1. Geology 5660/6660 Applied Geophysics Last time: Brief Intro to Seismology & began deriving the Seismic Wave Equation: Four types of seismic waves:  P (“Primary” = sound; a body wave )  S (“Secondary” = shear; also a body wave )  Surface waves (Love & Rayleigh: at free surface only)  Normal Modes (“Resonant tones” = standing waves) An abbreviated Derivation of the elastic wave equation:  Stress, strain and displacement waves propagate in a medium  Rheology is linear elastic:  = c  (Hooke’s Law)  Strain is the derivative of displacement:  = ∂u/∂x  Defined several elastic constants (e.g., E,, K ) 15 Jan 2014 © A.R. Lowry 2014 Read for Fri 17 Jan: Burger 21-60 (Ch 2.2–2.6)

2. A quick “review” of various strains and their elastic constants: Uniaxial compression:  xx  yy Elongation (change in length l ) Young’s modulus E :  = E  Poisson’s ratio : (0 < < 0.5) For dilatation  (change in volume  V / V 0 ): Bulk modulus K =  P /  (where P is pressure) Rigidity modulus  =  s /   = tan   applied  s applied  (strain) (elastic constant) (elastic constant) (elastic constant) (elastic constant) (strain)

3. In fact any of the five elastic constants can be expressed in terms of any two of the remaining elastic constants: I.e., there are only two independent elastic moduli for an isotropic solid. In seismology, it is most convenient to express elastic properties in terms of Lame’s constants  & These are not independent of the other elastic parameters we’ve already seen: e.g.,

4.  xx  yy  yx  xy For an isotropic solid, in two dimensions stress and strain are related simply as where the dilatation  =  xx +  yy. These look similar for 3D but with the addition of shear and normal stresses/strains in the z direction. Note tensor notation! Stress and strain are tensors…

5.  xx  yy  yx  xy The forces acting at the edges of a block of material within a medium must balance the movement of the block. If we consider force per unit volume F/V and express acceleration a in terms of displacement u, must balance with (from Green’s theorem) (substitute stress in terms of strains & substitute strains in terms of displacement gradients, e.g.

6. This is the “dynamic wave equation” or equivalently Here P -wave propagation velocity and S -wave propagation velocity

7. Rock properties that affect seismic velocity include:  Porosity  Rock composition  Pressure  Temperature  Fluid saturation V p, V s are much more sensitive to and  than to  Crustal Rocks Mantle Rocks

8. Seismic ground motions are recorded by a seismometer or geophone. Basically these consist of: A frame, hopefully well-coupled to the Earth, Connected by a spring or lever arm to an Inertial mass. Motion of the mass is damped, e.g., by a dashpot. Electronics convert mass movement to a recorded signal (e.g., voltage if mass is a magnet moving through a wire coil or vice-versa). Instrumentation M frame spring mass dashpot

9. isometric view cross-sectional view Geophone : Commonly-used by industry, less often for academic, seismic reflection studies Often vertical component only Often low dynamic range Undamped response of mechanical system Response after electronic damping 10 Hz “natural frequency” 1010020500200 A Seismometer differs mostly in cost/ componentry… 3-c, > dynamic range