Chapter 4

# Temperature wave

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

The basement is considered to be a half-infinite and homogeneous medium, with its thermal conductivity, its density, its mass thermal capacity and located in the half-space .

We suppose that the temperature of the ground ( ) is subjected to sinusoidal variations :

## Question

Determine the temperature at the depth (use the complex notations ) in stationary mode.

### Hint

• Solve the heat equation using the complex numbers method.

• This exercise deals with the equivalent of the skin effect in the domain of electromagnetism.

### Solution

The heat equation is :

We use the complex numbers method and we have :

The heat equation leads to :

That is to say :

Therefore :

We note :(skin thickness)

We go back to real numbers, and we keep only the solution that does not diverge in the infinite :

## Question

Calculate the velocity of the thermal wave that has been obtained.

### Solution

The velocity of the wave is :

## Question

We consider daily temperature changes, the one on the floor varying between at night and during the day.

From which depth are the temperature changes less than ?

Calculate .

We give :

Let us consider annual temperature changes, from to . Answer the same questions.

### Solution

First case : and .

second case : and  :

The temperature in an inter cave is fresh in summer and mild in winter.

Indeed, at a depth, the evolution of the temperature is the same than outside days late.

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Cooling fin
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Probabilistic approach of scattering