# I Why is the wave equation different from the heat equation

1. Jan 11, 2017

### fahraynk

I have been thinking about this. For a wave equation, the acceleration of a point on a drumhead is proportional to the height of its neighbors $$U_{tt}=\alpha^2\nabla^2U$$

The heat equation, change in concentration or temperature is equal to the average of its neighbors $$U_t=\alpha^2\nabla^2U$$

I was thinking its probably because the height of a point in the wave equation would need to keep going after it passes the average height of its neighbors, and then wobble back and forth. So even though it passes its neighbors it can still have a positive velocity and the acceleration will just flip signs.

But... why doesn't heat or concentration also do this? Why wouldn't the acceleration of change in temperature act more like a wave... or the velocity of a point on a drumhead just become 0 when its neighbors average is equal.

Do we not know, and the equations just model what we observe...or is there a reason?

2. Jan 11, 2017

### Staff: Mentor

Compare the behavior of a hot object touching a cold object with the behavior of a mass attached to a stretched spring. Heat will flow from the hot object to the cold object until the temperatures are the same; but then the flow stops because no heat flows between objects at the same temperature. The mass on the spring behaves differently because once it starts moving inertia makes it want to keep on moving; so it overshoots stretching the spring in the other direction.

3. Jan 12, 2017

### fahraynk

Why do they behave differently? Is it just because of boundary conditions? If you had a sinusoidal input on the heat equation should it then look like the wave equation?

4. Jan 13, 2017

### 256bits

If you look at your equations they are not the same on the left hand side.
One will have a single differentiation to solve, the other a double, hence the subscript "t" or "tt".

5. Jan 13, 2017

### 256bits

6. Jan 15, 2017

### Redbelly98

Staff Emeritus
They are different physical entities. A wave tends to continue traveling in whatever direction it is going. Heat tends to spread out or diffuse from hotter regions (i.e. regions containing more heat than the surroundings) to colder regions (containing less heat than the surroundings).

It shouldn't be surprising that these different behaviors are described by different equations.