Propagation Of Heat In An Infinite Rectangular Solid

163. PROBLEMS relative to the uniform propagation, or to the varied movement of heat in the interior of solids, are reduced, by the foregoing methods, to problems of pure analysis.

The progress of this part of physics will depend in consequence upon the advance which may be made in the art of analysis.

My differential equations:

• contain the chief results of my theory.
• express the necessary relations of numerical analysis to a very extensive class of phenomena
• they connect through mathematics one of the most important branches of natural philosophy.

I apply it to a problem below.

164. Suppose a homogeneous solid mass is:

• between 2 planes B and C vertical, parallel, and infinite.
• divided into 2 parts by a plane A perpendicular to the other two (fig. 7)

The temperatures of the mass BAC bounded by the 3 infinite planes A, B, C.

The other part BAC of the infinite solid is a constant source of heat.

All its points are maintained at the temperature 1, which cannot change.

The 2 lateral solids bounded, one by the plane C and the plane A produced, the other by the plane B and the plane A produced, have at all points the constant temperature 0, some external cause maintaining them always at that temperature.

The molecules of the solid bounded by A, B and C have the initial temperature 0.

Heat will:

• pass continually from the source into the solid BAC
• be propagated there in the infinite longitudinal direction
• turn towards the cool masses B and C, which will absorb most of it.

The temperatures of the solid BAC will be raised gradually.

But they will not be able to surpass nor even to attain a maximum of temperature, which is different for different points of the mass.

It is required to determine the final and constant state to which the variable state continually approaches.

If this final state were known, and were then formed, it would subsist of itself, and this is the property which distinguishes it from all other states.

Thus the problem is in determining the permanent temperatures of an infinite rectangular solid, bounded by:

• 2 masses of ice B and C
• a mass of boiling water A.

The answer to this simple and primary problem lets us discover the laws of nature.

To express more briefly the same problem,

Suppose a rectangular plate BAC, of infinite length, is heated at its base A, and to preserve at all points of the base a constant temperature 1, whilst each of the two infinite sides B and C, perpendicular to the base 4, is submitted also at every point to a constant temperature 0;

What is the stationary temperature at any point of the plate?

It is supposed that there is no loss of heat at the surface of the plate, or, which is the same thing, we consider a solid formed by superposing an infinite number of plates similar to the preceding the straight line Arz which divides the plate into two equal parts is taken as the axis of a, and the co-ordinates of any point m are x and y;

lastly, the width A of the plate is represented by 21, or, to abridge the calculation, by, the value of the ratio of the diameter to the circumference of a circle.

Imagine a point m of the solid plate BAC, whose co-ordinates are x and y, to have the actual temperature v, and that the quantities v, which correspond to different points, are such that no change can happen in the temperatures, provided that the temperature of every point of the base A is always 1, and that the sides B and C retain at all their points the temperature 0.

If at each point m a vertical co-ordinate be raised, equal to the temperature v, a curved surface would be formed which would extend above the plate and be prolonged to infinity. We shall endeavour to find the nature of this surface, which passes through a line drawn above the axis of y at a distance equal to unity, and which cuts the horizontal plane of xy along two infinite straight lines parallel to x.

166. The abstraction is d’v made of the co-ordinate z, so that the term dz must be omitted.

dv dt’ with respect to the first member it vanishes, since we wish to determine the stationary temperatures;

Thus the equation which belongs to the actual problem, and determines the properties of the required curved surface, is the following:

d’v d’v dy dot =0……………… ..(a).

The function of and y, (x, y), which represents the per- manent state of the solid BAC, must:

1. Satisfy the equation (a)
2. Become nothing when we substitute - or + for y, whatever the value of a may be
3. be equal to unity when we suppose x=0 and y to have any value included between - and +.

This function (x, y) should become extremely small when we give to a very large value, since all the heat proceeds from the source A.

167. We seek the simplest functions of a and y, which satisfy equation (a).

We shall then generalise the value of in order to satisfy all the stated conditions.

By this method the solution will receive all possible extension, and we shall prove that the problem proposed admits of no other solution.

Functions of 2 variables often reduce to less complex expressions, when we attribute to one of the variables or to both of them infinite values.

This is what may be remarked in algebraic functions which, in this particular case, take the form of the product of a function of a by a function of y.

If the value of v can be represented by such a product; for the function v must represent the state of the plate throughout its whole extent, and consequently that of the points whose co-ordinate z is infinite.

We shall then write = F(x) f(y); substituting in equation (a) and denoting df (y)

dF(x) da by F(x) and dy by f(y),

we shall have constant quantity, and as it is proposed only to find a particular value of v, we deduce from the preceding equations F(x)=eTMTM*, f(y) = cos my.

168. We could not suppose m to be a negative number, and we must necessarily exclude all particular values of v, into which terms such as e might enter, m being a positive number, since the temperature v cannot become infinite when is infinitely great.

In fact, no heat being supplied except from the constant source A, only an extremely small portion can arrive at those parts of space which are very far removed from the source.

The remainder is diverted more and more towards the infinite edges B and C, and is lost in the cold masses which bound them.

The exponent m which enters into the function ecos my is unknown, and we may choose for this exponent any positive number: but, in order that may become nul on making y= or y=+, whatever may be, m must be taken to be one of the terms of the series, 1, 3, 5, 7, &c.; by this means the second condition will be fulfilled.

169. A more general value of v is easily formed by adding together several terms similar to the preceding, and we have v=ae* cos y + be** cos 3y+ ce** cos 5y+de cos 7y+&c.. …(b).

It is evident that the function denoted by (x, y) satisfies d’v, d’v da+dy

the equation =0, and the condition (x, t)=0. A third condition remains to be fulfilled, which is expressed thus, (0, y)=1, and it is essential to remark that this result must exist when we give to y any value whatever included between - and +. Nothing can be inferred as to the values which the function (0, y) would take, if we substituted in place of y a quantity not included between the limits and +π. Equation (6) must therefore be subject to the following condition:

1 = a cos y + b cos 3y+c cos 5y + d cos 7y+ &c.

The coefficients, a, b, c, d, &c., whose number is infinite, are determined by means of this equation.

The second member is a function of y, which is equal to 1 so long as the variable y is included between the limits - and. It may be doubted whether such a function exists, but this difficulty will be fully cleared up by the sequel.

170. Before giving the calculation of the coefficients, we may notice the effect represented by each one of the terms of the series in equation (b).

Suppose the fixed temperature of the base 4, instead of being equal to unity at every point, to diminish as the point. of the line A becomes more remote from the middle point, being proportional to the cosine of that distance; in this case it will easily be seen what is the nature of the curved surface, whose vertical ordinate expresses the temperature v or (x, y).

If this surface be cut at the origin by a plane perpendicular to the axis of x, the curve which bounds the section will have for its equation va cos y; the values of the coefficients will be the following:

a= a, b = 0, c=0, d=0,

and so on, and the equation of the curved surface will be

v=ae" cos y.

If this surface be cut at right angles to the axis of y, the section will be a logarithmic spiral whose convexity is turned