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2. If the expression uda + v.dy + was be integrable either immediately or by a multiplier, the integral of the equation
which will furnish the surfaces alluded to in last article. But if the above expression should neither be integrable at once nor by a multiplier, the integral of the above equation will be of the form
and will denote, not a series of surfaces, but a series of curve lines.
3. It appears that all the particles through which the surface passes whose equation is uda + v.dy + widz = 0,
are connected by the common property proved in Art. 1; and as we have no other idea of a wave-surface than that it is the locus of particles in a similar state of disturbance, we may be permitted to take the above equation as the expression of that similarity which constitutes a wave-surface; or in other words, we may assume the equation uda + v.dy + widz = 0,
as the mathematical definition of a wave-surface, or of a wave-line, as the case may be.
By the assistance of this definition we may enunciate the proposition of Art. 1 in these terms;—
The motion of every particle of the fluid is perpendicular to the wave-surface in which it is situated.
4. It is proved by Pontécoulant in his “Théorie Analytique du système du Monde,” Tom. I. p. 163, and by most other writers on
Hydrodynamics, that if uda + v.dy + wax be at any one instant a complete
differential, it will be so as long as the motion lasts; this is the mathematical expression of the following physical fact;—
If, at any one instant, the motion of the fluid be in wave-surfaces, each surface will travel unbroken through the medium independently of all the rest; that is, as if the others did not exist.
Or, in other words, if the motion at any one instant be in wave-lines (Art. 2), then the motion can never resolve itself into wave-surfaces; and, conversely, if the motion at any one instant be in wave-surfaces, it can never break up into wave-lines.
5. If it happen that a particle be situated in two or more wavesurfaces at once, either the particle must be at rest, or the surfaces must have a contact at that point; for, if in motion, its direction must be perpendicular to all the wave-surfaces.
However complicated the motion of the fluid may be, it will always take place either in wave-lines or wave-surfaces. For the former will be the case when uda + v.dy + wax is not integrable per se or by a multiplier, and the latter when this expression is integrable.
Some of these remarks are illustrated in the following example.
Ex. Suppose the motion of the fluid to be such that uda + v.dy + wax = w” (yda — a dy).
In this case the differential equation of the wave surfaces is gyda – ardy = 0; and therefore, y = f'(t). a. is the general equation of a wave-surface in such a motion of the fluid.
Hence, all the wave-surfaces are planes passing through the axis of x, and the motion of the particles, being at right angles to them, will be in circular arcs parallel to the plane of ay.
All the particles in the axis of x will be at rest, for there the wave-surfaces intersect each other.
6. It does not appear possible to carry these investigations much farther in a perfectly general form ; it will be necessary therefore to introduce the hypothesis of the expression uda + v.dy + wal: being integrable per se. Denote its integral by p, then
p = constant = f(t).
will be the equation of a wave-surface.
The effect of this hypothesis will be, to exclude from our researches many cases of motion in wave-surfaces, and all motion in wave-lines.
FLUID MOTION OF TWO DIMENSIONS.
7. I have preferred commencing my investigations with this simple case because the results more frequently admit of perfect investigation, and are more easily and briefly expressed in words than in the case of
In this integral the forms of the functions F and F are perfectly arbitrary, to be adapted in any example to express the law of sequence (as to space) of coexistent wave surfaces, according to the nature of the original disturbance. The arbitrariness of these functions shews that the fluid can transmit a disturbance of any kind which does not violate the continuity of the fluid. a, 3 are arbitrary constants enabling us to fix the origin of co-ordinates in the most convenient position: they may besides contain functions of t, which depend upon the nature of the original agitation. The functions of f, g, t, which enter under F and F, enable us arbitrarily to fix the epoch from which the time is reckoned, and further to accommodate the wave-surfaces to any proposed form.
These observations will be fully illustrated in a subsequent part of this paper.
8. The object to which it will be necessary first to turn our attention in the above integral is the discovery of the meaning of the constants ..f. g. Whatever forms be given to F, F, whatever origin be taken for co-ordinates, whatever epoch for the time, still f and g are unaffected: and as an infinite number of quantities fulfilling the condition f* + g” = 0 may be invented, and any one set will satisfy the equation d.*q + do p = 0, which in a general view of the question is the only further condition to which they can be subjected, it follows that all imaginable values of f and g ought equally to appear in the general integral (see Art. 27); one set of values giving only a partial solution of the proposed differential equation. Hence the general integral of the equation of continuity of a moving fluid of two dimensions is
the quantities of f*, fo, so.... go, go, go.... embracing all values from — co to + co. It is manifest, however, that inasmuch as each set of values can be separately made to satisfy the equation of continuity, each set will represent a possible motion, i. e. a motion of such a nature that the fluid can transmit it. Hence the general integral just exhibited furnishes us with the following physical fact, which I believe has never * before been fully accounted for; —
and thence the superposition of disturbances has been inferred: but before this principle can be inferred, is it not necessary to shew that F (a + y V-1)+ F, (c 4-y V-1)+...
Any number of disturbances separately, though simultaneously, earcited in a fluid medium, will be separately, independently and simultaneously transmitted through the fluid, each as perfectly as though the others did not earist. See Art. 4.
9. Having ascertained this to be the meaning of the integral in its general form, it will be sufficient now to consider the transmission of one disturbance only; and if this investigation be carried on upon the general hypothesis of a single disturbance of any kind affecting the fluid, the results will be of a general character also. This point will be gained by keeping our integral under the form
From this we shall proceed to deduce the following results.
I. Motion cannot be represented by one of these functions alone. For, if possible, let motion be represented by
that is, the medium is at rest. Fo is used to denote the differential coefficient of F with regard to the quantity f'(a — a) + g (y – 3).
is a more general expression than one function F(r + y V-1)? In the Integral Calculus, we know that C, H. C., 4: C, ... represents only one constant C: are we certain that F + F + ... represents more than FP are we sure that F, F, F, ... are so essentially distinct that they
cannot be united in one function ?