The heat and wave equations have an important difference: heat: ∂u/∂t = k ∂²u/∂x² wave: ∂²u/∂t² = c² ∂²u/∂x² Note the extra time derivative on the wa

The heat and wave equations have an important difference: heat: ∂u/∂t = k ∂²u/∂x² wave: ∂²u/∂t² = c² ∂²u/∂x² Note the extra time derivative on the wave equation. The heat equation is dissipative, moving towards the average value, spreading energy out over time. The wave equation, on the other hand, allows for solutions that oscillate and maintain their shape while they travel. This means that waves such as electromagnetic (light) or gravitational can in theory be focused and beamed significant distances without much loss. A plane wave solution is lossless, but requires an infinite emitter. Finally, note that this does not apply to static fields, such as magnetic, electric, or gravitational fields, which if not oscillating will still spread out and dissipate over distance.