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Introduction to Linear Algebra
Glossary
3D wave equations
The wave equation for a function u(x1, …... , xn, t) = u(x, t) of n space variables x1, ... , xn and the time t is given by
\[
\square u = \square_c u \equiv u_{tt} - c^2 \nabla^2 u = 0 , \qquad \nabla^2 = \Delta = \frac{\partial^2}{\partial x_1^2} + \cdots + \frac{\partial^2}{\partial x_n^2} ,
\]
with a positive constant c (having dimensions of speed). The operator □ defined above is known as the d'Alembertian or the d'Alembert operator. The wave equation subject to the initial conditions is known as the initial value problem:
\[
\square u = 0, \qquad u({\bf x}, 0) = f_0 ({\bf x}), \quad u_t ({\bf x}, 0) = f_1 ({\bf x}) ,
\]
where f0(x) and f1(x) are given (smooth) functions in n-dimensional space ℝn. For n = 3, the solution of the initial value problem for wave equation is
\[
u(x_1 , x_2 , x_3 , t) = \frac{1}{4\pi c^2 t} \iint_{|{\bf x} - {\bf y}| = ct} f_1 (y_1 , y_2 , y_3 )\,{\text d} S_y + \frac{\partial}{\partial t} \, \frac{1}{4\pi c^2 t} \iint_{|{\bf x} - {\bf y}| = ct} f_0 (y_1 , y_2 , y_3 )\,{\text d} S_y ,
\]
where integration is taken along the surface of the sphere in the 3-dimensional space ℝ³. Upon introducing spherical coordinates and setting c = 1, the above formula becomes
\[
u(x_1 , x_2 , x_3 , t) = \frac{1}{4\pi} \int_0^{2\pi} {\text d}\phi \int_0^{\pi} {\text d}\theta \, f_1 (x_1 + t\,\sin\theta \,\cos \phi , x_2 +t\,\sin\theta \,\sin \phi , x_3 + t\,\cos \theta )\,\sin \theta .
\]
Maxwell's Equations
\[
\begin{split}
\nabla \cdot {\bf E} &= 0, \qquad c\nabla \times {\bf E} = -\frac{\partial {\bf B}}{\partial t} , \\
\nabla \cdot {\bf B} &= 0, \qquad c\nabla \times {\bf B} = \frac{\partial {\bf E}}{\partial t} ,
\end{split}
\]
where c ≈ 2.998 × 108 m/s is the speed of light in vacuum. This system of four partial differential equations---two vector equations and two scalar equations in the unknowns E and B---describes how uninterfered electromagnetic radiation propagates in three dimensional space.
The above equations were first published by the Scottish physicist James Clerk Maxwell (1831--1879) in his 1861 paper On Physical Lines of Force and again in a more unified manner in his 1864 paper A Dynamical Theory of the Electromagnetic Field. Maxwell is considered by many to be the most influential scientist on 19th century physics.
Taking the curl (∇×) of the curl equations, and using the curl of the curl identity we obtain
\[
\begin{split}
\frac{\partial^2 {\bf E}}{\partial t^2} &= \frac{\partial}{\partial t} \left( c\nabla \times {\bf B} \right) = c \left( \nabla \times \frac{\partial {\bf B}}{\partial t} \right)
\\
&= c \left( \nabla \times \left( -c\nabla \times {\bf E} \right) \right)
\\
&= - c^2 \left( \nabla \times \left( \nabla \times {\bf E} \right) \right) .
\end{split}
\]
This last expression seems intractable, but Lagrange's formula
\[
\nabla \times \left( \nabla \times {\bf E} \right) = \nabla \left( \nabla \cdot {\bf E} \right) - \left( \nabla \cdot \nabla \right) {\bf E}
\]
gives
\[
\frac{\partial^2 {\bf E}}{\partial t^2} = c^2 \Delta {\bf E} .
\]
That is, E satisfies the 3D wave equation. A similar argument shows that B solves the 3D wave equation as well:
\[
\frac{\partial^2 {\bf B}}{\partial t^2} = c^2 \Delta {\bf B} .
\]
Elastodynamic Equations
\[
\rho\,\frac{\partial^2 u_i}{\partial t^2} = f_i + \sum_{i=1}^3 \frac{\partial}{\partial x_j} \sigma_{i,j} , \qquad i=1,2,3,
\]
where ui is the particle displacement, fi is a body force term, and σi,j is the stress tensor.
Assuming the solid follows a linear elastic constitutive relations
\[
\sigma_{i,j} = \sum_{k,l} c_{ijkl} e_{kl} , \qquad e_{kl} = \frac{1}{2} \left( \frac{\partial u_k}{\partial x_l} + \frac{\partial u_l}{\partial x_k} \right) ,
\]
then
\[
\rho \,\ddot{u}_i = f_i + \left( c_{ijkl}u_{k,l} \right)-{,j} .
\]
For isotropic material, we have
\[
\sigma_{i.j} = \lambda \theta \delta_{i,j} + 2\mu\,e_{ij} , \qquad \theta = e_{11} + e_{22} + e_{33} .
\]
Then the equations of motion become
\[
\rho\,\frac{\partial^2 u_i}{\partial t^2} = f_i + \lambda \,\frac{\partial}{\partial x_i} \,\theta + 2\mu\,\sum_i \frac{\partial e_{ij}}{\partial x_j} .
\]
Here we have assumed for simplicity that density ρ, and Lamé parameters (also called the Lamé coefficients, Lamé constants or Lamé moduli) λ, and μ are constants and the equation for a homogeneous medium.
We can rewrite these equations in more convenient form:
\[
\rho\,\frac{\partial^2 u_i}{\partial t^2} = f_i + \left( \lambda + \mu \right) \frac{\partial}{\partial x_i} \,\theta + \mu\,\nabla^2 u_i , \qquad i=1,2,3 .
\]
They also can be written in vector form:
\[
\rho\,\frac{\partial^2 {\bf u}}{\partial t^2} = {\bf f} + \left( \lambda + \mu \right) \nabla \left( \nabla \cdot {\bf u} \right) + \mu \nabla^2 {\bf u} , \qquad {\bf u} = ( u_1 , u_2 , u_3 ) .
\]
It turns out that the dilatation θ satisfies the wave equation
\[
\rho\,\frac{\partial^2 \theta}{\partial t^2} = \left( \lambda + 2 \mu \right)
\nabla^2 \theta \qquad\mbox{or} \qquad \frac{\partial^2 \theta}{\partial t^2} = c_p \nabla^2 \theta ,
\]
where
\[
c_p = \sqrt{\frac{\lambda + 2 \mu}{\rho}}
\]
is the dilatational or longitudinal speed.
Similarly, we can find an equation for the rotation \( \Omega = \frac{1}{2}\,\nabla \times {\bf u} \) by taking the curl (which is ∇× in modern notation) of the isotropic electrodynamics equation:
\[
\rho\,\frac{\partial^2 \Omega}{\partial t^2} = \mu \,
\nabla^2 \Omega \qquad\mbox{or} \qquad \frac{\partial^2 \Omega}{\partial t^2} = c_s \nabla^2 \Omega ,
\]
where
\[
c_s = \sqrt{\frac{\mu}{\rho}}
\]
is the transverse speed. For the Earth model PREM (Dziewonski and Anderson, 1981), the physical properties for several depths in the Earth are
| Depth | cp (km/s) | cs (km/s) | ρ (kg/m³) | |
|---|---|---|---|---|
| Ocean | 3.0 | 1.45 | 0.0 | 1020 |
| Upper Crust | 15.0 | 5.79 | 3.19 | 2600 |
| Lower Crust | 25.0 | 6.79 | 3.89 | 2900 |
| Uppermost mantle | 80 km | 8.07 | 4.38 | 3375 |
| Mid-mantle | 1071 km | 11.55 | 6.41 | 4621 |
| Lowermost mantle | 2891 km | 13.69 | 7.23 | 5566 |
| Outer core | 3871 km | 9.38 | 0.0 | 11,982 |
| Inner core | 5671 km | 11.14 | 3.54 | 12,982 |
\[
{\bf u} = \nabla \phi + \nabla \times {\bf \psi} ,
\]
where \( \phi \) and ψ are called the scalar and vector potentials, respectively. These potentials are solutions of the wave equations:
\[
\ddot{\phi} = c_p^2 \nabla^2 \phi \qquad\mbox{and} \qquad \ddot{\bf \psi} = c_s^2 \nabla^2 {\bf \psi} .
\]
There \( \nabla \cdot \phi \) and ∇ × ψ are the P-wave and S-wave components of u.
- Biazar, J. and Islam, R., Solution of wave equation by Adomian decomposition method and the restrictions of the method, Applied Mathematics and Computation, 2004, Vol. 149, No. 3, pp. 807--814.
- Biazar, J. and Ghazvini, H., An analytical approximation to the solution of a wave equation by a variational iteration method, Applied Mathematics Letters, 2008, Vol. 21, pp. 780–785; https://doi.org/10.1016/j.aml.2007.08.004
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