Return to computing page for the first course APMA0330
Return to computing page for the second course APMA0340
Return to Mathematica tutorial for the first course APMA0330
Return to Mathematica tutorial for the second course APMA0340
Return to the main page for the
first course APMA0330
Return to the main page for the
second course APMA0340
Return to Part VI of the course APMA0340
Introduction to Linear Algebra
The wave equation for a function u(x_{1}, …... , x_{n}, t) = u(x, t) of n space variables x_{1}, ... , x_{n} 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:
where f_{0}(x) and f_{1}(x) are given (smooth) functions in n-dimensional space ℝ^{n}. For n = 3, the solution of the initial value problem for wave equation is
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
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(x_{1}, (x_{2}, (x_{3}, t). If the particles are also in motion, a magnetic vector field B(x, y, z, t) or B(x_{1}, (x_{2}, (x_{3}, 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):
where c ≈ 2.998 × 10^{8} 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.
where u_{i} is the particle displacement, f_{i} is a body force term, and σ_{i,j} is the stress tensor.
Assuming the solid follows a linear elastic constitutive relations
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:
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:
Return to Mathematica page
Return to the main page (APMA0340)
Return to the Part 1 Matrix Algebra
Return to the Part 2 Linear Systems of Ordinary Differential Equations
Return to the Part 3 Non-linear Systems of Ordinary Differential Equations
Return to the Part 4 Numerical Methods
Return to the Part 5 Fourier Series
Return to the Part 6 Partial Differential Equations
Return to the Part 7 Special Functions