Engineering Math Series 21: Heat, Wave, and Laplace Equations

There are three equations you should become familiar with first in introductory PDEs. Rather than just memorizing their names, **understanding what phenomenon each describes** is far more important.

Heat Equation

The representative form is

$$u_t = k u_{xx}$$

This models phenomena where a value spreads from high to low and gradually becomes uniform, such as temperature or concentration. The key feature is that abrupt changes smooth out over time.

Intuition

When there are hot and cold parts, the difference gradually decreases. The heat equation is the mathematical expression of this "flattening" process.

Wave Equation

The representative form is

$$u_{tt} = c^2 u_{xx}$$

This deals with vibrations propagating through space. Think of plucking a string and watching the shape spread outward.

Intuition

While the heat equation causes sharp changes to spread and vanish, the wave equation allows shapes to travel, reflect, and maintain oscillation.

Laplace Equation

The representative form is

$$u_{xx}+u_{yy}=0$$

Since time does not appear at all, it represents steady-state problems. It frequently appears when dealing with potential distributions or temperature distributions that have already reached equilibrium.

Intuition

The Laplace equation describes a state where "interior values are harmoniously matched with surrounding values."

Comparing the Three

- Heat equation: spreading and relaxation

- Wave equation: oscillation and propagation

- Laplace equation: equilibrium and static distribution

Keeping this difference in mind makes it much faster to read the character of a problem when first encountering a PDE.

A Short Example by Hand

Consider the steady-state temperature distribution of a rod of length 1. Suppose the temperatures at both ends are 0 and 100. In the steady state, there is no time change, so

$$u_{xx}=0$$

Therefore

$$u(x)=Ax+B$$

With boundary conditions

$$u(0)=0, \quad u(1)=100$$

we get

$$B=0, \quad A=100$$

so

$$u(x)=100x$$

This example shows that even though PDEs can look complex, depending on the situation they can give very simple and meaningful results.

Engineering Applications

Semiconductors and Chip Cooling

The heat equation is the foundation of packaging and cooling design.

Communications and Acoustics

The wave equation provides the fundamental intuition for transmission lines, acoustics, and electromagnetic wave propagation.

Electric Fields and Electrostatic Potential

The Laplace equation appears repeatedly in potential distribution and potential flow problems.

Common Mistakes

Viewing all three equations with the same intuition

Although they look similar on the surface, the solution behaviors are quite different. Heat is smoothing, waves are propagation, and Laplace is equilibrium.

Looking at equations without boundary conditions

Especially for PDEs, the boundary conditions must be present for the physical problem to be complete.

Applying separation of variables too early and mechanically

At the introductory level, understanding the physical meaning and representative forms first is more important.

One-Line Summary

The heat equation deals with spreading, the wave equation with propagation, and the Laplace equation with equilibrium.

Next Post Preview

In the next post, we will go beyond the real axis and enter **complex numbers and analytic functions**, which appear centrally in circuits and signal processing.

References

- Erwin Kreyszig, _Advanced Engineering Mathematics_, 10th Edition

- Richard Haberman, _Applied Partial Differential Equations_

- Lawrence C. Evans, _Partial Differential Equations_