Why is the wave equation different from the heat equation

• fahraynk
In summary: The wave equation is used for describing physical phenomena like sound and water waves, while the heat equation is used to describe things like the transfer of heat or changes in temperature.
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?

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.

Nugatory said:
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.

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?

fahraynk said:
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?
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".

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.

Nugatory

1. Why are the wave equation and heat equation different?

The wave equation and heat equation are different because they describe different physical phenomena. The wave equation describes the propagation of a disturbance or wave through a medium, while the heat equation describes the diffusion of heat through a medium.

2. Can the wave equation and heat equation be used interchangeably?

No, the wave equation and heat equation cannot be used interchangeably. They have different mathematical forms and are applicable in different situations. Trying to use one equation to describe the other phenomenon would lead to incorrect results.

3. How are the wave equation and heat equation related?

The wave equation and heat equation are both partial differential equations and can be derived from similar principles, such as conservation of energy and mass. However, they differ in the specific physical processes they describe.

4. Are there any real-life applications of the wave equation and heat equation?

Yes, both the wave equation and heat equation have numerous real-life applications in fields such as physics, engineering, and mathematics. The wave equation can be used to model the behavior of sound waves, electromagnetic waves, and seismic waves, while the heat equation is used to model heat transfer in various systems, such as buildings, engines, and the Earth's crust.

5. Is there a way to combine the wave equation and heat equation?

Yes, there are ways to combine the wave equation and heat equation to create more complex models that account for both phenomena. One example is the telegraph equation, which combines the wave equation with a diffusion term to model the propagation of electrical signals along a wire.

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