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I am a mathematician. I'd like to know the physical interpretation of some terms for example if we have $ u:\Omega\times [0,T]\rightarrow \mathbb{R}^{d}$ ($\Omega$ bounded domain and $[0,T]$ is the time interval with $T>0$ and $d \in\mathbb{N}$ equals or greater then $1$) is the displacement so $u_{t}$ the derivative of $u$ with respect to $t$ is the the velocity and $u_{tt}$ the second derivative with respect to $t$ is the acceleration and $$ u_{tt}+Au_{t}+Bu=0$$ is the equation of motion ($A, B$ some operators).

My question is if we consider $ \theta:\Omega\times [0,T]\rightarrow \mathbb{R}$ is the temperature, what is the interpretaion of $\theta_{t}$ and $\theta_{tt}$ and $$\theta_{tt}+C\theta_{t}+D\theta=0$$ ($C, D$ some operators).

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  • $\begingroup$ That T is a temperature (field?) doesn't seem to enter here at all. $\endgroup$ Commented Jun 6, 2023 at 18:37
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    $\begingroup$ Can you clarify your question to provide context? Is there a reason to replace the displacement with the temperature, or are you just curious about inserting one parameter into an equation derived for another? Are you asking whether the equation has any physical meaning, or whether a system could be assembled to produce the response given in the equation, or something else? The "acceleration" of temperature isn't really in common use in physics—left alone, simple thermal systems don't overshoot or oscillate. However, higher temperature derivatives are used in engineering control theory. $\endgroup$ Commented Jun 6, 2023 at 18:41
  • $\begingroup$ I have edited the post [0,T] is the interval of time. $\endgroup$
    – ran
    Commented Jun 6, 2023 at 19:10
  • $\begingroup$ I don't replace ..I take an example for the different interpretation if we consider the displacement and I would to know if we have temperature and not the displacement what are the new interpretations! $\endgroup$
    – ran
    Commented Jun 6, 2023 at 19:14

2 Answers 2

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$\theta$ is a temperature field, telling you the temperature at every neighbourhood of a point at every time. $\theta_t$ is the rate of change of temperature field, and $\theta_{tt}$ is the rate of change of that, and does not have a simple interpretation. The last thing would be an equation of motion. I really think you should only have one time derivative in the heat equation. Check how the PDE is differing.

If you have a second derivative, then the interpretation of it would be a second sound. It is relevant in very specific scenarios.

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Physically, 2nd derivative terms represent inertial effects while first derivative terms represent response rates. This is highly typical of a linear system. To physically arrive at the equation, one needs a relations between 3 terms:

I) Temperature

II)Physical Variable causing delay in Temperature Change( Heat flow)

III) Further inertial effect.

This entire thing can be traced to simple harmonic oscillator and is typical of Linear Systems used in Engineering.


Also if one looks at the first order term for dissipation,

The double derivative is a kind of inertial term which leads to oscillations. There is a part proportional to temperature and there is a part proportional to temperature derivative representing in-phase and 90 degree out of phase components. Continuing the simple harmonic oscillator analogy, the temperature term is kind of like stiffness and the temperature derivative like damping. ( Damping leads to finite response rates)


To semi-rigorously derive these equations from PDEs, some kind of normal mode expansion is done usually. Called Lumped- Parameter - Modeling.

Thermal Resistance and Thermal Capacitance are known from Diffusion Equation.

But Thermal Inductance is non trivial.

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