On the partial differential equations of mathematical physics

On the partial differential equations of mathematical physics

By E. T. Whittaker, in Cambridge. Mathematische Annalen, Band 57, pages 333 to 355.

1. Introduction

The object of this paper is the solution of Laplace's potential equation — the sum of the second derivatives of the potential with respect to the three rectangular coordinates equated to zero — and of the general differential equation of wave-motions — the same sum equated to a constant multiplied by the second derivative with respect to the time — and of other equations derived from these.

In section 2, the general solution of the potential equation is found.

In section 3, a number of results are deduced from this, chiefly relating to particular solutions of the equation, and expansions of the general solution in terms of them.

In section 4, the general solution of the differential equation of wave-motions is given.

In section 5, a number of deductions from this general solution is given, including a theorem to the effect that any solution of this equation can be compounded from simple uniform plane waves, and an undulatory explanation of the propagation of gravitation.

2. The general solution of the potential equation

We shall first consider the equation which was originally given by Laplace in the Mémoire sur la théorie de l'anneau de Saturne, 1787.

This equation is satisfied by the potential of any distribution of matter which attracts according to the Newtonian Law. We shall first obtain a general form for potential-functions, and then shall shew that this form constitutes the general solution of Laplace's equation.

From the identity — the reciprocal of the distance between the point x, y, z and the point a, b, c written as one over two pi times the integral, over u from zero to two pi, of du divided by the quantity formed from z minus c, plus i times x minus a multiplied by cosine u, plus i times y minus b multiplied by sine u — we see that the potential at any point x, y, z of a particle of mass m situated at the point a, b, c is an integral over u from zero to two pi which, considered as a function of x, y and z, is an expression of the type the integral, over u from zero to two pi, of a function of the two arguments z plus i x cos u plus i y sin u, and u.

It follows that the potential of any number of particles situated at any number of points is an expression of the same type, since a sum of such functions is again a function of the same two arguments. In this way we see that the potential of any distribution of matter which attracts according to the Newtonian Law can be represented by an expression of that type.

The question now naturally suggests itself, whether the most general solution of Laplace's equation can be represented by an expression of this type. We shall shew that the answer to this is in the affirmative.

For let V be any solution, single-valued or many-valued, of the equation. Let a point be chosen at which some branch of the function V is regular. Then writing the coordinates as that point's coordinates plus the increments X, Y and Z, it follows that for all points situated within a finite domain surrounding the point, this branch of the function V can be expanded in an absolutely and uniformly convergent power series in X, Y and Z. Substituting this expansion in Laplace's equation and equating to zero the coefficients of the various powers of X, Y and Z, we obtain an infinite number of linear relations between the constants in the expansion.

There are one half of n times n minus one of these relations between the one sixth of n plus one times n plus two coefficients of the terms of any degree n in the expansion of V; so that only two n plus one of the coefficients of terms of degree n in the expansion of V are really independent. It follows that the terms of degree n in V must be a linear combination of two n plus one linearly independent particular solutions of Laplace's equation, which are of degree n in X, Y and Z.

To find these solutions, consider the expansion of the quantity Z plus i X cos u plus i Y sin u, raised to the power n, as a sum of sines and cosines of multiples of u. Now the coefficient functions in that expansion are together characterised by the fact that the highest power of Z contained in them is fixed; moreover the cosine coefficients are even functions of Y whereas the sine coefficients are odd functions of Y; and hence the two n plus one quantities so obtained are linearly independent of each other. They are clearly homogeneous polynomials of degree n in X, Y and Z, and each of them satisfies Laplace's equation, since the quantity raised to the power n does so. They may therefore be taken as the two n plus one linearly independent solutions of degree n of Laplace's equation.

Now since by Fourier's Theorem each of these coefficient functions is itself an integral, over u from zero to two pi, of the same quantity raised to the power n multiplied by the cosine or the sine of a multiple of u, it follows that each of these two n plus one solutions can be expressed in the form of an integral over u from zero to two pi of a function of the two arguments; and therefore any linear combination of these solutions can be expressed in this form. That is, the terms of any degree n in the expansion of V can be expressed in this form; and therefore V itself can be expressed in the form of an integral over u from zero to two pi of a function of the two arguments, since the constant part of the argument contributed by the chosen point can be absorbed into the second argument u.

Now V was taken to be any solution of Laplace's equation, with no restriction beyond the assumption that some branch of it was at some point a regular function — an assumption which is always tacitly made in the solution of differential equations; and thus we have the result, that the general solution of Laplace's equation is the integral, over u from zero to two pi, of an arbitrary function of the two arguments z plus i x cos u plus i y sin u, and u. Moreover, it is clear from the proof that no generality is lost by supposing that this function is a periodic function of u.

This Theorem is the three-dimensional analogue of the theorem that the general solution of Laplace's equation in two dimensions is a function of x plus i y added to a function of x minus i y.

3. Deductions from the theorem of section 2; particular solutions; expansions of the general solution

1. Interpretation of the solution. We may give to the general solution just obtained a concrete interpretation, as follows.

Since a definite integral can be regarded as the limit of a sum, we can regard V as the sum of an infinite number of terms, each of the type a function of z plus i x cos u-sub-r plus i y sin u-sub-r, each term corresponding to some value of u-sub-r. But this term is a solution of a two-dimensional Laplace equation in the coordinates X-sub-r and z, where X-sub-r equals x cos u-sub-r plus y sin u-sub-r and Y-sub-r equals minus x sin u-sub-r plus y cos u-sub-r, so that these represent coordinates derived from x, y and z by a rotation of the axes through an angle u-sub-r round the axis of z.

Thus we see that the general solution of Laplace's equation can be regarded as the sum of an infinite number of elementary constituents, each constituent being the solution of a two-dimensional equation, the axes being derived from the original axes by a simple rotation round the axis of z.

2. The particular solutions in terms of Legendre functions. It is interesting to see how the well-known particular solutions of Laplace's equation in terms of Legendre functions can be obtained as a case of the solution given in section 2. The particular solutions in question are of the form r to the power n multiplied by an associated Legendre function of cos theta and by the cosine or the sine of m times phi, where r, theta and phi are the polar coordinates corresponding to the rectangular coordinates x, y and z.

Now the Legendre function can be expressed by an integral, and thus each of these solutions is seen to be a numerical multiple of the integral, over u from zero to two pi, of z plus i x cos u plus i y sin u, raised to the power n, multiplied by the cosine or the sine of m times u.

From this it is clear that in order to express any solution of Laplace's equation as a series of harmonic terms of that form, it is only necessary to expand the arbitrary function as a Taylor series with respect to its first argument, and as a Fourier series with respect to its second argument.

As an example of this procedure, we shall suppose it required to find the potential of a prolate spheroid in that form, and to expand this potential as a series of harmonics. Let the surface of the spheroid be a homogeneous attracting body of mass M. To find its potential, we can make use of the theorem that the potential at external points is the same as that of a rod joining the foci, of a stated line-density. Expanding the integrand in ascending powers gives the potential in the required form, and this gives the required expansion of the potential of the spheroid in Legendre functions. This result may be extended to the case of the potential of an ellipsoid with three unequal axes, by using a formula for the potential of an ellipsoid given by Laguerre in 1878.

3. The particular solutions of Laplace's equation which involve Bessel functions. We shall next shew how the well-known particular solutions of Laplace's equation in terms of Bessel functions can be obtained as a case of the general solution. The particular solutions in question are of the form an exponential in z multiplied by a Bessel function of k times rho and by the cosine or the sine of m times phi, where k and m are constants and rho, phi and z are the cylindrical coordinates corresponding to the rectangular coordinates.

Now if in that solution we replace the Bessel function by its value as the integral, over theta from zero to pi, of the cosine of m theta minus k rho sine theta, divided by pi, we find after a few simple transformations that the solution is a numerical multiple of an integral, over u from zero to two pi, of an exponential of k times the quantity z plus i x cos u plus i y sin u, multiplied by the cosine or the sine of m times u.

It follows from this that in order to express any solution of Laplace's equation as a sum of terms of the Bessel type, it is only necessary to expand the arbitrary function in terms of exponentials of its first argument, and as a Fourier series with respect to its second argument.

As an example of the use which may be made of these results, we shall suppose it required to express the potential function one plus the sum, over the positive integers, of an exponential of minus n z multiplied by the Bessel function of order zero of n rho as a series of harmonic terms of the type involving Legendre functions, and also to find a distribution of attracting matter of which this is the potential. Writing each term as an integral over u and summing the geometric series, the potential becomes a single integral. But if a variable is different from zero and its modulus is less than two pi, the reciprocal of one minus its exponential can be expanded in a series whose coefficients are Bernoulli's numbers. Therefore, so long as z is positive and the modulus of the argument is less than two pi — that is, so long as the distance from the origin is less than two pi — we obtain the required expansion of the potential as a series of harmonics involving Legendre functions.

Next, resumming the series in a different way, the potential can be written as a sum of terms each of the form of the reciprocal distance from a point on the axis of z displaced by an imaginary multiple of two pi i. Therefore the potential may be regarded as due to a set of attracting masses placed at equal imaginary intervals of two pi i along the axis of z.

4. The differential equation of wave-motions

We shall next consider the general differential equation of wave-motions — the sum of the second derivatives of V with respect to the three rectangular coordinates, equated to a constant k squared multiplied by the second derivative of V with respect to the time — where k is a constant. Writing k t for t, this becomes the equation in which the sum of the three spatial second derivatives equals the second derivative with respect to the new time variable, which we shall take for the present as the standard form of the equation.

In order to find the general solution of this equation, we follow a procedure analogous to that of section 2. Let V be any solution, single-valued or many-valued, of the equation, and let a place be chosen at which some branch of the function V is regular. Then writing the coordinates and the time as that place's values plus the increments X, Y, Z and T, it will be possible to expand this branch of the function V as a power-series in X, Y, Z and T, which will be absolutely and uniformly convergent for a certain finite domain of values.

Substituting this expansion in the differential equation and equating to zero the coefficients of the various powers, we obtain an infinite number of linear relations between the constants in the expansion. There are one sixth of n minus one times n times n plus one of these relations between the one twenty-fourth of n plus one times n plus two times n plus three coefficients of terms of any degree n in the expansion of V; so that only n plus one, squared of the coefficients of terms of degree n in the expansion of V are really independent. It follows that the terms of degree n in V must be a linear combination of n plus one, squared linearly independent particular solutions of degree n in X, Y, Z and T.

To find these solutions, consider the expansion of the quantity X sin u cos v plus Y sin u sin v plus Z cos u plus T, raised to the power n. If we first take the expansion as a sum of sines and cosines of multiples of v, we have seen in section 2 that the coefficient functions are linearly independent functions of X, Y, Z and T. Moreover, each coefficient is of the form of a power of sine u multiplied by a polynomial in cosine u, and therefore each of them contains a determinate number of independent polynomials in X, Y, Z and T. Thus the total number of independent polynomials in the expansion, in sines and cosines of multiples of u and v, comes to n plus one, squared.

Now each of these polynomials must satisfy the equation, since the quantity raised to the power n does so; and therefore they may be taken as the n plus one, squared linearly independent solutions of the equation which are homogeneous of degree n in X, Y, Z and T.

Now by Fourier's theorem each coefficient is an integral of the same quantity raised to the power n multiplied by a cosine of a multiple of v; and since each is of the form of a power of sine u multiplied by a polynomial in cosine u, it is clear that it can be expressed as a sum of sines or cosines of multiples of u, according as the index is even or odd. It follows that each of the polynomials in question can be expressed as a double integral over u and v of the same quantity raised to the power n, multiplied by a periodic function of u and a cosine of a multiple of v.

It follows from this that each of the n plus one, squared polynomial solutions of degree n can be expressed as a double integral, over u and v, of the quantity raised to the power n multiplied by some periodic function of u and v; and therefore the terms of degree n in V can be expressed in this form. The function V itself can therefore be expressed as a double integral of a function of the three arguments X sin u cos v plus Y sin u sin v plus Z cos u plus T, and u, and v; and that function may without loss of generality be supposed to be periodic in u and v.

Now the projection formed with the increments differs from the projection formed with the original coordinates and time only by the constant projection of the chosen point, and that constant can be absorbed into the arguments u and v; moreover V was taken to be any solution of the partial differential equation. We have, therefore, on restoring the original time variable, the result that the general solution of the partial differential equation of wave-motions is the double integral, over u from zero to pi and v from zero to two pi, of an arbitrary function of the three arguments: x sin u cos v plus y sin u sin v plus z cos u plus the time divided by k, together with u and v.

5. Deductions from the general solution of section 4

1. The analysis of wave-motions. We shall now deduce from the general solution thus obtained a result relating to the analysis of those phenomena which are represented by solutions of the equation.

If we revert to the fundamental idea of the definite integral as the limit of a sum of an infinite number of terms, we see that the general solution can be interpreted as meaning that V is the sum of an infinite number of terms, there being one of these terms corresponding to every direction in space given by the direction-cosines sine u cosine v, sine u sine v, cosine u. The solution V can therefore be regarded as the sum of constituent solutions, each a function of the projection along one direction plus the time divided by k, where the function varies from one direction to another.

Now let us fix our attention on one of these constituent solutions. If for some range of values of that argument the function is finite and continuous, we can for this range of values express it by Fourier's integral formula; or, supposing the integration with respect to the auxiliary variable to be performed, as an integral over a wavenumber of the cosine of that wavenumber multiplied by the argument, weighted by some function of the wavenumber.

Now let us again revert to the idea of the definite integral as the limit of a sum. Then this latter integral can be regarded as the sum of an infinite number of terms of the type cosine, or sine, of the wavenumber multiplied by the quantity x sin u cos v plus y sin u sin v plus z cos u plus the time divided by k, each term being multiplied by some factor depending on the wavenumber.

The solution can therefore be regarded as constituted by the superposition of terms of this last type. But a term of this type represents a simple uniform plane wave; for on transforming the axes so that the new axis of x is the line whose direction-cosines are sine u cosine v, sine u sine v, cosine u, the term becomes an expression which represents a simple plane wave whose direction of propagation is the new axis of x.

We see therefore that the general finite solution of the differential equation of wave-motions can be analysed into simple plane waves, represented by terms of that type.

It is interesting to observe that Dr. Johnstone Stoney in 1897, in the Philosophical Magazine, shewed by physical reasoning, and without any reference to the equation, that all the disturbances of the luminiferous ether arising from sources of certain kinds can be resolved into trains of plane waves.

2. Solution of the reduced equation. If a solution of the wave equation be of the form of a function of x, y and z only, which does not involve the time, multiplied by an exponential in the time, then that function clearly satisfies the reduced equation in which the sum of the three spatial second derivatives is equated to minus the function itself. Therefore, on reference to the general solution of the wave-motion equation found in section 4, we see that the general solution of the reduced equation is the double integral, over u and v, of an exponential of i times the projection x sin u cos v plus y sin u sin v plus z cos u, multiplied by an arbitrary function of u and v.

3. Deduction of the known particular solutions of the reduced equation. It is known that particular solutions of the reduced equation exist which are of the form r to the power minus one half, multiplied by a Bessel function of order n plus one half of r, multiplied by an associated Legendre function of cosine theta and by the cosine or the sine of m times phi, where r, theta and phi are the polar coordinates. We shall now shew how these may be derived from the general solution just found.

Let the general solution be written with the arbitrary function of u and v multiplied by sine u in the element of integration. Now let that function be expanded in surface-harmonics of u and v. If the direction is regarded as a point in space, then each term of the expansion is a homogeneous polynomial of the corresponding degree satisfying Laplace's equation.

Next, let the variables be changed by a substitution which takes as the new axis the line whose direction-cosines are sine theta cosine phi, sine theta sine phi, cosine theta. A surface-harmonic of any order remains a surface-harmonic of the same order under any transformation of axes in which the origin is unchanged; and therefore each term can be expanded in Legendre functions of the new polar angle, with coefficients which are functions of theta and phi. Substituting this value in the integral, and performing the integration with respect to the new azimuth, and using a known relation between the integral of an exponential against a Legendre function and the half-odd-order Bessel function — a proof of which, and of several related results, will be found in a paper shortly to be published by the author — the solution can be written as a sum over n of terms in which the radial factor is r to the power minus one half multiplied by the Bessel function of order n plus one half, and the angular factor is some function of theta and phi.

Since the surface-harmonics were independent of each other, these angular functions will be independent of each other, and therefore each of the resulting quantities will be a solution of the equation. But on transforming the equation to polar coordinates and substituting that expression, we find that the angular function must satisfy the differential equation for a surface-harmonic of order n in theta and phi. It follows that it can be expanded in associated Legendre functions with cosines and sines of multiples of phi, and thus the particular solutions are obtained.

Moreover, it is clear from the above proof that in order to expand any solution of the reduced equation as a series of such terms, it is only necessary to expand the arbitrary function of u and v in surface-harmonics of u and v.

4. Expression of the solution as a series of generalised Bessel functions. Another analysis of the solutions of the reduced equation, entirely different from that given above, can be found in the following way. Consider an exponential expression built from two auxiliary variables s and t. If this expression be regarded as a function of s and t, it can for finite non-zero values of s and t be expanded as a series of positive and negative integral powers of s and t, the coefficients in this series being functions of x, y and z. Let the coefficient of a given pair of powers be denoted by a two-index function. This equation can be regarded as a generalisation of the equation which defines the ordinary Bessel functions; and we shall consequently call these coefficients generalised Bessel functions.

We now proceed to establish some properties of these functions, which are similar to those of the ordinary Bessel functions.

In the first place, since the exponential expression satisfies the reduced equation, it follows that each of the generalised Bessel functions satisfies that same equation.

In the second place, we shall obtain an expression for them as a definite integral. By Laurent's theorem, we know that the coefficient of one power is a contour integral in the s-plane surrounding the origin; and again applying Laurent's theorem, the coefficient of the second power in this expression is seen to be a further contour integral in the t-plane surrounding the origin. Now write each auxiliary variable as an exponential of an angle. Thus we have a double integral over two angles from zero to two pi, which may be regarded as the analogue of Bessel's integral — the integral, over u from zero to pi, of the cosine of n u minus z sine u, divided by pi.

The generalised functions likewise possess an addition-theorem: from the identity obtained by displacing the coordinates, and equating coefficients on both sides, we have an addition-theorem for the generalised Bessel functions, which is the analogue of the well-known result for the ordinary ones.

We shall now shew how the generalised Bessel functions furnish an analysis of the general solution of the reduced equation. For the general solution is, by point 2 above, the double integral of the exponential against an arbitrary function which can without loss of generality be taken to be periodic in u and v. Now let that function be expanded by the extended form of Fourier's theorem as a double series in exponentials of multiples of u and v. Comparing the result with the form just found for the generalised Bessel functions, we see that the general solution of the reduced equation can be written as a double series of generalised Bessel functions with arbitrary constant coefficients. This furnishes an alternative analysis of the solution to that given above.

5. Gravitation and electrostatic attraction explained as modes of wave-disturbance. The result of point 1, namely that any solution of the wave equation can be analysed into simple plane waves, throws a new light on the nature of those forces, such as gravitation and electrostatic attraction, which vary as the inverse square of the distance. For if a system of forces of this character be considered, their potential, or their component in any given direction, satisfies Laplace's equation, and therefore a fortiori it satisfies the equation of wave-motions, where the constant is any constant whatever.

It follows from point 1 that this potential, or force-component, can be analysed into simple plane waves in various directions, each wave being propagated with constant velocity. These waves interfere with each other in such a way that, when the action has once been set up, the disturbance at any point does not vary with the time, and depends only on the coordinates of the point.

It is not difficult to construct, synthetically, systems of coexistent simple waves having this property that the total disturbance at any point, due to the sum of all the waves, varies from point to point but does not vary with the time. A simple example of such a system is the following.

Suppose that a particle is emitting spherical waves, such that the disturbance at a distance r from the origin, at time t, due to those waves whose wave-length lies between two pi over mu and two pi over mu plus d mu, is represented by two d mu over pi, multiplied by the sine of mu V t minus mu r, divided by r, where V is the velocity of propagation of the waves. Then after the waves have reached the point r, so that V t minus r is positive, the total disturbance at this point, due to the sum of all the waves, is the integral of that expression over all values of mu. Taking mu V t minus mu r as a new variable, this disturbance becomes an integral of the sine of that variable divided by the variable; and since the integral, from zero to infinity, of the sine of a variable divided by that variable is one half of pi, the total disturbance is simply one divided by r.

The total disturbance at any point, due to this system of waves, is therefore independent of the time, and is everywhere proportional to the gravitational potential due to the particle at the point.

It is clear from the foregoing that the field of force due to a gravitating body can be analysed, by a "spectrum analysis" as it were, into an infinite number of constituent fields; and although the whole field of force does not vary with the time, yet each of the constituent fields is of an undulatory character, consisting of a simple wave-disturbance propagated with uniform velocity. This analysis of the field into constituent fields can most easily be accomplished by analysing the potential of each attracting particle into terms of the type sine of mu V t minus mu r, divided by r, as in the example already given. To each of these terms will correspond one of the constituent fields.

In each of these constituent fields the potential will be constant along each wave-front, and consequently the gravitational force in each constituent field will be perpendicular to the wave-front, that is, the waves will be longitudinal.

But these results assimilate the propagation of gravity to that of light: for the undulatory phenomena just described, in which the varying vector is a gravitational force perpendicular to the wave-front, may be compared with the undulatory phenomena made familiar by the magnetic theory of light, in which the varying vectors consist of electric and magnetic forces parallel to the wave-front. The waves are in other respects exactly similar, and it seems probable that an identical property of the medium ensures their transmission through space.

This undulatory theory of gravity would require that gravity should be propagated with a finite velocity, which however need not be the same as that of light, and may be enormously greater.

Of course, this investigation does not explain the cause of gravity; all that is done is to shew that in order to account for the propagation across space of forces which vary as the inverse square of the distance, we have only to suppose that the medium is capable of transmitting, with a definite though large velocity, simple periodic undulatory disturbances, similar to those whose propagation by the medium constitutes, according to the electromagnetic theory, the transmission of light.

(Every displayed formula in the original is given above as a named result, and the two most equation-dense passages — the worked spheroid expansion in section 3 point 2 and the construction of the generalised Bessel functions in section 5 point 4 — are given in outline; the complete text is in the digitised volume of Mathematische Annalen, Band 57. Whittaker’s companion paper of 1904, reducing the whole electromagnetic field to two scalar potential functions, is at /library/stm-db53a69c6c.)