LECTURE 5 - Wave Equation Hrvoje Tkalčić " 2 # "t = ( $ + 2µ ) & 2 # 2 % " 2 (& ' u r ) = µ "t 2 % & 2 (& ' u r ) *** N.B. The material presented in these lectures is from the principal textbooks, other books on similar subject, the research and lectures of my colleagues from various universities around the world, my own research, and finally, numerous web sites. Some colleagues to whom I am grateful for the material I used in this lecture are: P. Wu, E. Garnero, D. Russell and B. Kennett. I am thankful to many others who make their research and teaching material available online; sometimes even a single figure or an idea about how to present a subject is a valuable resource. Please note that this PowerPoint presentation is not a complete lecture; it is most likely accompanied by an inclass presentation of main mathematical concepts (on transparencies or blackboard).***
What is a Wave? A wave is a disturbance or variation which travels through a medium. The medium may experience some local oscillations as the wave passes, but the particles in the medium do not travel with the wave. The disturbance may take any of a number of shapes, from a finite width pulse to an infinitely long sine wave.
Body Waves & Surface Waves Body waves are waves that propagate through the bulk of the Earth Surface waves are waves that are trapped near the surface (i.e. the amplitude of the wave decays exponentially away from the surface). For example, Earth s surface, core-mantle boundary, or any internal boundaries within the Earth.
The most general form of Hooke s law * from the previous lecture * " ij = C ijkl # kl The constants of proportionality, C ijkl are elastic moduli. We saw that the both strain and stress tensors are second-order tensors, which are symmetric and have 6 independent elements. C ijkl is thus a third-order tensor and in its most general form consists of 81 elements. However, since we showed that the strain and stress tensors only have 6 independent elements, the number of independent elements in C ijkl can be reduced to 36. The first stress element is related to the strain elements by: " ij = C 1111 # 11 + C 1112 # 12 + C 1113 # 13 + C 1121 # 21 + C 1122 # 22 + C 1123 # 23 + C 1131 # 31 + C 1132 # 32 + C 1133 # 33 For an isotropic medium (material properties independent on direction or orientation of sample), the number of elastic moduli can conveniently be reduced to only 2. These elastic moduli are called the Lamé constants λ and µ. " ij = #$% ij + 2µ& ij where δ ij is Krönecker delta function (δ ij =0 when i j and δ ij =1 when i=j). This was formulated by Navier in 1821 and Cauchy in 1823.
+σ 31, +σ 32, +σ 33 Derivation of the equations of motion +σ 21, +σ 22, +σ 23 In equilibrium, the stresses on this face are +σ 11, +σ 12, +σ 13 while the stresses on the opposite face are -σ 11, -σ 12, -σ 13 If we add a small differential stress δσ ij to the front face perpendicular to x 1, this small element of volume will be out of equilibrium and the net force will be different from 0.
LECTURE 5 - Wave Equation (acting on face x1) (forces acting on face x 2 and face x 3 ) (note that these are the 3 contributions of the force in x1 direction, coming from all 3 faces)
LECTURE 5 - Wave Equation displacement U r = u r r i + v r r j + w r k r, where r i, r j and k r are the normal vectors in x 1,x 1 and x 3 directions. a component of displacement in x 1 direction, in other words
LECTURE 5 - Wave Equation
LECTURE 5 - Wave Equation Obviously, we are differentiating this way because we want to get the rotational components But, how certain can we be that the inner core has finite rigidity?
3-D Grid for Seismic Wave Animations Courtesy of D. Russell & P. Wu No attenuation (decrease in amplitude with distance due to spreading out of the waves or absorption of energy by the material) dispersion (variation in velocity with frequency), nor anisotropy (velocity depends on direction of propagation) is included.
Compressional Wave (P-Wave) Animation Deformation propagates. Particle motion consists of alternating compression and dilation. Particle motion is parallel to the direction of propagation (longitudinal). Material returns to its original shape after wave passes.
Shear Wave (S-Wave) Animation Deformation propagates. Particle motion consists of alternating transverse motion. Particle motion is perpendicular to the direction of propagation (transverse). Transverse particle motion shown here is vertical but can be in any direction. However, Earth s layers tend to cause mostly vertical (SV; in the vertical plane) or horizontal (SH) shear motions. Material returns to its original shape after wave passes.
Wave Type (and names) Seismic Body Waves Particle Motion Other Characteristics P, Compressional, Primary, Longitudinal S, Shear, Secondary, Transverse Alternating compressions ( pushes ) and dilations ( pulls ) which are directed in the same direction as the wave is propagating (along the raypath); and therefore, perpendicular to the wavefront. Alternating transverse motions (perpendicular to the direction of propagation, and the raypath); commonly approximately polarized such that particle motion is in vertical or horizontal planes. P motion travels fastest in materials, so the P-wave is the firstarriving energy on a seismogram. Generally smaller and higher frequency than the S and Surfacewaves. P waves in a liquid or gas are pressure waves, including sound waves. S-waves do not travel through fluids, so do not exist in Earth s outer core (inferred to be primarily liquid iron) or in air or water or molten rock (magma). S waves travel slower than P waves in a solid and, therefore, arrive after the P wave.
Tides Glacial Rebound 10 10 s The frequency of your birthday