Preface

This tutorial was made solely for the purpose of education and it was designed for students taking Applied Math 0340. It is primarily for students who have some experience using Mathematica. If you have never used Mathematica before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA 0330 tutorial. As a friendly reminder, don't forget to clear variables in use and/or the kernel. The Mathematica commands in this tutorial are all written in bold black font, while Mathematica output is in normal font.

Finally, you can copy and paste all commands into your Mathematica notebook, change the parameters, and run them because the tutorial is under the terms of the GNU General Public License (GPL). You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately. The tutorial accompanies the textbook Applied Differential Equations. The Primary Course by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043

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Introduction to Linear Algebra

Glossary

Derivation of Wave Equation

We present a derivation of one dimensional the wave equation as it applies to the transverse vibrations of an elastic string. Such string may be thought of as a piano string or a waire in an electric power line.

The wave equation for real-valued function \( u(x_1, x_2, \ldots , x_n , t) \) of n spatial variables and a time variable t is

\[ \frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u , \qquad \mbox{or} \qquad \square u =0 , \]
where c is a positive constant (having dimensions of speed) and
\[ \nabla^2 u \equiv \Delta u = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \cdots + \frac{\partial^2 u}{\partial x_n^2} \qquad\mbox{and} \qquad \square u \equiv \square_c u = \frac{\partial^2 u}{\partial t^2} - c^2 \Delta u . \]
The differential operator □ is called the d'Alembertian and △ is the Laplacian.

Suppose we have a medium whose displacement may be described by a scalar function u(x,t), where \( {\bf x} \in \mathbb{R}^n , \quad t\in\mathbb{R} . \) Suppose that the system is conservative and it has the Lagrangian \( {\cal L} = \mbox{K} - \Pi , \) where the kinetic energy K and potential energy Π of the medium are

\[ \mbox{K} \left( u_t \right) = \frac{1}{2} \int \rho\,u_t\,{\text d}{\bf x} , \qquad \Pi \left( u \right) = \frac{1}{2} \int k \left\vert \nabla u \right\vert^2 {\text d}{\bf x} . \]
Here ρ(x) is a mass-density and k(x) is a stiffness, both assumed positive, and \( u_t = \partial u/\partial t . \) The corresponding action becomes
\[ S \left( u \right) = \int {\text d}t \, {\cal L} \left( u, u_t \right) = \int {\text d}t \int {\text d}{\bf x} \,\frac{1}{2} \left\{ \rho\,u_t^2 - k\left\vert \nabla u \right\vert^2 \right\} . \]
Note that the kinetic and potential energies and the Lagrangian are functions of the spacial field and velocity at each fixed time, whereas the action is a functional of the space-time field u(x, t), obtained by integrating the Lagrangian with respect to time.

The Euler--Lagrange equation satisfied by a stationary point (which is a function u(x, t)) of this action becomes

\[ \rho\, u_{tt} - \nabla \cdot \left( k\,\nabla u \right) =0 . \]
If &rho: and k are constants, then we get the wave equation
\[ \square_c u \equiv u_{tt} - c^2 \Delta u =0 \qquad\mbox{or}\qquad \frac{\partial^2 u}{\partial t^2} - c^2 \nabla^2 u =0 . \]
The action functional for the wave equation is not positive definite. therefore, we cannot expect a solution of the wave equation to be a minimizer of the action, in general, only a critical point.   ■
Note that \( a_0 = u(x,t) . \)

 

 

 

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