。</p><p>　　2.2平面波方程和Euler’s方程</p><p>　　讓滿(mǎn)足平面波方程的其均衡值成為無(wú)窮小的變化聲波的壓力p ( x，y，

52、z，t )</p><p><b>　　(2.1)</b></p><p>　　因?yàn)闆](méi)有粘性.c的一種同類(lèi)的流體在水中和在空氣中的速度指的都是中等的.時(shí)在空氣中和在水中的聲速分別是c=343 m/s和c=1481 m/s。Eq.(2.1)的右邊方程表明在其中有效的區(qū)域中沒(méi)有來(lái)源。笛卡兒派認(rèn)為</p><p>　　將作為第二方程在整個(gè)書(shū)中被使用稱(chēng)

53、為Euler’s方程，</p><p><b>　　(2.2)</b></p><p>　　用組成部分,,代表速度矢量</p><p><b>　　(2.3)</b></p><p>　　i j和k在該處表示x，y，和z方向矢量， </p><p><b>　　(2

54、.4)</b></p><p>　　我們使用關(guān)于置換的一個(gè)點(diǎn)的結(jié)論來(lái)作為表明速度數(shù)量在Junger和Feit中被完成。 三個(gè)置換協(xié)調(diào)方向用u，v，和w表示。</p><p>　　Eq.(2.2)的由來(lái)在對(duì)p的物理意義的理解時(shí)是有用的。 讓我們朝著這方向進(jìn)行研究。</p><p>　　Figure2.1： 用無(wú)窮小的元素說(shuō)明Euler’s方程</p&

55、gt;<p>　　式(2.1)中用x,y,z表示無(wú)窮小的元素量，用x表示壓力p在流體中方向.六個(gè)面都受力。 總壓力是一等級(jí)的數(shù)量。 但沒(méi)有與它有關(guān)的方向.力在每個(gè)單元區(qū)域會(huì)隨著壓力變化，</p><p><b>　　P﹥0→壓縮</b></p><p><b>　　P﹤0→稀少</b></p><p>　　流

56、體內(nèi),在壓力下a的一個(gè)具體的點(diǎn)的周?chē)囊粺o(wú)窮小的區(qū)域內(nèi)是壓縮的,力在這點(diǎn)方向是向外。 如果在第2.1圖中的立方體的左面的壓力是向里的，那么一種力將朝著 p(x,y,z)yz.的x方向延伸。在相反面p(x+x,y,z)的壓力朝著負(fù)值x方向被延伸。 我們首先命令p(x+x,y,z)一個(gè)Taylor系列作為被引入量.力的箭頭方向表示力的正方向.，力在x的正方向延伸.</p><p>　　我們現(xiàn)在調(diào)用Newton’s方程

57、, f =ma =m ，在其中f是力，是流體密度，得出</p><p>　　在y和z方向?qū)嵭型瑯拥姆治龅贸鋈缦聝蓚€(gè)方程：</p><p><b>　　和，</b></p><p>　　我們把上述的三個(gè)方程結(jié)合到在上面的出Eq(2.2)就得出Euler’s方程。</p><p>　　2.3瞬時(shí)聲學(xué)的強(qiáng)度</p>

58、<p>　　它的研究重點(diǎn)在理解聲學(xué)中的一定的能量關(guān)系。重要的是大多數(shù)聲學(xué)的強(qiáng)度矢量.在時(shí)間領(lǐng)域中被使用并且被定義為瞬時(shí)聲學(xué)</p><p>　　, (2.5)</p><p>　　能量在單元時(shí)間(力)和單元區(qū)域，單位為(焦耳/ 秒)/或者 watts/ .</p><p>　　聲學(xué)強(qiáng)度其分歧與能量密度有關(guān)，<

59、/p><p>　　, (2.6)</p><p><b>　　分歧所在的地方</b></p><p><b>　　(2.7)</b></p><p><b>　　能量密度</b></p><p><b>　

60、　(2.8)</b></p><p><b>　　流動(dòng)的可壓縮性，</b></p><p><b>　　(2.9)</b></p><p>　　方程( 2.6 )表明在一些的能量密集的流體中強(qiáng)度矢量是減弱的.聲學(xué)強(qiáng)度矢量的分歧表明； 強(qiáng)度矢量在能量密度低的區(qū)域是增加的。 式(2.2)表達(dá)的很清楚。</p&

61、gt;<p>　　如果我們?cè)趫D2.2中翻轉(zhuǎn)箭頭方向，中心的能量密度一定減少,分歧也可能減少，﹤0 這種情況表明中心明顯是一個(gè)能量來(lái)源。</p><p>　　式2.2:表示聲學(xué)的強(qiáng)度的分歧。在該地區(qū)中心有一個(gè)增加的能量密度時(shí)間,，該時(shí)間內(nèi)是能量是減弱的.</p><p><b>　　2.4平衡狀態(tài) </b></p><p>　　在頻

62、率領(lǐng)域,我們考慮通過(guò)解決轉(zhuǎn)變現(xiàn)象獲得穩(wěn)定的狀態(tài).</p><p><b>　　(2.10)</b></p><p><b>　　穩(wěn)定的狀態(tài)</b></p><p><b>　　(2.11)</b></p><p>　　方程( 2.10 )關(guān)于時(shí)間得出重要的關(guān)系</p>

63、;<p><b>　　因此</b></p><p><b>　　(2.12)</b></p><p>　　Calligraphic理論中f代表傅立葉轉(zhuǎn)變，得出Helmholtz關(guān)于時(shí)間的平面波方程，Eq,(2.1 )</p><p><b>　　(2.13)</b></p>

64、<p>　　的波的波速為k = w / c ,頻率為。 如果在頻率中或者在時(shí)間領(lǐng)域中，討論的結(jié)果是正確的。傅立葉轉(zhuǎn)變得出的Euler’s方程，Eq .(2.2 )在頻率領(lǐng)域中和Eq.(2.12)在時(shí)間領(lǐng)域中被生物研究充分利用.</p><p><b>　　(2.14)</b></p><p>　　2.5平均時(shí)間的強(qiáng)度</p><p>

65、;　　現(xiàn)在考慮強(qiáng)度關(guān)系在穩(wěn)定的狀態(tài)領(lǐng)域.在這時(shí)間區(qū)域被定義為強(qiáng)度的平均值，在其中T=1/f和f是激勵(lì)頻率：</p><p><b>　　(2.15)</b></p><p>　　使用可變符號(hào)得出關(guān)系式為</p><p><b>　　(2.16)</b></p><p>　　用可變符號(hào)表示的地方為真正

66、的部分.從時(shí)間開(kāi)始和到一半時(shí)間的平均過(guò)程, 穿過(guò)單元區(qū)域的平均功率超過(guò)平均功率。 例如，x的組成部分的穿過(guò)該區(qū)域的一個(gè)元素的力的功率。</p><p>　　在這章中重點(diǎn)是平面散熱器.特殊的地方該力是橫越一個(gè)無(wú)窮的平面的力量流。 例如，在z=0時(shí)瓦特和焦耳認(rèn)為整個(gè)平面的被該力量橫越.我們認(rèn)為全部能量通過(guò)該平面，用符號(hào)表示橫越邊界的力：</p><p><b>　　(2.17)<

67、;/b></p><p>　　如果沒(méi)有上面的一半空間中的能量，那么被放射的整個(gè)的能量就會(huì)有穿過(guò)它的相同的能量，當(dāng)流體在邊界上沒(méi)有能量的來(lái)源時(shí)。</p><p>　　連續(xù)性的方程，Eq.(2.6 )，變成</p><p><b>　　(2.18)</b></p><p>　　stesdy理論在時(shí)間T內(nèi)能量密度在零時(shí)間

68、與液體密度一樣，這樣我們有 (2.19)</p><p>　　這意味著在一個(gè)自由領(lǐng)域中平均時(shí)間內(nèi)聲波的強(qiáng)度必須總是零。 如果強(qiáng)度能量有非零的唯一的原因是能量源在媒介內(nèi)減弱，能量在媒介的損失。</p><p><b>　　2.6平面波擴(kuò)展</b></p><p>　　我們現(xiàn)在開(kāi)始研究笛卡兒的平面波

69、方程.該結(jié)論將在研究諸如振動(dòng)的幾何學(xué)中平面的來(lái)源時(shí)是有用的.類(lèi)似于Eq.(2.1)振動(dòng)的方程；</p><p><b>　　(2.20)</b></p><p>　　在置換串的地方是而不是波度， 對(duì)于方程的一種解決辦法</p><p><b>　　(2.21)</b></p><p>　　對(duì)于解決滿(mǎn)

70、足Eq.(2.20 )我們必須有</p><p><b>　　(2.22)</b></p><p>　　我們解釋Eq.(2.1)時(shí)。 Eq.(2.21 )朝著x的方向被定義為波速。</p><p>　　在Eq(2.21)中考慮階段時(shí)期. 我們把通過(guò)選擇階段的值作為峰值.在該位置一個(gè)波到達(dá)頂峰然后下降。 這樣，在x方向上到達(dá)的頂峰的速度被定義波的