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we shall obtain all possible functions, and amongst others 7 functions similar to those mentioned before (Art. 21). This function, therefore, may be considered as the representative of all, and consequently p is now properly represented by
This equation is the representative of any continuous wave, whatever be its form. The integral of the equation of continuity furnishes no other general means of representing a continuous wave. Hence, whatever be the nature of the original disturbance which gives rise to a continuous wave, the above equation teaches, that the wave may be hypothetically resolved into an infinite number of plane-waves moving with the same velocity a.
Plane-waves are therefore shewn to be the proper components of currilinear waves.
And since the values of f, g, h are the cosines of the inclinations of these component-waves to the co-ordinate axes, if the original disturbance produce a single wave, f, g, h will follow some law, that law being in fact the condition that the original disturbance may be single.
The disturbance being thus resolved into component plane-waves, each component is to be supposed transmitted parallel to itself with the velocity a, and at any time we may compound them into a single wave, by finding the surface to which they are all simultaneously tangent planes.
27. In confirmation of these views, I shall make a few observations upon general and singular solutions of common differential equations of two variables, the theory of which is well understood and allowed.
Suppose, for instance, it were required to find a curve, such that the rectangle of the perpendiculars drawn from two given points upon any tangent shall be constant. This problem produces a differential equation whose general solution is
f being an arbitrary constant. This quantity gives rise to a difficulty. For though the proposed problem is sufficiently specific, yet it affords no data for the determination of f, and consequently the curve sought remains as undetermined, and the problem apparently as unsolved, as ever; all that we can gather from the above integral being, that a straight line whose position depends on the value of f will fulfil the proposed conditions.
It may be said, however, that any value given at pleasure to f will determine a line answering the conditions of the question; but it is clear that a line so found can only be considered as a partial solution; inasmuch as the fixing upon a particular value of f tacitly implies the possession of data enabling us to decide upon that value in preference to all others. Now such a decision cannot be received, unless the data which led to it are supplied by the conditions of the proposed problem; and, as we have seen, no such data exist; consequently no particular value of f can be received. Thus it appears, that though the integral
furnishes partial solutions of the proposed problem without number, it does not present us with the required curve. The only resource left is, to employ equally all possible values of f from — oc to + c ; all having an equal claim to have weight in the general interpretation of the above integral. This will present us with an infinite number of straight lines, not drawn at random, but according to a law expressed by $y = fa + vajo To,
and by the intersections of consecutive lines forming a curve to which
they all have an equal relation, being tangents, which is allowed to be the curve required.
* Since f admits the sign – as well as +, we have, by taking all the variations of
sign, four straight lines; one in each portion of space comprehended between the coordinate axes. See Arts. 18, 21.
The general inference from this reasoning appears to be this: that when the integral of a differential equation contains constants (introduced by integration), the values of which the proposed conditions of the problem are not sufficient to determine, the singular solution” is the proper integral for that particular problem.
28. Upon these grounds I state the following general principle,
A wave may, at any moment, be resolved into plane component waves, each of which is a tangent to the original wave. These components may be supposed to be uniformly transmitted with the velocity a, and at any time they may be compounded into a single wave by taking that surface to which they are all simultaneously tangents.
Hence if the form of a wave-surface at any one instant be known, its form at any other time will be determinable from geometrical prin
The thickness (or, as it is sometimes called, the breadth) of a wave is never altered by transmission.
Ex. 1. In a quiescent medium let us suppose one of its particles to expand, pushing equally from its centre on every side the adjacent particles. The effect of such a disturbance will be a sudden condensation in its neighbourhood, which we may divide into concentric spherical wave-surfaces, for each one of which p is constant; though from surface to surface p may vary. Now resolve any one of these surfaces into its components by drawing an infinite number of tangent planes to it: each one is transmitted with the velocity a parallel to its edge, and thus at the end of any period they will be equidistant from the centre of original disturbance, and be tangent planes to a spherical surface, which is therefore the form of the wave at any moment.
Ex. 2. In a quiescent medium let all the particles situated in a given
* Or rather the solution determined by the usual method of finding the singular solution, for as is well known such a one may not happen to be a singular solution but a particular integral.
Wol. VI. PART II. G G.
straight line suddenly expand equally. In this case each surface of equal density is a cylinder with hemispherical ends; the given line being the axis, and the extremities of the line being the centres of the hemispheres.
By resolving each wave-surface into component waves, by drawing tangent planes to the cylinder and hemispheres, and supposing these components transmitted, each parallel to itself with the velocity a, it will appear that at any time each wave-surface is of the form of a cylinder with hemispherical ends. The radius of the cylinder increases with the uniform velocity a.
29. It is sufficiently manifest, from what has already been done, that the form of a wave-surface depends only on the form of it at any given moment, or indeed only on its initial form ; its magnitude, but not its nature, depends on the time. This is equivalent to saying, that the nature of the equation of a wave-surface depends only on the form of the given disturbance, while the parameters of that equation depend upon the time. This will sometimes enable us to determine the properties of the wave when it is curved, without employing the general integral
Having determined the form - of the wave-surface upon the principles of Art. 28, let its equation be x (w, y, x; A, B, C...) = 0; in which A, B, C ... are parameters depending on the time only; also let the state of the fluid be expressed by the equation
Now for any point in a wave-surface p = constant; that is, any values of a, y, z which make x = 0, will make N = constant. N, therefore can only be made to change in value by writing different values of A, B, C. We may therefore say that N is a function of A, B, C... only, which are connected by the equation x = 0. We may therefore consider a, y, z as functions of A, B, C... by virtue of x = 0; and having found the values of d.”q, d, p, d.”q in terms of the partial
differential coefficients of p with regard to A, B, c... we may change the equation d’op = a (d."p + d, p + d.o.op)
into another not containing ar, y, x. It is to be observed that 4, B, C. are dependent only on t and quantities which are absolutely constant; B, C, ... are therefore expressible in terms of A, and consequently there is only one independent parameter A. Hence the process above pointed out will give us an equation between d’op, d.o.op, d, p and A: that is, p will be a function of A and t only.
30. As one of the simplest examples of this process, let us suppose the form of the waves to be spherical. Then x = 0 is a" + y + 2* = r",
r being the parameter A. Now p being a function of t and r only, and r, y, z being functions of r only, we have
which expresses the state of the fluid when the motion is in spherical waves: the former term F(r – at) shews that spherical waves may diverge from a fixed centre; and the latter f(r + at) that they may converge towards a fixed centre. The velocity of either wave is a. The motion of each particle is directed towards or from the fixed
centre Art. 3, and its velocity = d.op