Understanding Waves in Fluids: Geol 4068 Class Presentation
This presentation accompanies the reading of Chapter 2 on Waves in Fluids from "Elements of 3D Seismology" by Christopher Liner. It covers topics like fluid properties, elastic moduli, acoustic wave equations, seismic materials, and key physical parameters of acoustic waves. The importance of velocity over density in wave dynamics is highlighted. Different approaches in understanding wave mechanics, including Hooke's Law and Lagrangian Mechanics, are discussed.
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Geol 4068 Class presentation to accompany reading of Chapter 2- Waves in Fluids Elements of 3D Seismology, 2nd Edition by Christopher Liner September 6, 2005 Juan M. Lorenzo
Outline What is a Fluid? Elastic Moduli Acoustic Solution to the Wave equation Density Velocity
Outline What is a Fluid? Elastic Moduli Acoustic Solution to the Wave equation Density Velocity
TYPES OF SEISMIC MATERIALS Fluid (no shear strength) Porous solid (Solid+fluid - BIOT THEORY) Solid (shear strength and high and finite resistance to compressibility
Outline What is a Fluid? Elastic Moduli Acoustic Solution to the Wave equation Density Velocity
Key Physical Parameters of the Acoustic Wave Equation V = P P (Pa) o P V = V o V Bulk Modulus or modulus of incompressibility
Key Physical Parameters of the Acoustic Wave Equation V = P f P P V = (Pa) f V V Bulk Modulus or modulus of incompressibility
P P = V V V = P f o V f o ref . kgm N m = = Pa 2 2 s m 2 Example: Ocean Water density = 1,035 kg.m^3 V = = 1500 / V P m s 2 P Incompressibility modulus is of order 10^9 Pa
Outline What is a Fluid? Elastic Moduli Acoustic Solution to the Wave equation Density Velocity
Different Approaches Hooke s Law: stress (Pa)= Y (Pa). strain + Newton s Law (F=ma) Lagrangian Mechanics (Energy Relationships)
Outline What is a Fluid? Elastic Moduli Acoustic Solution to the Wave equation Density Velocity
Density is given minor importance with respect to velocity. Usually velocity and density increase together. Gardner s Rule (Gardner et al., 1974) density = a .V^0.25, where a=0.31 Salt: 4500-5000 m/s; 2.0-2.1 g/cc
Outline What is a Fluid? Elastic Moduli Acoustic Solution to the Wave equation Density Velocity
1/2 j j = Vt = 2 i i V t i i ( ) j average V i V 1 ( ) rms j j j t t i i i 1 ( , ) i V ( ) V ( ) V backus V V average V V rms i i i backus 1/2 j 2 Vt i i j = 1 Vt backus V j j Vt V (Find typo) i i i i i 2 i i 1 1
j = Vt Mean velocity; traditional i i i ( ) j average V j t i i 1/2 j Very important for basic seismic processing. Can be obtained directly from seismic field data or GPR field data. Errors ~10% = 2 i V t i V 1 ( ) rms j j t i 1
V=330 m/s, rho =0 z=100000m i=1 s = 200m; V=1000 m/s, rho =1.6 i=2 V=1500 m/s, z= 500m rho =1.8 i=3=j
layer V rho z(m) time thickness(s) 100000 303.0303 200 500 0.333333 V*t V*V*t rho*V*t z*z z/(VVvrho) 1 2 3 330 1000 1500 0.013 1.6 1.8 100000 33000000 200 500 1300 320 900 1E+10 70.63643 40000 0.000125 250000 0.000123 0.2 200000 750000 z/(Vvrho) sums TIMES Vt VVt rhoVt ZZ 100700 303.5636 100700 33950000 2520 1E+10 70.63668 VrmsVrms Vrms 111838.2 334.4221 Vavg 331.7262 VbacVbac Vbac 56180 237.0232 Excel macro
1/2 j j = Vt = 2 2 i i V t i i ( ) j average V i rms j V 1 ( ) j j t t i i i 1 ( , ) i V ( ) V ( ) V backus V V average V V rms i i i backus 1/2 j 2 Vt i i = 1 backus V j j j Vt i i i Vt i i 2 2 i i V t i 1 1
