Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Earthquake Engineering GE / CEE - 479/679 Topic 11. Wave Propagation 1 John G. Anderson Professor of Geophysics

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Wave Propagation What is the physics behind propagation of seismic waves? Seismic waves propagate due to the elastic properties of the medium. Equation of motion in a homogeneous, linear elastic medium Solution in terms of P- and S- waves

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Derivation of the wave equation Starting point: F=ma Let u(x,t) be the infinitesimal motion of a particle in an elastic medium. For motion in the i th direction, the right hand side is:

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Stress Force/unit area Use on/in notation Thus is the stress on the plane normal to the unit vector in the i direction, acting in the j direction. xixi xjxj

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Infinitesimal strain: By definition, sum over repeated indices. Consider only that part of the motion that does not include whole-body rotation. Define

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Hooke’s Laws 2nd main assumption Stress proportional to strain The Lamé constants are λ and μ. The dilatation is

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Combining in F=ma In this equation, X i is a body force acting on the point, if any.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Key Concept 1 General description of a propagating wave:

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Key Concept 2 If: Then: Where: is the wavelength, f is frequency, and v is wave velocity.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Trial Solution Number 1 Suppose Then: And:

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Substitute in the equation of motion This is true if General case, define as the speed of shear waves. The solution describes a shear wave traveling in the x 1 direction.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Trial Solution Number 2 Suppose Then: And:

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Substitute in the equation of motion

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Substitute in the equation of motion This is true if General case, define as the speed of compressional waves. The solution describes a compressional wave traveling in the x 1 direction.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Notes If λ=μ, which is the usual assumption for crustal rocks, then

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 The Free Surface Most observations are made on the surface. Structures are mostly built on the surface. So it is important to understand what happens to a seismic wave when it impacts the free surface. Since incoming waves cannot propagate into the air, energy is reflected back downward.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 The Free Surface: Key Concept S-waves can have two polarizations: –SH - wave motion is parallel to the surface. Causes only horizontal shaking. –SV - wave motion is oriented to cause vertical motion on the surface. SH SV Motion in and out of the plane of this figure - hard to draw. Motion perpendicular to the direction of propagation causes vertical motion of the free surface.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 The Free Surface Consider an incoming, vertically-propagating S-wave. There must be a downgoing wave, as the upgoing wave alone cannot satisfy the boundary conditions. At the free surface, the stress is zero. Use this boundary condition to solve for u down x3x3 Shear wave x1x1

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 The Free Surface At the free surface, the stress is zero. τ 31 = τ 32 = τ 33 = 0 For this S-wave, only τ 31 can be non-zero anyplace. θ=0 Thus τ 31 =0 implies e 31 =0 x3x3 Shear wave x1x1

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 The Free Surface e 31 =0 Since and u 3 =0, The steps at the left show how this is used to solve for u down. The condition is only met if g(t)=h(t) at all times. x3x3 Shear wave x1x1 Let At x 3 =0,

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 The Free Surface Since g(t)=h(t), at x 3 =0, u 1 =2g(t). For a vertically incident S- wave, the amplitude at the free surface is double the amplitude of the incoming wave. This result holds for vertically incident P-waves also. x3x3 Shear wave x1x1 So

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 The Free Surface: Summary At the free surface, waves are reflected back downwards. For a vertically incident S-wave, the amplitude at the free surface is double the amplitude of the incoming wave. This holds for vertically incident P-waves also. Waves are still approximately doubled in amplitude when incident angles are near vertical. If there is a structure, the boundary conditions are changed. Some energy enters the structure instead of being reflected back.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact Observe that when waves travel from a solid of one velocity to different velocity, the direction changes. This has a large impact on the nature of seismic waves, since the Earth is highly variable. i1i1 i2i2

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact Boundary conditions: “welded contact” This implies that displacement is continuous across the boundary. Also, that stress is continuous across the boundary. i1i1 i2i2

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact Envision a wave of frequency f. It cannot change frequency at the boundary. Wavefronts are drawn perpendicular to direction of wave travel. Note how angle of incidence, i 1 and i 2 are defined. i1i1 i2i2 wavefront i1i1 i2i2 A B

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Snell’s Law: Two Media in Contact Line segment AB is common to two right triangles. The geometry leads to Snell’s Law: i1i1 i2i2 wavefront i1i1 i2i2 A B

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact This way of drawing is consistent with horizontal layers in the Earth. Lower velocities near the surface imply wave propagation direction is bent towards the vertical as the waves near the surface. i1i1 i2i2

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Example of a 3-component ground motion record. Note how the S-wave is dominantly showing up on the horizontal components, and the P-wave is strongest on the vertical component. P S

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact In addition to the “refraction” of energy into the second medium, some energy is reflected back. The angle of reflection is equal to the angle of incidence. This brings up the issue: how is the energy partitioned at the interface? i1i1 i2i2 i2i2 Incoming SH

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact The energy partitioning is determined by “reflection” and “transmission” coefficients. The coefficients are determined by matching boundary conditions For incoming SH waves, the form is relatively simple. i1i1 i2i2 i2i2 A T R Transmission coefficient Reflection coefficient Incoming SH

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact These coefficients are not a function of frequency. At most, the transmitted wave has an amplitude of 2 x the amplitude of the incoming wave. Going from a stiffer to a softer material, the transmission coefficient is never less than 1.0. i1i1 i2i2 i2i2 A T R Incoming SH

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact Going from a softer to a stiffer material, the transmission coefficient is never more than 1.0. If there is a large impedence contrast from softer to stiffer, the transmission coefficient approaches zero. i1i1 i2i2 i2i2 A T R Incoming SH

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact The reflection coefficient is always less than 1.0. In the limit of the two media being identical, the transmission coefficient is 1.0 and the reflection coefficient is 0.0. In the limit of a reflection from a much stiffer or much softer medium, the reflection coefficient approaches 1.0. i1i1 i2i2 i2i2 A T R Incoming SH

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact An important case is when waves in a soft medium contact a stiff boundary. In this case, the reflection coefficient is almost 1.0 (actually -1.0), meaning that the energy is trapped in the softer material. This applies to energy in a sedimentary basin. i1i1 i2i2 i2i2 A T R Incoming SH

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact For an incoming SV wave, the situation gets even more complex. In this case, both P- and SV-waves are transmitted and reflected from the boundary. The P- and SV-waves are coupled by the deformation of the boundary. i1i1 i2i2 i2i2 Incoming SV Reflected SV Transmitted SV Transmitted P Reflected P j2j2 j1j1 Generalized Snell’s Law

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact For an incoming P wave, the situation is similar to incoming SV. In this case also, both P- and SV-waves are transmitted and reflected from the boundary. The P- and SV-waves are again coupled by the deformation of the boundary. i1i1 j2j2 i2i2 Incoming P Reflected SV Transmitted SV Transmitted P Reflected P j2j2 j1j1 Generalized Snell’s Law

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Two Media in Contact Lower velocities near the surface also imply that waves are bent towards the horizontal at depth. i1i1 i2i2

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Realistic Earth Model Eventually, as the velocity increases with depth, rays are bent back towards the surface. Waves cannot penetrate into layers where β is too large. i1i1 i2i2 p is the “ray parameter. It is constant along the ray β increases

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Body Waves: Discussion The travel time curves of body waves can be inverted to find the velocity structure of the path.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Realistic Earth Model Due to Snell’s law, energy gets trapped near the surface. This trapped energy organizes into surface waves. i1i1 i2i2 β increases

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Four types of seismic waves Body Waves P Waves Compressional, Primary S Waves Shear, Secondary Surface Waves Love Waves Rayleigh Waves

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Surface Waves Love waves: trapped SH energy. Rayleigh waves: combination of trapped P- and SV- energy.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Surface Waves For surface waves, geometrical spreading is changed. –For body waves, spreading is ~1/r. –For body waves, energy spreads over the surface of a sphere, but for surface waves it spreads over the perimeter of a circle. –Thus, for surface waves, spreading is ~1/r 0.5.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Surface Waves Depth of motion: body waves can penetrate into the center of the Earth, but surface waves are confined to the upper 1’s to 10’s of kilometers.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Surface Waves Body waves are not dispersed. Surface waves are dispersed, meaning that different frequencies travel at different speeds. Typically, low frequencies travel faster. These have a longer wavelength, and penetrate deeper into the Earth, where velocities are faster. Typically, Love waves travel faster than Rayleigh waves.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Surface Waves Body waves amplitudes do not diminish so rapidly with depth in the Earth. Surface waves amplitudes decrease rapidly, especially below a few kilometers (depending on the period). Surface wave dispersion curves can be inverted to find the velocity structure of the path crossed by the surface waves.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Surface Waves Particle motion in S-waves is normal to the direction of propagation. This is also true of Love waves. However, Love waves would show changes in phase along the direction of propagation that would not appear in vertically propagating S waves.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Surface Waves Motion of Rayleigh waves is “retrograde elliptical”.

Feb 26, John Anderson: GE/CEE 479/679: Lecture 11 Surface Waves These examples have all been from surface waves seen at teleseismic distances. Later on, we will see examples of surface waves seen at short distances, on strong ground motion records.