An Elementary Treatise on Fourier's Series: and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical

CHAPTER I.

INTRODUCTION.

1.In many important problems in mathematical physics we are obliged to deal with partial differential equations of a comparatively simple form.

For example, in the Analytical Theory of Heat we have for the change of temperature of any solid due to the flow of heat within the solid, the equation

where u represents the temperature at any point of the solid and t the time.

In the simplest case, that of a slab of infinite extent with parallel plane faces, where the temperature can be regarded as a function of one coördinate, [I] reduces to

a form of considerable importance in the consideration of the problem of the cooling of the earth’s crust.

In the problem of the permanent state of temperatures in a thin rectangular plate, the equation [I] becomes

In polar or spherical coördinates [I] is less simple, it is

In the case where the solid in question is a sphere and the temperature at any point depends merely on the distance of the point from the centre [IV] reduces to

In cylindrical coördinates [I] becomes

In considering the flow of heat in a cylinder when the temperature at any point depends merely on the distance r of the point from the axis [VI] becomes

In Acoustics in several problems we have the equation

for instance, in considering the transverse or the longitudinal vibrations of a stretched elastic string, or the transmission of plane sound waves through the air.

If in considering the transverse vibrations of a stretched string we take account of the resistance of the air [VIII] is replaced by

In dealing with the vibrations of a stretched elastic membrane, we have the equation

or in cylindrical coördinates

In the theory of Potential we constantly meet Laplace’s Equation

which in spherical coördinates becomes

and in cylindrical coördinates

In curvilinear coördinates it is

represent a set of surfaces which cut one another at right angles, no matter what values are given to ρ1, ρ2, and ρ3; and where

and, of course, must be expressed in terms of ρ1, ρ2, and ρ3.

If it happens that ∇²ρ1 = 0, ∇²ρ2 = 0, and ∇²ρ3 = 0, then Laplace’s Equation [XV] assumes the very simple form

2.A differential equation is an equation containing derivatives or differentials with or without the primitive variables from which they are derived.

The general solution of a differential equation is the equation expressing the most general relation between the primitive variables which is consistent with the given differential equation and which does not involve differentials or derivatives. A general solution will always contain arbitrary (i. e., undetermined) constants or arbitrary functions.

A particular solution of a differential equation is a relation between the primitive variables which is consistent with the given differential equation, but which is less general than the general solution, although included in it.

Theoretically, every particular solution can be obtained from the general solution by substituting in the general solution particular values for the arbitrary constants or particular functions for the arbitrary functions; but in practice it is often easy to obtain particular solutions directly from the differential equation when it would be difficult or impossible to obtain the general solution.

3.If a problem requiring for its solution the solving of a differential equation is determinate, there must always be given in addition to the differential equation enough outside conditions for the determination of all the arbitrary constants or arbitrary functions that enter into the general solution of the equation; and in dealing with such a problem, if the differential equation can be readily solved the natural method of procedure is to obtain its general solution, and then to determine the constants or functions by the aid of the given conditions.

It often happens, however, that the general solution of the differential equation in question cannot be obtained, and then, since the problem if determinate will be solved if by any means a solution of the equation can be found which will also satisfy the given outside conditions, it is worth while to try to get particular solutions and so to combine them as to form a result which shall satisfy the given conditions without ceasing to satisfy the differential equation.

4.A differential equation is linear when it would be of the first degree if the dependent variable and all its derivatives were regarded as algebraic unknown quantities. If it is linear and contains no term which does not involve the dependent variable or one of its derivatives, it is said to be linear and homogeneous.

All the differential equations collected in Art. 1 are linear and homogeneous.

5.If a value of the dependent variable has been found which satisfies a given homogeneous, linear, differential equation, the product formed by multiplying this value by any constant will also be a value of the dependent variable which will satisfy the equation.

For if all the terms of the given equation are transposed to the first member, the substitution of the first-named value must reduce that member to zero; substituting the second value is equivalent to multiplying each term of the result of the first substitution by the same constant factor, which therefore may be taken out as a factor of the whole first member. The remaining factor being zero, the product is zero and the equation is satisfied.

If several values of the dependent variable have been found each of which satisfies the given differential equation, their sum will satisfy the equation; for if the sum of the values in question is substituted in the equation each term of the sum will give rise to a set of terms which must be equal to zero, and therefore the sum of these sets must be zero.

6.It is generally possible to get by some simple device particular solutions of such differential equations as those we have collected in Art. 1. The object of the branch of mathematics with which we are about to deal is to find methods of so combining these particular solutions as to satisfy any given conditions which are consistent with the nature of the problem in question.

This often requires us to be able to develop any given function of the variables which enter into the expression of these conditions in terms of normal forms suited to the problem with which we happen to be dealing, and suggested by the form of particular solution that we are able to obtain for the differential equation.

These normal forms are frequently sines and cosines, but they are often much more complicated functions known as Legendre’s Coefficients, or Zonal Harmonics; Laplace’s Coefficients, or Spherical Harmonics; Bessel’s Functions, or Cylindrical Harmonics; Lamé’s Functions, or Ellipsoidal Harmonics, &c.

7.As an illustration, let us take Fourier’s problem of the permanent state of temperatures in a thin rectangular plate of breadth π and of infinite length whose faces are impervious to heat. We shall suppose that the two long edges of the plate are kept at the constant temperature zero, that one of the short edges, which we shall call the base of the plate, is kept at the temperature unity, and that the temperatures of points in the plate decrease indefinitely as we recede from the base; we shall attempt to find the temperature at any point of the plate.

Let us take the base as the axis of X and one end of the base as the origin. Then to solve the problem we are to find the temperature u of any point from the equation

We shall begin by getting a particular solution of [III], and we shall use a device which always succeeds when the equation is linear and homogeneous and has constant coefficients.

Assume* u = eαy + βx, where α and β are constants, substitute in [III] and divide by eαy + βx, and we have α² + β² = 0. If, then, this condition is satisfied u = eαy + βx is a solution.

Hence u = eαy ± αxi† is a solution of [III], no matter what value may be given to a.

This form is objectionable, since it involves an imaginary. We can, however, readily improve it.

Take u = eαy eαxi, a solution of [III], and u = eαy e−αxi, another solution of [III]; add these values of u and divide the sum by 2 and we have eαy cos ax. (v. Int. Cal. Art. 35, [1].) Therefore by Art. 5

is a solution of [III]. Take u = eαy eαxi and u = eαy e−αxi, subtract the second value of u from the first and divide by 2i and we have eay sin ax. (v. Int. Cal. Art. 35, [2]). Therefore by Art. 5

is a solution of [III].

Let us now see if out of these particular solutions we can build up a solution which will satisfy the conditions (1), (2), (3), and (4).

It is zero when x = 0 for all values of a. It is zero when x = π if a is a whole number. It is zero when y = ∞ if a is negative. If, then, we write u equal to a sum of terms of the form Ae− my sin mx, where m is a positive integer, we shall have a solution of [III] which satisfies conditions (1), (2) and (3). Let this solution be

A1, A2, A3, A4, &c., being undetermined constants.

When y = 0 (7) reduces to

If now it is possible to develop unity into a series of the form (8), our problem is solved; we have only to substitute the coefficients of that series for A1, A2, A3, &c. in (7).

It will be proved later that

for all values of x between 0 and π; hence our required solution is

for this satisfies the differential equation and all the given conditions.

If the given temperature of the base of the plate instead of being unity is a function of x, we can solve the problem as before if we can express the given function of x as a sum of terms of the form A sin m x, where m is a whole number.

The problem of finding the value of the potential function at any point of a long, thin, rectangular conducting sheet, of breadth π, through which an electric current is flowing, when the two long edges are kept at potential zero, and one short edge at potential unity, is mathematically identical with the problem we have just solved.

EXAMPLE.

Taking the temperature of the base of the plate described above as 100° centigrade, and that of the sides of the plate as 0°, compute the temperatures of the points

8.As another illustration, we shall take the problem of the transverse vibrations of a stretched string fastened at the ends, initially distorted into some given curve and then allowed to swing.

Let the length of the string be l. Take the position of equilibrium of the string as the axis of X, and one of the ends as the origin, and suppose the string initially distorted into a curve whose equation y = f(x) is given.

We have then to find an expression for y which will be a solution of the equation

while satisfying the conditions

the last condition meaning merely that the string starts from rest.

As in the last problem let* y = eαx + βt and substitute in [VIII]. Divide by eαx + βt and we have β² = a² a² as the condition that our assumed value of y shall satisfy the equation.

is, then, a solution of (VIII) whatever the value of a.

It is more convenient to have a trigonometric than an exponential form to deal with, and we can readily obtain one by using an imaginary value for a in (5). Replace a by ai and (5) becomes y = e(x ± αt)αi, a solution of [VIII]. Replace α by − αi and (5) becomes y = e(−x ± αt)αi, another solution of [VIII]. Add these values of y and divide by 2 and we have cos a(x ± αt). Subtract the second value of y from the first and divide by 2i and we have sin a(x ± at).

are, then, solutions of [VIII]. Writing y successively equal to half the sum of the first pair of values, half their difference, half the sum of the last pair of values, and half their difference, we get the very convenient particular solutions of [VIII].

If we take the third form

it will satisfy conditions (1) and (4), no matter what value may be given to a, and it will satisfy (2) if where m is an integer.

If then we take

where A1, A2, A3 … are undetermined constants, we shall have a solution of [VIII] which satisfies (1), (2), and (4). When t = 0 it reduces to

If now it is possible to develop f(x) into a series of the form (7), we can solve our problem completely. We have only to take the coefficients of this series as values of A1, A2, A3 … in (6), and we shall have a solution of [VIII] which satisfies all our given conditions.

In each of the preceding problems the normal function, in terms of which a given function has to be expressed, is the sine of a simple multiple of the variable. It would be easy to modify the problem so that the normal form should be a cosine.

We shall now take a couple of problems which are much more complicated and where the normal function is an unfamiliar one.

9.Let it be required to find the potential function due to a circular wire ring of small cross section and of given radius c, supposing the matter of the ring to attract according to the law of nature.

We can readily find, by direct integration, the value of the potential function at any point of the axis of the ring. We get for it

where M is the mass of the ring, and x the distance of the point from the centre of the ring.

Let us use spherical coördinates, taking the centre of the ring as origin and the axis of the ring as the polar axis.

To obtain the value of the potential function at any point in space, we must satisfy the equation

subject to the condition

From the symmetry of the ring, it is clear that the value of the potential function must be independent of ϕ, so that [XIII] will reduce to

We must now try to get particular solutions of (2), and as the coefficients ere not constant, we are driven to a new device.

Let* V = rmP, where P is a function of θ only, and m is a positive integer, and substitute in (2), which becomes

Divide by rm and use the notation of ordinary derivatives since P depends upon θ only, and we have the equation

from which to obtain P.

Equation (3) can be simplified by changing the independent variable. Let x = cos θ and (3) becomes

Assume * now that P can be expressed as a sum or as a series of terms involving whole powers of x multiplied by constant coefficients.

Let P = Σ anxn and substitute this value of P in (4). We get

where the symbol Σ indicates that we are to form all the terms we can by taking successive whole numbers for n.

As (5) must be true no matter what the value of x, the coefficient of any given power of x, as for instance xk, must vanish. Hence

If now any set of coefficients satisfying the relation (7) be taken, P = Σ akxk will be a solution of (4).

Since it will answer our purpose if we pick out the simplest set of coefficients that will obey the condition (7), we can take a set including am.

Let us rewrite (7) in the form

We get from (8), beginning with k = m − 2,

If m is even we see that the set will end with a0, if m is odd, with a1.

where am is entirely arbitrary, is, then, a solution of (4). It is found convenient to take am equal to

and it can be shown that with this value of am P = 1 when x = 1.

P is a function of x and contains no higher powers of x than xm. It is usual to write it as Pm(x).

We proceed to compute a few values of Pm (x) from the formula

We have:

We have obtained P = Pm(x) as a particular solution of (4) and P = Pm (cos θ) as a particular solution of (3). Pm (x) or Pm (cos θ) is a new function, known as a Legendre’s Coefficient, or as a Surface Zonal Harmonic, and occurs as a normal form in many important problems.

V = rmPm (cos θ) is a particular solution of (2) and rmPm (cos θ) is some times called a Solid Zonal Harmonic.

We can now proceed to the solution of our original problem.

where A0, A1, A2, &c., are entirely arbitrary, is a solution of