Interpretation of an integral transform from the wave equation to the heat equation

I'm having troubles with understanding the physical meaning of a certain transform.

If $u$ is a solution to the wave equation $$\partial_t^2u-\Delta u=0\ \mathrm{in}\ \mathbb{R}^n\times(0,\infty)\\u=g,\ \partial_tu=0\ \mathrm{on}\ \mathbb{R}^n\times\{0\}$$ then $$v(x,t)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^\infty{\exp\left(-\frac{s^2}{4t}\right)u(x,s)\,\mathrm{d}s}$$ solves the heat equation $$\partial_tv-\Delta v=0\ \mathrm{in}\ \mathbb{R}^n\times(0,\infty)\\v=g\ \mathrm{on}\ \mathbb{R}^n\times\{0\}$$

What is the physical interpretation of this behaviour?

And why is that so?

Solution 1:

Remark

This is actually more general than it may seem. Consider an equation $$Tf-\Delta f=0\tag1$$ where $T$ is an operator acting on the time variable. (For instance if $T=\partial_t^2+2\partial_t$, the equation is called the telegrapher's equation). Let $h$ be the Green's function of this equation. We can define $$v(x,t)=\int h(s,t)u(x,s)\mathrm ds$$ where $u$ is a solution of the wave equation with initial condition $g(x)$. Note that $v(x,0)=u(x,0)=g(x)$ and $$T v(x,t)=\int T h(s,t)u(x,s)\mathrm ds=\int \Delta h(s,t)u(x,s)\mathrm ds.$$ One can now integrate two times by parts, with respect to the variable $s$. If $h$ and $\partial_xh$ tend to $0$ at infinity (this is required by causality in physics), we obtain $$Tv(x,t)=\int h(s,t)\partial_t^2 u(x,s)\mathrm ds=\int h(s,t)\Delta u(x,t)=\Delta v(x,t).$$ Consequently, it follows that $v$ satisfies the equation (1) and was constructed using a solution a the wave equation with the same initial condition. You should note that one can replace $\Delta$ by any polynomial in $\Delta$, this would still work.

Attempt for a physical interpretation

As you observed, provided that an equation possesses a Green's function, the same construction can be done. The fundamental property that makes this work is the fact that the wave equation is no more that a change of reference frame.

The solution to the wave equation describes, in our reference frame, how some disturbance propagates in a medium. But if we move together with this disturbance, at the same speed, we will observe it as static. Physical equations describing another phenomenon, like in your case heat transfer or diffusion, should not be changed in another referential. This principle is called the relativity principle in Galileo's sense. The relativity principle states that equations of physics must satisfy the same equations in reference frames under uniform translation with respect to each other. Equation (1) describes therefore acceptable equations for physical phenomena. They all have the same property as the one you discovered for the heat equation.

All of this would be nice if there were no limit for the speed. As Einstein first understood, this is not the case and physical equations must be preserved under Galileo's transform only for speed small compared to $c$, the speed of light in vacuum. But this is another story.