2.2 Wave propagation and computation of surface wave
2.2.1 Basic theory of elastic wave propagation
¾ Definition and notation of stress
Fig. 2-4 Stress notation of an element in an x-y-z Cartesian coordination
σij is usually used as the symbol representing stress. In the subscript, the first letter denotes the axis perpendicular to the plane in which the stress acts and the second denotes the direction of stress. For a small element in an x-y-z Cartesian coordination, there are totally nine components acting on its face (as shown on Fig. 2-4).
The normal stresses are denoted as:
σxx , σyy , σzz
The shear stresses are denoted as:
σxy , σxz , σyx , σyz , σzx , σzy
For the moment equilibrium of the element, it requires that σxy =σyx , σxz =σzx , σyz =σzy. So there are only six independent components required for describing the state of stresses of an element.
¾ Definition and notation of displacement and strain
Let ui represent the displacement in i direction. The relationship between displacements and strains can be defined as:
⎟⎟⎠
¾ Stress-strain relationship
For a homogeneous and isotropic linear elastic body, the stress-strain constitutive law can be described by two Lame’s constants, λ and μ, and Hooke’s law. The generalized form can be writen as: function. For example, substituting (2-2) into (2-3) gives
⎟⎠
2.2.1.2 Equation of motion for elastic solid
Consider the infinitesimal element shown on Fig. 2-4 with dimensions of dx, dy and dz respectively in x, y and z direction. In the x direction, the unbalanced external force must be balanced by an inertial force in that direction. So,
( )
operation in y and z directions givesz
(2-6) represent the three dimensional equations of motion of solid. The equations are derived on the basis of equilibrium, therefore it is valid for solids of any constitutive model.
Substituting (2-4) derived from Hooke’s law into the equations of motion and repeating the operation in y and z directions, the equations of motion expressed in term of displacement are shown as following:
⎟⎟⎠
If the expression of vector is applied, equations (2-7) can be expressed as,
u represents the Laplacian operator. The Laplacian operator has another alternative expression:
) (
)
2u=∇(∇⋅u − ∇×∇×u
∇ (2-9)
Then (2-8) can be written as following:
)
According to the Helmholtz decomposition theorem, any vector field u can be considered to be generated by a pair of potentials: a scalar potential ψ and a vector potential Ψ.
Substituting (2-11) into (2-10) gives, 0
Substituting (2-14) into (2-13), then (2-13) has solutions if the (2-15) are satisfied.
1 0
(2-15) are the wave equations, in which vP and vS are the propagating velocities of P wave and S wave respectively. As it shown, the displacements of solid after disturbance can be categorized into those related to P wave and those related to S wave. The propagating velocities of body waves in a solid are not correlative with the frequency of excitation but
correlative only with the Lame’s constants of the medium. From (2-14), it is obvious that vP is greater than vS. It can be observed on the seismogram of earthquake that the displacements caused by P wave always occur earlier than those caused by S wave.
The separation of variables is applied here for solution of wave equations (2-15).
Suppose
Substituting (2-16) into (2-15), the general solution of the wave scalar potential for plane waves propagating in any direction is obtained,
( )
(2-18) defines the surface of plane waves propagating in directions of (kpx, kpy, kpz) with a P wave velocity of vp in a Cartesian coordinate. The constant A represents the amplitude. In the same way, the general solution of the wave vector potential for plane waves propagating in any direction can be obtained.
( )
(2-19) defines the surface of plane waves propagating in directions of (ksx, ksy, ksz) with an S wave velocity of vs in a Cartesian coordinate. The constant B represents the amplitude.
Base on the hypothesis of no inference when P and S wave propagating, the displacement field u caused by excitation can be represented by the linear combination of the vibrations caused by P wave and S wave. According to the Helmholtz theorem, the
displacement field u can be decomposed into:
in which up, us represent the displacement caused by P wave and S wave respectively. So,
∧ vectors in x, y, z directions. For simplifying the problem, the plane wave is supposed that kpy, ksy =0, which means the wave is steady in y direction, ∂φ ∂y→0 and ∂Ψi ∂y→0. Then us
can be expressed as:
∧
The equation (2-21) can be re-written by substituting (2-22) and (2-23).
∧
From the result of (2-24), the displacement in y direction depends only on S wave. But displacements in x and z direction is a combination provided by P wave and S wave and the interference of S wave here is different from the interference of S wave in y direction.
Distinctly, the displacement field u can be discussed separately in the x-z plane and y direction.
The displacement in y direction is represented by SH wave and the displacement in x-z plane is represented by P-SV wave.
2.2.1.3 Rayleigh wave in a homogeneous halfspace
A free surface is necessary for occurrence of surface waves. The main characteristic of surface waves is that the carried energy decays with increasing depth. Surface wave in a homogeneous halfspace is the focus of this section. From (2-24), it is clear that the components of the displacement field u in x, y, z directions are:
x
As mentioned above, the uy depends only on the vector potential of S wave. Keeping it in term of uy does not affect the form of general solution. For a wave with a frequency of ω and velocity of vR traveling through a homogeneous halfspace, its φ, Ψy and uy can be supposed as:
( )
in which, f(z), g(z), h(z) are used for the phenomenon that energy decays gradually with depth (z) increasing; k=w/ vR represents spatial frequency. Putting (2-26) into (2-15) leads to
0
The solutions of (2-27) can be expressed as:
( ) ( )
in which, A, A’, B, B’, C, C’ are all arbitrary constants. The constants A’, B’, C’ equal to zero due to the behavior of energy decay with depth increasing. Both of r and s are imaginary numbers, which implies vp> vs> vR. Substituting (2-29) into (2-26) gives,
( )
The boundary conditions are necessary for resolving the constants A, B, C. On the free surface, the stresses in z direction must be zero, σzz =0, σzx =0, σzy =0. The boundary conditions here are:
0
Substituting (2-24) and (2-29) into (2-31), that gives:
(
2)
2 0Substituting the third of (2-30) into the second of (2-32) resolves that the constant C equals zero which means no displacement in y direction (uy=0). It implies the surface wave in a homogeneous halfspace does not contain Love wave. Next, substituting the first and the second of (2-30) into the first and third of (2-32) respectively obtains a homogeneous system
of linear equations.
If nontrivial solutions of (2-33) exist, the following requirement should be satisfied,
(
2 21)
2 (12 2 ) 0Calculating the determinant gives
( )
[
vp r2 +1 −2vs]
(1−s2)−4rsvs2 =0 (2-35) Substituting (2-28) into (2-35) and rearranging,2
(2-36) is first presented by Rayleigh in 1887 for the wave velocity of Rayleigh wave.
The equation clearly shows that the propagating velocity of Rayleigh wave in a homogeneous halfspace is irrelevant to frequency (which means non-dispersive).
Let
Substituting (2-37) into (2-36) and rearranging,
(
3 2)
16( 1) 0 88 2
3− ξ + − qξ+ q− =
ξ (2-38)
in which, q is a constant once the Lame’s constants of the homogeneous halfspace are assured.
ξ is the only unknown which have three nontrivial solutions for the cubic equation. The propagating velocity of Rayleigh wave, vR, wave should satisfy the requirement of vp > vs > vR. Therefore the solution satisfyingξ<1 is the one for the propagating velocity of Rayleigh wave.
One can express (2-39) in terms of Poisson’s ratio and solve the cubic equation. For
typical values of Poisson’s ratio, 0.2<ν<0.4, the velocity of Rayleigh wave ranges from 0.9 vs to 0.95 vs. The ratios between vR, vp and vs as a function ofνis shown in Fig. 2-5. The particle motion of a propagating Rayleigh wave in a homogeneous half space transits from retrograde to prograde elliptical with depth as shown in Fig. 2-6 and Fig. 2-7. The Rayleigh wave comprises both compressional and rotational components and the eccentricity of the ellipse for locus of particle motion depends on Poisson’s ratio.
Fig. 2-5 The ratio of Rayleigh wave velocity, vR, verse body wave velocities as a function of Poisson ratio,ν (Sheriff et al, 1982)
Fig. 2-6 Rayleigh wave particle motion in a homogeneous, isotropic half space; retrograde at the surface, passing through purely vertical at about λ/5 then becoming prograde at depth
(Cuellar, 1997)
Fig. 2-7 Particle motions of Rayleigh wave over one wavelength along the surface and as a function of depth (Sheriff et al, 1982)
2.2.1.4 Rayleigh wave in a vertically heterogeneous halfspace
The surface wave applied in this study is Rayleigh wave due to the interference of P wave and SV wave. Its most important characteristic is dispersion due to the stiffness variation with depth of the tested strata. Dispersion means the propagating velocity of wave varies with frequency.
Fig. 2-8 The model of a vertically heterogeneous halfspace
Considering a vertically heterogeneous halfspace as shown in Fig. 2-8, the elastic parameters depend only on the depth z. The following discussion focuses only on the part related to P-SV wave, so the component of displacement on y direction, uy, is assumed to be zero. For a plane wave propagating in +x direction, it can be expressed as:
( ) [ ( ) ]
in which, r1(k,z,ω) and r2 (k,z,ω) represent the amplitudes of components of the displacement in x and z directions respectively. Both have characteristics of decay with z increasing and each different frequencyωwith a corresponding spatial frequency k.
Due to the continuity of stress between layers, so let
( ) [ ( ) ]
directions respectively. Both have characteristics of decay with z increasing and each different frequencyωwith a corresponding spatial frequency k.For solutions of r1 (k,z,ω), r2 (k,z,ω), r3 (k,z,ω) and r4 (k,z,ω), it is necessary to obtain four equations for four unknowns. Two can be obtained from the stress-strain relationship,
( ) ( )
Substituting (2-39) into (2-41) gives
( ) ( )
Then substituting (2-40) into (2-42) and rearranging,
( )
(2-43) are two of four equations for solutions. The other two can be found by means of the equations of motion (2-7),
( ) ( ) ( ) ( )
Substituting (2-39) and (2-43) into (2-44) and rearranging,
( ) ( ) ( ) ( ) ( )
Combining equations (2-43) and (2-45), the motion-stress vector (Aki et al, 2002) can be formed, functions of stiffness, wavenumber and frequency. (2-47) can be expressed in the form of vector as,
( )
A( ) ( )
z z dzz
df = f (2-48)