2020地球物理学基础作业05及参考答案

1.

When a bell is struck with a hammer, it vibrates freely at a number of natural frequencies. The combination of natural oscillations that are excited gives each bell its particular sonority. In an analogous way, the sudden release of energy in a very large earthquake can set the entire Earth into vibration, with natural frequencies of oscillation that are determined by the elastic properties and structure of the Earth’s interior. The free oscillations involve three-dimensional defo rmation of the Earth’s spherical shape and can be quite complex. Before discussing the Earth’s free oscillations it is worth reviewing some concepts of vibrating systems that can be learned from the one-dimensional excitation of a vibrating string that is fixed at both ends.

Any complicated vibration of the string can be represented by the superposition of a number of simpler vibrations, called the normal modes of vibration. These arise when travelling waves reflected from the boundaries at the ends of the string interfere with each other to give a standing wave. Each normal mode corresponds to a standing wave with frequency and wavelength determined by the condition that the length of the string must always equal an integral number of half-wavelengths (Fig. 3.16).As well as the fixed ends, there are other points on the string that have zero displacement; these are called the nodes of the vibration. The first normal (or fundamental)mode of vibration has no nodes. The second normal mode (sometimes called the first overtone) has one node; its wavelength and period are half those of the fundamental mode. The third normal mode (second overtone) has three times the frequency of the first mode, and so on.Modes with one or more node are called higher-order modes.

当用一把锤子敲击一个钟时，钟会以一系列的固有频率自由的颤动。被激

发的固有震动的联合给每个一钟独特的音响。与此相似，在一个大地震中能量

的突然释放可以使整个地球颤动，这种颤动的固有频率决定于弹性性质和地球

内部的结构。自由振荡涉及地球球面形状的三维变形，可能相当复杂。在讨论

地球的自由振荡之前，有必要回顾一下振动系统的一些概念，这些概念可以从

两端固定的一维振动的激发中学习。

弦的任何复杂的弦振动都可以用一些简单振动的叠加来表示，称为简正振动。当从两端的边界反射出的行波相互干涉以产生驻波时，就会产生这种现象。

每一个简正模态对应于一个驻波，它的频率和波长取决于长度必须等于半波长的

整数的弦（图3.16)。在弦上还存在一些除固定端外的具有零位移的其他点，这

些被称为振动的节点。第一个简正（或基本）模态振动没有节点。第二个简正

模态（有时称为第一谐波）有一个节点，它的波长和周期是基态的一半。第三

个简正模态（第二谐波）的频率是第一模态的三倍，一个或多个节点的模态称

为高阶模态。

2. Explanation of nouns (20points)

surface wave（面波）：沿界面及界面一定深度范围内传播的一类地震波，振幅随

深度增加而衰减，能量集中在介质分界面并沿分界面传播，包括瑞利波，勒夫

波和斯通利波。dispersion（频散）：面波速度随着周期（或频率）而变化而

变化，成为面波频

散。在记录中面波是很多列波的叠加，随着到时的先后，各相位的周期逐渐改变。

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除均匀半空间的瑞利面波无频散外，所有的地震面波都具有频散特性。

group velocity （群速度）：由于频散，各种频率的波以各自速度传播，相互叠加， 形成合震动，其振幅不断变化，用其极大值速度表征其传播速度，该速度为群速 度，即波群整体移动速度。波传播时能量与振幅平方成正比，绝大部分能量集中 于振幅极大处，因此，群速度也是能量和信号的传播速度。

phase velocity （相速度）：单色简谐波传播时，其同相面传播的速度。

3.Assuming a homogeneous elastic plate covered on the homogeneous elastic semi-infinite space, the top interface is z=0 and the bottom interface is z=h. Solve the dispersion equation of love wave propagation and discuss its dispersion characteristics.(25points)

取 x 、 y 坐标轴在自由表面上( z = 0 )， z 轴垂直向下，令层中横波速度为 β1 ，密度为 ρ1 ， 半空间中横波的速度为 β2 ，密度为 ρ2 ，且有 β1 < β2 ，Love 波振幅在层中为 v 1 ，在半空间 为 v 2 ，x 轴为波的传播方向，振幅垂直与 x 轴且平行于分界面，即振动沿 y 轴方向，v 1 ，v 2 满足方程：

?2v 1 + ?2v 1 = 1 ?2v 1 ?

2 2 2

? ?x ?z 2 ?t β1 ?

?2v 2 ?2v 2 ?(1) + = 1 ?2v 2 ?

? ?x 2 ?z 2 2 ?t 2

β2 ?

解得，

v 1 = (Ae b 1z + Be -b 1z )e i ( kx -wt ) ? ,0 < z < h ? -b 2 z i ( kx -wt )

?(2) v 2 = Ce e , z > h ? ?

期中，

= ω ?

b =

k 2 + k 2 , k β1 1

β1 β1 ? ?

ω ?(3)

?

b 2 = k 2 + k β22 , k β2 = β2 ?

? k = ω (4)

c

因要满足 z → ∞ 的收敛条件，在 v 2 中去掉了 e -b 2 z 项， k 为波数， c 为面波速度。 式(2)因满足自由表面边界条件和层与半空间界面连续条件

当 z = 0 时， ε xy = μ1 ?v 1 = 0(5)

v 1 = v 2 , ?z

? 当 z = h 时， ?v ?v ?

μ ?(6)