The heat equation bears some resemblence to the wave equation.
The heat equation is ∂u/∂t = k∇²u, while the wave equation is
∂²u/∂t² = c²∇²u, where u is the scalar function that
describes either heat content or wave amplitude.
Heatwave treats both k and c as constant (except at the edges of the sheet).
The divergence of the gradient of a function (∇²u) (also called the Laplacian) is a measure
of how quickly the gradient is changing, or how much the value at some point is above or below the (tilted) plane
defined by the points immediately adjacent to that point.
And ∂u/∂t (from the heat equation) is the speed at which the heatcontent or temperature is changing.
So the speed at which the temperature at any point is changing is directly proportional to the Laplacian in the heat equation.
In other words, every point is smoothing itself out relative to its neighbors at a rate that's directly proportional
to how far it's sticking out.
And since it slows down as it moves closer to equilibrium, it never actually gets to equilibrium in finite time.
It gets closer and closer, but never finishes.
It's like Zeno's paradox, except the time slices are not getting smaller. It's asymptotic.
With the wave equation it's ∂²u/∂t², instead of ∂u/∂t,
that's directly proportional to the Laplacian.
And whereas ∂u/∂t is velocity, ∂²u/∂t² is acceleration.
So the wave equation is a=c²L where a is acceleration, L is the Laplacian,
and c is the speed of wave propagation (which we treat as constant).
For sound waves, the wave function describes the displacement of the material at each point,
and ∂²u/∂t² is the acceleration of the material at each point.
To accurately describe sound waves in a solid (which propagates both longitudinal and transverse waves),
the displacement function u(x,y,z,t) should be a vector function, and the velocity and acceleration at each point are also a vectors.
The wave equation is sometimes called the vector wave equation in these situations.
Heatwave solves the scalar, not vector, heat equation.
This accurately models sound waves in liquids and gases, which don't propagate transverse waves, and electromagnetic waves
which don't have a longitudinal component.
If we compare a=c²L to Newton's second law, F=ma restated as a=F/m, it implies that c²L
is analogous to F, the force resisting deformation.
This makes sense because c² is proportional to the stiffness of the material, and L is a measure of
how far the material is (locally) bent out of shape.
Let's look at the heat equation, ∂u/∂t = k∇²u, again.
We can approximate ∂u/∂t as (u(t+dt)u(t))/dt and get
This brings up damping. Since u(t)  u(tdt) is a term carrying over speed or momentum,
we can think of it as speed * dt.
And one way to model friction is
When damping==0, we get an undamped wave equation. When damping==1 we get a fullydamped wave equation, which
is equivalent to the heat equation. When damping>1 (Heatwave allows this so you can experiment) we get oscillations and exploding values.
And when damping<1 (Heatwave also allows this) we get accelerating and quickly growing values. Which brings up the next topic:
instabilities and exploding values.
