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Richa Nandra, Lovely Professional University Unit 23: Wave and Diffusion Equations by Separation of Variable

Unit 23: Wave and Diffusion Equations by Notes
Separation of Variable

CONTENTS
Objectives

Introduction
23.1 On Solution of Wave Equation

23.1.1 Solution of One Dimensional Wave Equation
23.1.2 Two Dimensional Wave Equation
23.1.3 The Vibrations of a Circular Membrane

23.2 Boundary Value Problems (Heat Conduction or Diffusion)
23.2.1 Variable Heat Flow in One Dimension

23.2.2 Heat Flow in Two Dimensional Rectangular System
23.2.3 Temperature Inside a Circular Plate
23.3 Summary

23.4 Keywords
23.5 Review Questions

23.6 Further Readings

Objectives

After studying this unit, you should be able to:
 Note that it finds its applications in almost all branches of applied sciences.
 Understand how heat flows in solids
 See how the electrical current and potentials are distributed in certain medias.
 Know how the diffusion problem is tackled by means of diffusion equation.
Introduction

It is seen that Laplace equation plays an important role in the solution of wave equation as well
as conduction of heat.

The problems occurring in this unit are based on boundary values of the waves as well as the
temperature distribution of the substance.
Depending upon the symmetry of the problem the Laplace equation is solved in Cartesian or
spherical polar co-ordinates or cylindrical co-ordinates.

23.1 On Solution of Wave Equation

When a stone is dropped into a pond, the surface of the water is disturbed and waves of
displacement travel radially outward, when a tuning fork or a bill is struck, sound waves are

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