Does the Wave Equation Describe Termination?

The short answer is: No, the classical wave equation itself does not describe how far or when a wave terminates.

The wave equation, in its fundamental form, describes an idealized, lossless system where a wave propagates forever without diminishing in energy or stopping.

∂²y/∂t² = v² · (∂²y/∂x²)

The Ideal World of the Classical Wave Equation

This standard wave equation makes these key assumptions:

  • No Energy Loss: There is no friction, air resistance, or internal dissipation.
  • No Boundaries: The medium is imagined to be infinitely long with no ends to reflect or terminate the wave.

In this idealized model, a wave never terminates. Once created, it will continue propagating in both directions at speed v for all time, and its amplitude never changes.

How We Describe Termination and Decay in the Real World

Real-world waves do terminate or die out. To describe this, physicists add terms to the classical wave equation to model specific mechanisms of energy loss.

A) Termination by Dissipation (Decay)

Waves lose energy due to friction, viscous forces, or resistance. This causes the wave's amplitude to decay exponentially over distance and time.

We model this by adding a damping term proportional to the velocity of the medium:

∂²y/∂t² = v² · (∂²y/∂x²) - γ · (∂y/∂t)

Where γ is the damping coefficient. The larger γ is, the faster the wave's energy is converted to heat and the quicker it dies out.

B) Termination by Boundaries

Waves often encounter the end of their medium. The wave doesn't just "stop"; its energy is reflected or transmitted.

  • Fixed End: When a wave pulse hits a fixed end, it inverts and reflects back.
  • Free End: The wave reflects without inverting.
  • Different Medium: Part of the wave reflects back, and part is transmitted into the new medium.

C) Termination by Dispersion

In some media, the wave speed v depends on the frequency of the wave. This is called dispersion.

A wave packet or pulse will spread out as it travels. The pulse becomes wider and lower in amplitude until it effectively vanishes into the background.

Summary

While the classical wave equation is the fundamental starting point, physicists "turn on" various real-world effects by adding terms or changing conditions to accurately model how a wave's journey eventually ends.