Partial Differential Equations: Wave Heat Laplace Learn

A partial differential equation, or PDE, is an equation that involves partial derivatives of a function of several variables, such as temperature depending on both position and time. The three most famous PDEs in physics are the wave equation, the heat equation, and Laplace's equation, and each one describes a different fundamental process in nature.

These three equations matter because they appear almost everywhere: the wave equation governs sound, light, and vibrating strings; the heat equation governs how warmth spreads through a metal rod; and Laplace's equation governs steady situations like the electric potential around a charged object or the final, settled temperature distribution in a plate.

Key Concepts

• Wave equation: second time derivative proportional to second space derivative. Describes disturbances that travel without changing shape, like a pulse on a rope.
• Heat (diffusion) equation: first time derivative proportional to second space derivative. Describes spreading and smoothing, like ink diffusing in water.
• Laplace's equation: sum of second space derivatives equals zero. Describes steady states with no sources, like electrostatic potential in empty space.
• Classification: wave is hyperbolic (signals travel at finite speed), heat is parabolic (disturbances spread instantly but decay), Laplace is elliptic (solution at every point depends on the whole boundary).

Formulas

• Wave equation (one dimension): second partial of u with respect to t equals c squared times second partial of u with respect to x, where c is the wave speed in meters per second.
• Heat equation: partial of u with respect to t equals alpha times second partial of u with respect to x, where alpha is the thermal diffusivity in square meters per second.
• Laplace's equation: del squared u = 0, where del squared is the sum of second partial derivatives in all space directions.
• Wave speed on a stretched string: v = square root of (T divided by mu), with tension T in newtons and linear mass density mu in kilograms per meter.

Worked Example

Find the fundamental frequency of a guitar string of length L = 2 meters, tension T = 100 newtons, and linear density mu = 0.01 kilograms per meter, with both ends fixed.

1. Compute the wave speed: v = square root of (100 / 0.01) = square root of 10000 = 100 meters per second.
2. Fixed ends force the string into standing waves with nodes at both ends, so the longest allowed wavelength is lambda = 2L = 4 meters.
3. The fundamental frequency is f = v / lambda = 100 / 4 = 25 hertz.
4. Higher modes vibrate at 50, 75, 100 hertz and so on, which is why a plucked string sounds rich rather than a pure tone.

Common Misconceptions

• "Wave speed is how fast the string material moves." No. The wave speed c is how fast the disturbance travels along the string; each bit of string only wiggles up and down in place.
• "The heat equation works the same forward and backward in time." It does not. Heat flow is irreversible: a hot spot spreads out and never spontaneously re-concentrates, unlike wave motion which can be run backward.
• "Laplace's equation peaks inside the region." A harmonic function always takes its maximum and minimum on the boundary, never inside.

Real-World Applications

• Musical instruments and concert hall acoustics are designed using the wave equation.
• Heat sinks, building insulation, and cooking times are modeled with the heat equation.

The wave, heat, and Laplace equations are the three pillars of classical field physics: one carries disturbances, one smooths them out, one describes balance. Recognizing which fits is the first step to solving a problem.