A seismic pulse travels along a borehole as a one-dimensional wave,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Wave motion, the wave equation and wave energy

Conditions of Validity, Ch The three equations, their characteristics and what makes a problem well posed

Problem

u_{tt} = c^2 u_{xx}

, with

u(x,t)

the displacement,

x

the distance along the borehole axis and

c = 6000\,\mathrm{m\,s^{-1}}

the compressional wave speed in the rock. ((a)) Write down the two families of characteristic directions of this equation. ((b)) A geophone sits at

x_0 = 3.0\,\mathrm{km}

. Find the domain of dependence of the point

(x_0, t_0)

with

t_0 = 0.40\,\mathrm{s}

, i.e. the interval on the line

t=0

that determines the displacement recorded there.

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