Seismic Wave Motion in Anelastic Media

Abstract

Consider a mechanical linear oscillator with a single degree of freedom driven by an external time-dependent acceleration f(t). The time behavior of this system is governed by the equation

$$\ddot x + \frac{1}{Q}{\omega _0}\dot x + \omega _0^2x = f\left( t \right),\left( {Q \gg 1,\dot x = \frac{{dx}}{{dt}}} \right)$$

here Q is a dimensionless parameter to be studied later. Mathematically, the solution of Eq. (10.1), under the initial conditions x(0) = x ₀, ẋ(0) = ẋ₀ can formally be considered as the sum x(t) = x ₁(t) + x ₂(t) where

$${\ddot x_1}\left( t \right) + \frac{1}{Q}{\omega _0}{\dot x_1}\left( t \right) + \omega _0^2{x_1}\left( t \right) = 0,{\rm{ }}{x_1}\left( 0 \right) = {x_0},{\rm{ }}{\dot x_1}\left( 0 \right) = {\dot x_0},{\rm{ }}{\ddot x_2}\left( t \right) + \frac{1}{Q}{\omega _0}{\dot x_2}\left( t \right) + \omega _0^2{x_2}\left( t \right) = f\left( t \right),{\rm{ }}{x_2}\left( 0 \right) = 0,{\rm{ }}{\dot x_2}\left( 0 \right) = 0.{\rm{ }} $$

Men argue, Nature acts.

(Voltaire)