Answer to Question #144976 in Molecular Physics | Thermodynamics for Joel George

Question

Compare the partial differential equation of Wave and Heat; state their similari ties and differences.

Answer 1

The wave equation has a form

$\displaystyle \frac{\partial^2u}{\partial t^2} = c^2 \Delta u$ (1)

The heat equation is following

$\displaystyle \frac{\partial u}{\partial t} = \alpha \Delta u$

Similarities:

1) both are second-order PDE

2) both have $\Delta u$ on the right-hand side with some coefficient

Differences:

1) (1) is hyperbolic equation, (2) is parabolic equation

2) Coefficients have different physical sense and different power: c in (1) is speed of the wave, $\alpha$ in (2)is thermal diffusivity

3) they have different order of time derivative, and, therefore, the solutions depend on time in a different way (for (1) it is $\sim e^{-t}$, for (2) it is $\sim ct$)