# Preface

This section about three dimensional wave equation.

Introduction to Linear Algebra

# 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

In the theory of electromagnetism, the effects of charged particles in the three dimensional space ℝ³ acting on one another result in an electrical vector field E(x, y, z, t) or E(x1, (x2, (x3, t). If the particles are also in motion, a magnetic vector field B(x, y, z, t) or B(x1, (x2, (x3, t) is also generated. The theory of electromagnetism rests on the principle that these vector fields E and B obey Maxwell's equations (in vacuum):
$\begin{split} \nabla \cdot {\bf E} &= 0, \qquad c\nabla \times {\bf E} = -\frac{\partial {\bf B}}{\partial t} , \\ \nabla \cdot {\bf B} &= 0, \qquad c\nabla \times {\bf B} = \frac{\partial {\bf E}}{\partial t} , \end{split}$
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

The linearized equation of motion is
$\rho\,\frac{\partial^2 u_i}{\partial t^2} = f_i + \sum_{i=1}^3 \frac{\partial}{\partial x_j} \sigma_{i,j} , \qquad i=1,2,3,$
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
Another direct decomposition of the original vector displacement can be written as
${\bf u} = \nabla \phi + \nabla \times {\bf \psi} ,$
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.

1. 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.
2. 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