固體地球物理學(xué)導(dǎo)論7

1、第七章,第七章 地球內(nèi)部的地震波場,地震與介質(zhì)的彈性性質(zhì) 地震波及其特征 地震體波的傳播 地震面波及其特征 地球的自由振蕩 天然地震,第七章,地震波場在地球物理學(xué)中占有重要地位，當(dāng)今在研究地球內(nèi)部，地震活動的機(jī)制、資源與能源的地震勘探以及海、陸工程建設(shè)中均主要依賴于人工源或天然源激發(fā)的地震波場效應(yīng)。 震源(包括人工源或天然源)激勵出來的各種類型的地震波，在地球內(nèi)部各圍層介質(zhì)中或沿其表面?zhèn)鞑?，依?jù)這些波動的走時，頻率和振幅特性或波的頻散，可以推測地球內(nèi)部各圈層介質(zhì)的速度分布和結(jié)構(gòu)。 根據(jù)地震臺站紀(jì)錄的地震事件，可推斷震源的參數(shù)(震源深度、震中位置、發(fā)震時刻、地震震級和震中距離等)和震源機(jī)制，并進(jìn)

2、一步了解產(chǎn)生這種機(jī)制的應(yīng)力狀態(tài)。如果發(fā)生的地震足夠大，則地球作為一個整體會激發(fā)出各種振型的振蕩，并可通過它來探討地球內(nèi)部的性質(zhì)。,地震學(xué)研究及其意義,第七章,7.1 地震與介質(zhì)的彈性性質(zhì) 7.1.1 地震震源及地震波 Earthquake sources Physically, earthquake sources are the abrupt release of the potential elastic energy stored in rocks over a period ranging from a few years to thousands of years. Only a s

3、mall part of the energy converts heat to the surround rocks near the source, the most of energy is radiated away as elastic waves. In fact, rocks at an earthquake source generate plastic deformation but elastic deformation at the moment of earthquakes occurring. Up to now, no earthquake that locates

4、 deeper than 670 km has been observed.,地震震源,第七章,Seismic waves Seismic waves are the elastic wave from the source. They are of various types. While traveling through the earth, these waves are influenced by the properties of the media they pass. We are able to understand and analyze the influence by

5、applying mathematical and physical methods. In addition, we can determine the earths structures: the crust, the mantle, the outer core, the inner core, as well as the lateral changes near the surface.,地震波,第七章,Receiving The main task of receiving is to record the seismic waves at desired positions to

6、 study earthquakes. It includes the sampling technology and the receiving system consisting of seismographs(地震儀), geophones(檢波器), and other instruments. The quality of recorded data is important to analysis and interpretation of the seismic waves and determination of earthquake sources.,地震波的接收,第七章,7

7、.1.2 板塊構(gòu)造與地震 At present the theory of plate tectonics is accepted by most of geoscientists, even though many of its details are still unclear or controversial. We can use a simplified dynamic model to describe the movement of continent. When the material in the mantle is heated, it expands and becom

8、es lighter. In spite of its high viscosity（粘性）, it rises more or less vertically in some places, especially under the oceanic ridges. With its losing pressure and heat during traveling upward, the material is forced to travel horizontally. They drag the lithosphere motion. The results of continent c

9、olliding form mountain chains (Himalayas) , and the results of their separating form ocean rifts (East Africa). So major earthquakes often cause near their collided boundaries. In the region of oceanic ridges, where new lithosphere is growing, small earthquakes occur frequently.,板塊構(gòu)造與地震,第七章,板塊構(gòu)造與地震分

10、布圖,第七章,7.1.3 巖石彈性性質(zhì)基本概念 (1)形變 A material occurs deformation(形變） under a force act on itself. If it recovers as the force disappears, it is called elastic material. The shape change is called as elastic deformation. Otherwise, it is called as non-elastic deformation. Whether elastic deformation occur

11、s depends on the magnitude of acting force, the acting period as well as the surrounding temperature. For most materials in the earth, this elastic property only exists in a short period.,彈性概念形變,第七章,(2)應(yīng)力 Stress tensor（張量） Definition: Here stress means a force acts on unit area of a body against the

12、 elastic deformation caused by the action of an external force. Describe any stress needs consider two factors, direction and outer normal(法向）of a face. We generally express it by pst. Here s means the direction of the force and t the outer normal direction on a face. In three dimension orthogonal c

13、oordinate system, we can define stress p as (pxx pxy pxz pyx pyy pyz pzx pzy pzz).,彈性概念應(yīng)力,第七章,The stresses are symmetrical（對稱的）, i.e. only six components of the stress tensor p are independent because pxy= pyx , pyz= pzy , pzx= pxz For a cubic body in x-y-z coordinate system, when the face edges of

14、the body are parallel to coordinate planes, pxx , pyy , pzz are normal stresses and pxy , pxz , pyz , are shear(剪切） stresses.,彈性概念應(yīng)力 (續(xù)),第七章,Pressure At a given point the sum of the normal stresses on any three orthogonal(直角的）planes is a constant (a scalar). The pressure P is defined as P = - (pxx+

15、pyy+ pzz)/3 This is a general definition of the “pressure”. In the special case of a liquid at rest, pxx= pyy= pzz = - P, this is the hydrostatic pressure. In geology, lithostatic pressure is often estimated by using P=gh where is the density, g is the acceleration of gravity, and h is the depth. Bu

16、t it is not always correct near the surface.,彈性概念壓強(qiáng),第七章,(3) Strain（應(yīng)變） tensor Definition: In general, the relative change in the length or in the shape of an object acted by forces is called as strain. This kind of length or shape changes should be recovered after the forces disappear - Elastic defo

17、rmation. Linear strain A rod is 50cm long initially in the direction x-axis. When a force or forces are applied to it, its length increases to 50.2cm in the x-direction. The relative change in length is (50.2-50)/50.,彈性概念應(yīng)變,第七章,To measure the relative change, we define u as the length change in x, x

18、 as the original length of the rod. On any point inner the rod, the linear strain can be defined as exx = u/x u/x,第七章,In XYZ-coordinate system, in same way we can obtain eyy = v/y, ezz = w/z Here exx , eyy ,ezz are normal strains. shear strain,Y f X,第七章,Suppose that the graph shown is as a result of

19、 external forces, the cross section of the body is deformed to the rhombus(菱形) shown by dashed lines, and in the procession all points move parallel to the x-axis. The area of the cross section has not changed, but the shape has. The angle is a measure of this distortion, called shear strain. Here t

20、an = u/ y At the limit while y0 = u/y Consider the variation at another direction = v/x,第七章,The shear strain in the x-y plane is defined as exy = (+)/2 = (u/y+v/x)/2 For three dimension orthogonal coordinates, we also have eyz = (v/z+w/y)/2 exz = (u/z+w/x)/2 Their symmetry gives exy= eyx , exz= ezx

21、, eyz= ezy .,第七章,Dilatation (體膨脹) The sum of normal strains is defined as dilatation = exx+ eyy+ezz The dilatation is a measure of the relative change in volume. For a homogeneous bulk applied by external forces, the relative change in volume is V/(xyz). V=x(1+exx)y(1+eyy)z(1+ezz) -xyz Hence V/(xyz)

22、=exx+eyy+ezz+exxezz+eyyezz+exx eyy +exxeyyezz exx+ eyy+ezz,第七章,(4)Elastic modulus and equations Suppose a body is homogeneous and isotropic, i.e. its properties are independent of both spatial coordinates and directions. Hookes law tell us the stresses in the body are linear combinations of the stra

23、ins. For instance, pxy=aexx+beyy+cezz+dexy+fexz+geyz (a,b,c,d,f,g are constants) According to elasticity theory, we have pxy=2exy , pxz=2exz , pyz=2eyz where is the modulus of rigidity or shear modulus.,第七章,The shear stresses are proportional to the shear strains. On the other hands, the relations o

24、f the normal stresses and normal strains are, e.g., pxx=+2exx where is another elastic modulus. and are called Lam elastic constants. They are difficult to measure directly. For this reason, they are often computed from other elastic parameters.,第七章,All the relations to describing stresses and strai

25、ns can be written in pij=ij+2eij where ij is Kronecker delta-a function. i and j represent x, y or z. when i=j the value of ij is 1, otherwise, 0.,第七章,Other elastic modulass Youngs modulus, E Youngs modulus measures the resistance to extension in a direction. It is defined as E=pxx/exx (if the force

26、 only applied in x-direction) Since pxx=+2exx pyy=+2eyy=0 pzz=+2ezz=0,第七章,To add them Pxx=3+2=(3+2) By symmetry, eyy=ezz =exx+2 eyy , +2eyy=0 so E=(3+2)/ (+),第七章,Incompressibility, This parameter measures the resistance to a change in volume under pressure. It also called bulk modulus. = - dP/d sinc

27、e P= - (pxx+ pyy+ pzz)/3 = - (3+2)/3 = - (3+2) /3 thus = +2 /3,第七章,Poissons ratio, is the ratio of the lateral contraction to the longitudinal extension. Suppose only normal stress pxx acts on the body. Thus = - eyy/exx or = - ezz/exx Because pyy=+2eyy= (exx+eyy+ezz) +2eyy =0 By symmetry, eyy=ezz ,

28、and (exx+2eyy) +2eyy =0 Hence =/(2(+),第七章,In the earth, Poissons ratio ranges from 0.1 to 0.38 near the surface. At hydrostatic pressure equivalent to a depth of 13km it ranges from 0.23 to 0.31, except for quartzite =0.15. In the absence of any other information, it is often assumed that =0.25. Oth

29、er relations =E/(1+)/(1-2) =E/(2(1+) =E/(3(1-) /=2/(1-2),第七章,Wave velocities In a infinite , homogeneous, isotropic and elastic medium, only two kinds of waves can propagate, P-wave and S-wave. Their travel velocities Vp and Vs are given respectively by,第七章,Where is the density of the medium. The te

30、rm P-wave means “primary wave or pressure wave,” since it arrives first or it is caused by pressure. S-wave stands for “secondary wave or shear wave,” because it travel slower than P-wave or it is generated by shear strain. From the equations above, if increases the velocity Vp and Vs should decreas

31、e. In fact, it is not real in most cases. Generally the heavier materials have the higher velocities than the lighter do, because the and increase faster than . For the materials in the earth ,assume =0.25, thus =. So we have Vp=1.73Vs,第七章,在均勻各向同性介質(zhì)中，質(zhì)點(diǎn)的運(yùn)動方程為 在無體應(yīng)力（| f |=0）的情況下，上式變?yōu)?從場論可知，任何一個場，均有 且

32、有 則,波動方程,第七章,根據(jù) 有 討論： 1)由于速度是位移對時間的偏導(dǎo)數(shù)。因此縱波和橫波的速度滿足波動方程； 2)由于無旋位移場的散度是無旋應(yīng)變，無散位移場的旋度是無散應(yīng)變。因此無旋應(yīng)變或正應(yīng)變與無散應(yīng)變或切應(yīng)變均滿足波動方程； 3)由于無旋場可用標(biāo)量位來表示，無散場可以用矢量位來表示，并分別設(shè)為標(biāo)量位和為矢量位，即有,波動方程 (續(xù)),第七章,縱波、橫波,7.2 地震波及其特征 7.2.1 地震波的類型 (1)縱波 (2)橫波,第七章,瑞雷波與勒夫波面波,(3)瑞雷波 (4)勒夫波,第七章,轉(zhuǎn)換波,(5)SH-wave and SV-wave For the anisotropic me

33、dia, S-waves may decomposed into two components SH- and SV-waves in their propagation. At the boundaries between different media with the differences of elastic properties, S-wave can generate polarization, i.e. the particles are restricted in a special plane. SH- and SV-wave have slight difference

34、in velocity.,第七章,地震子波,7.2.2 地震子波 Seismic waves are mechanical waves. They behave the kinds of particle vibration. Since the earthquakes have limit energy so that they only last rather limit time like pulses. We often say that seismic waves signals are wavelets and their periods are irregular.,第七章,By

35、 applying Fourier analysis we can decompose the wavelets in sine and cosine sequences. The seismic wavelets can be considered the sum of components of the simple waves with regular periods and various amplitudes. Therefore, we generally use a sine or cosine wave to discuss the feature of wave.,子波的分解

36、,第七章,A sine wave can be considered in two ways: at one point, they are periodic in time; at one instant, they are periodic in space. Suppose the motion of the particles is along the y-axis, we have y= A sin2 (x/ - t/T) where T is the period, A is the amplitude of particle vibration, and is wave leng

37、th, the distance at one instant between two crests or troughs, or between any two adjacent points having same phase. We usually use the terms wave number and frequency. The wave number is k=1/ and the frequency is f=1/T. Form the above, we know that the wave velocity can be obtained by V= /T. y=A si

38、n2 k(x-Vt),波數(shù)與頻率,第七章,7.2.2 平面波 Assume that the wave propagates only in x-direction and the particles move only in y-z plane. This means that all particles move in phase-they form wave fronts. We say the waves are plane waves. The movement of the particles may be described by f(x,t)=A sin2k(x-Vt),平面波

39、,第七章,7.2.3 球面波 For a point source, waves propagate in all directions. If particles move in phase and they constitute spherical wave fronts. We say they are spherical wave. Practically, for the far away point sources the spherical wave can be approximately considered as the plane waves. f(r,t)=A sin2

40、k(r-Vt),球面波,第七章,Dispersion of body waves From a point source the P- and S-waves spread radially from the source along a straight line. The spherical wave fronts still dilate, so that the energy of vibrating particles on the spheres decreases continually. Assume E is the energy of a seismic source, r

41、 is the radial distant from the source to a sphere. At one instant the energy per unit area of the sphere E can be written in E=E/(4r2) At any point out of source, the energy is proportional to the inverse of the square of r and the amplitude is proportional to the inverse of r. They both decay with

42、 the increasing of the distance from the source.,球面波的擴(kuò)散,第七章,Absorption and attenuation of body waves Up to now, we have assumed that rock or other materials are perfectly elastic. In fact, pure elastic material does not exist. The energy of the waves transform to heat due to the friction of vibratin

43、g particles. The energy and the amplitudes of the waves decay with the traveled distance and the frequency of the wave. A =A0 e-fr,地震波的吸收與衰減,第七章,地震波射線理論,7.3 地震體波的傳播 研究地震波傳播通常有兩種途徑，一個是依據(jù)波動方程的動力學(xué)理論，另一個是依據(jù)地震波走時的射線理論。 7.3.1 地震波射線理論 (1)費(fèi)瑪原理 射線理論的基礎(chǔ)是費(fèi)馬原理。費(fèi)馬原理指出：在連續(xù)介質(zhì)中，擾動沿著一條走時穩(wěn)定的路徑傳播。若以 t 表示擾動從P點(diǎn)沿著一條路徑傳到Q

44、點(diǎn)所用的時間，以v(x,y,z)表示擾動的傳播速度，以l表示該路徑的弧長，則費(fèi)馬原理可以表示為,第七章,地震波射線理論 (續(xù)),換句話說，擾動沿任一射線S傳播所用時間t與沿其它路徑傳播用時一樣，即 顯然，在均勻介質(zhì)中，射線為直線，上式可寫為 而在非均勻介質(zhì)中，射線方程的積分形式可寫成,第七章,Snell定律,(2)Snell定律 在均勻介質(zhì)中，地震波射線是直線，在連續(xù)介質(zhì)則為曲線。在非均勻介質(zhì)中，當(dāng)射線到達(dá)速度的不連續(xù)界面時，其方向會發(fā)生偏折，在界面上出現(xiàn)反射波、折射波和轉(zhuǎn)換波。 Snell定律指出了入射與反射和透射射線之間的關(guān)系。Snell定律是費(fèi)馬原理的延伸。,第七章,Snell定律 (續(xù)

45、),如果介質(zhì)中有界面存在，界面兩邊的彈性參數(shù)及密度各不相同。因為界面兩邊介質(zhì)的彈常數(shù)和密度都不相同，所以彈性波的速度也不相同。一部分彈性被能量穿過界面，產(chǎn)生透射；另一部分彈性被能量由界面反射回來?？v波經(jīng)過界面時產(chǎn)生縱波反射與透射，還可以轉(zhuǎn)換成橫被的反射與透射。 假設(shè)界面是一個平面，當(dāng)一個單純的縱波P1入射到界面時，便有四個不同的波同時產(chǎn)生，P1 P1和P1 S1表示反射的縱波和橫波（SV），P1P2和P1S2表示透射縱波和橫波（SV）。 橫波的質(zhì)點(diǎn)運(yùn)動可有兩個方向，質(zhì)點(diǎn)運(yùn)動與界面垂直的稱為SV波，質(zhì)點(diǎn)運(yùn)動與界面平行的稱為SH波。入射的SV波在界面上同樣可以產(chǎn)生上述四種波，而SH波因質(zhì)點(diǎn)運(yùn)動在與

46、界面平行的面上，所以沒有縱波產(chǎn)生。,第七章,單一水平界面地震波走時,7.3.2 水平層狀介質(zhì)中的地震波 (1)勻速層狀介質(zhì)中體波的走時 單個水平界面 在距離振動源不同的地點(diǎn)設(shè)置觀測儀器，接收某種地震波到達(dá)的時刻，以距離x為橫軸，到達(dá)時刻t為縱軸，所得的曲線稱為走時曲線(或稱時距曲線)。,xc,第七章,單一水平界面地震波走時方程,假定振動源位于地面A點(diǎn)，地下存在一個水平界面，其深度為h，在地面B點(diǎn)接受到的直達(dá)波、反射波和折射走時可分別寫成 直達(dá)波 反射波 折射波,第七章,多水平界面地震波走時,多層水平界面反射波 設(shè)Vk, hk,為第k層的速度，則有地震波向下傳播的射線走時和距離分別為 根據(jù)Sne

47、ll定律，有 則,第七章,連續(xù)介質(zhì)地震波走時方程,(2)垂向連續(xù)介質(zhì)（橫向均勻） 若 hk很小, n很大，則有地震波向下傳播的射線走時和距離分別為 由此可見，在連續(xù)介質(zhì)中，射線為一條曲線。,第七章,連續(xù)介質(zhì)地震波射線曲率,根據(jù)曲率的定義， 射線曲率為 若速度隨深度呈線性變化，即 則有 射線曲率為常數(shù)，且射線是半徑為1/Pa的圓弧。,第七章,地震波的能量分配,(3)地震波的能量分配 直達(dá)波的擴(kuò)散與衰減 r為震源到接收點(diǎn)的距離，為相應(yīng)的頻率衰減系數(shù)。 反射波能量分配與擴(kuò)散 若不考慮吸收因素，反射波的能量取決于反射波能量的分配和擴(kuò)散。其中能量分配系數(shù)反射系數(shù)為 能量擴(kuò)散與射線路徑有關(guān)，不考慮能量分配

48、的情況下，擴(kuò)散函數(shù)為,第七章,地震波的能量分配 (續(xù)),透射波的能量分配與擴(kuò)散 若不考慮吸收因素，透射波的能量取決于反射波能量的分配和擴(kuò)散。其中能量分配系數(shù)透射系數(shù)為 在不考慮能量分配的情況下，且折射波接收距較大時，擴(kuò)散函數(shù)為 其中xc為折射波臨界距離。,第七章,球坐標(biāo)中Snell定律的形式,7.3.3 球?qū)ΨQ介質(zhì)中的地震波 (1)球坐標(biāo)中Snell定律的形式 假設(shè)地球由數(shù)個厚度不等的同心球殼組成，每層內(nèi)波速均勻，根據(jù)Snell定律，有 根據(jù)正弦定理，有 所以,第七章,球坐標(biāo)中Snell定律的形式 (續(xù)),多層情況下，球坐標(biāo)中Snell定律的形式為 徑向連續(xù)介質(zhì)中Snell定律：,第七章,本多

49、夫定律,(2)射線參數(shù)方程 本多夫定律 假定射線PQ1的參數(shù)為P，走時為t，角距離為，相鄰射線PQ2的相應(yīng)數(shù)值為t+dt，+d，作Q1N垂直于PQ2，則 V* 為視速度，R為地球半徑，V0為地球表面速度，有 本多夫定律,第七章,射線曲率,射線曲率與臨界條件 設(shè)射線曲率半徑為，則 上式表示了速度隨深度變化的條件。dV/dr可以為正，也可以為負(fù)。為負(fù)時表示速度隨深度而增加，射線向下彎曲；為正時表示速度隨深度而減小，射線向上彎曲。,第七章,射線臨界條件,當(dāng)射線達(dá)到最低點(diǎn)時，有i=90，若=r，則有 射線臨界條件 表示該處射線曲率與地球曲率相同。 當(dāng) 時，無論速度隨深度增加還是減小，射線曲率小于地 球

50、曲率，射線的另一端都能在地面出現(xiàn)；當(dāng) 時，且速度 隨深度減小，射線曲率大于地球曲率，射線的另一端不能在地面出 現(xiàn)，除非當(dāng)更深處出現(xiàn)速度隨深度增加的情況。,第七章,射線臨界條件 (續(xù)),第七章,走時曲線方程,連續(xù)介質(zhì)中的走時曲線方程,第七章,走時曲線方程 (續(xù)),設(shè) rc 為最低點(diǎn)半徑，R為地球半徑，則有走時方程,第七章,近震與遠(yuǎn)震走時曲線,7.3.4 近震與遠(yuǎn)震走時曲線 由于地球表面并非一個平面，因此，觀測點(diǎn)距震中的遠(yuǎn)近不同，研究問題的方法也有所不同。震中距小于100km叫地方震，在100km至1000km范圍內(nèi)叫近震。對地方震和近震而言，地面可近似看為平面，直達(dá)波可以直接切于莫氏面到達(dá)接收點(diǎn)

51、。如果震源O位于地殼中(即在莫氏面以上)，不難證明，下式成立 這里H為莫氏面的深度，R為地球的半徑，若取H= 40km，R=6271km，則有712kmOS1424km，故取1000k皿作為遠(yuǎn)震和近震的界線。換言之，從震源出發(fā)，直達(dá)波不經(jīng)過莫氏面的反射，可直接到達(dá)的區(qū)域所觀測到的地震叫地方震或近震，而遠(yuǎn)于1000km，直達(dá)波不能直接到達(dá)，故稱為遠(yuǎn)震,第七章,近震與遠(yuǎn)震走時曲線 (續(xù)一),7.3.4 近震與遠(yuǎn)震走時曲線 (1)近震與地方震走時曲線 首波 震中距,第七章,近震與遠(yuǎn)震走時曲線 (續(xù)二),(2)遠(yuǎn)震走時曲線 由于實(shí)際地球的速度結(jié)構(gòu)復(fù)雜，不僅是由于地球各圈層的成分不同，水平方向的不均勻，

52、而且還由于物質(zhì)態(tài)的變化。為了區(qū)分經(jīng)過不同路徑的地震波，在地震學(xué)中常用以下符號： P縱波 S橫波 K在外核中的P波 I在內(nèi)核中的P波 J在內(nèi)核中的S波C在核幔邊界上的反射 f在內(nèi)外核邊界上的反射 用以上符號可以表示各種通過地核的地震波，如PKIKP,PKJKP，PKP，SKP以及在地球內(nèi)部各分界面上發(fā)生的反射波PCP，PCS，PfP等。,第七章,地震面波及其特征,7.4 地震面波及其特征 面波有兩類：即勒夫波和瑞雷波。勒夫波的振動為水平橫向(與傳播方向相垂直)，它與SH波相似。瑞雷波的振動為水平縱向(與傳播方向平行)和鉛直方向，它的軌跡為逆進(jìn)橢圓。 一個擾動在半無限的均勻介質(zhì)中，不會產(chǎn)生勒夫波，

53、而可以產(chǎn)生瑞雷波，但所產(chǎn)生的瑞雷波沒有頻散。地震記錄中出現(xiàn)勒夫波以及有頻散的瑞雷波，這說明地下的介質(zhì)是不均勻的或是呈層狀的。 不同周期的面波，其滲透深度不同，周期愈大，其滲透深度愈大。因此利用頻散曲線可以求得地球內(nèi)部速度隨深度的變化。 盡管目前面波的形成機(jī)制尚不清楚，但一般認(rèn)為，勒夫波是SH波在層間的傳播的一種形式，與SH波不同的是存在頻散現(xiàn)象；而瑞雷波是由P波與SV波干涉的結(jié)果。,第七章,地震面波及其特征 (續(xù)),勒夫波： 瑞雷波：,第七章,面波的波動方程,7.4.1 面波的波動方程 波動方程的一般形式： 考慮簡化問題，設(shè)擾動信號為一諧波。在無限半空間中，解得形式為 其中c為波速，2/k為波

54、長，U，V，W為z的函數(shù)。,第七章,面波的波動方程 (續(xù)一),(1)勒夫波頻散方程 假設(shè)一個單層半空間介質(zhì)，層厚為h。上層頂面z=-h，上層介質(zhì)參數(shù)為1、1、和V1，下半空間參數(shù)為2、2和V2。根據(jù)勒夫（1911年）的解答，有 這里，x為波的傳播方向。利用邊界條件,第七章,面波的波動方程 (續(xù)二),可求得 由于這是一個多解方程，寫成更一般的形式為 當(dāng)n=0時，稱為基模式，n=1時，稱為二階模式，。 上式表明，勒夫波具有頻散特性，不同的k值對應(yīng)得頻率不同，所對應(yīng)的速度c也不同，這里的速度為勒夫波的相速度（phase velocity）。,第七章,面波的波動方程 (續(xù)三),勒夫波具有的特點(diǎn) 勒夫波

55、產(chǎn)生在層狀介質(zhì)表面，且有Vs1Vs2； 勒夫波是一種SH型波，其振動方向與界面平行； 其速度c滿足Vs1cVs2，存在頻散現(xiàn)象； 勒夫波具有多模式，其中，基模式能量占優(yōu)； 基級模式波長 n階模式波長,第七章,面波的波動方程 (續(xù)四),(2)瑞雷波頻散方程 由于P波的入射，會轉(zhuǎn)換為P波和SV波，所以瑞雷波問題要更復(fù)雜些。對于上述介質(zhì)中在x-z平面內(nèi)的瑞雷波問題，1886年瑞雷已證明，可選擇 當(dāng)z=0時，在外應(yīng)力的情況下，有,第七章,面波的波動方程 (續(xù)五),求解可得波速方程 這里 。顯然，方程存在一個實(shí)根。 假設(shè)一個特定條件下，在半無限空間介質(zhì)中，有2=32，則c=0.92，1=0.85，2=0

56、.39。由此，可得擾動方程 其中a為常數(shù)。,第七章,面波的波動方程 (續(xù)六),當(dāng)z0時，自由面上的瑞雷波擾動為 該擾動的軌跡是一個橢圓因為k(x-ct)是隨時間增大而減小的參量，橢圓是逆進(jìn)的。由擾動方程還可看出，在w值隨z的增加而單調(diào)下降到零。u值在z等于1.21/k左右時改變符號，在此深度上橢圓為一條垂直直線。超過這個深度，橢圓成為“前進(jìn)”的，而不是“逆進(jìn)”的。,第七章,面波的波動方程 (續(xù)七),瑞雷波具有的特點(diǎn) 瑞雷波產(chǎn)生在介質(zhì)的自由表面； 瑞雷波是一種橢圓極化波，其振動方沿橢圓逆進(jìn)（在界面附近),當(dāng)離開界面一定深度時為前進(jìn)； 瑞雷波的速度c滿足cVs，當(dāng)介質(zhì)為非均勻時，有頻散現(xiàn)象； 瑞雷波也具有多模式，其中，基模式能量占優(yōu)；,第七章,面波的群速度與相速度,7.4.2 面波的群速度與相速度 (1)群速度和相速度的概念 利用面波研究地球內(nèi)部構(gòu)造時，主要利用它