Heat Wave Equation Solution Integral

[James McVittie]

After we determined the solution to the heat wave equation, we took the integral from negative infinity to some x. The solution was in the form of a Normal Distribution function where in statistics the integral indicates the probability that we randomly choose a value within that interval. I was wondering what are the physical implications of this same integral in the wave equation? Does the area under the solution indicate the energy in the system up to that point or does it have some other meaning?

[Victor Ivrii]

After we determined the solution to the heat wave equation, we took the integral from negative infinity to some x. The solution was in the form of a Normal Distribution function where in statistics the integral indicates the probability that we randomly choose a value within that interval. I was wondering what are the physical implications of this same integral in the wave equation? Does the area under the solution indicate the energy in the system up to that point or does it have some other meaning?

Heat wave equation? WTH are you talking about? There is a wave equation (and its ilk) and the heat equation  (and its ilk) describing either very different processes or the same process under very different assumption. The properties of these equations are really different.

PS Sure more complex phenomena can be described by systems containing both equations but for the purpose of this class ...

[James McVittie]

Sorry I meant the Heat Equation. What does the integral of the solution to that equation represent?

[Victor Ivrii]

Sorry I meant the Heat Equation. What does the integral of the solution to that equation represent?

If we talking about heat equation then it is a total heat energy in $(-\infty,x)$. But for us it mainly a trick to get a a formula for a solution.