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Remark on the Origin of Harmony.
represented by 1, 2, 3, &c.; in that of a rod they are represented by the squares of these numbers 1, 4, 9, &c., which correspond to double the former intervals. In all other cases the series is still less simple.
125. This alone suffices to show the insufficiency of any attempt to establish, as some have wished to do, the whole theory of harmony and music on the aliquot subdivision of a vibrating string.
Had vibrating rods or steel springs (which yield an exquisite tone) been always used instead of stretched cords, such an idea would never have suggested itself; yet no doubt our notions of harmony would have been what they now are. The same remark applies still more forcibly to the modes of subdivision of vibrating surfaces, which in many cases have their harmonics altogether irreducible to any musical scale.
126. The most simple modes of vibration of a rectangular surface are those, which exhibit quiescent or nodal lines parallel to one of its edges.
a. A rectangular plate may be regarded as an assemblage of straight rods of equal length, ranged parallel to each other. Supposing such an assemblage all set in vibration similarly and at once, they will retain their parallel juxtaposition during their vibration, and may, therefore, be supposed to adhere and form a plate. Consequently, among the possible series of vibrations of a rectangular plate will be found all those of a rigid rod, and the harmonics will be the same as those of the rod.
b. When the same mode of vibration of different plates is compared, the number of vibrations is inversely as the square of the length of the plate; but increase of breadth occasions no difference in the sound; and the distance from a free end to a nodal line is rather less than half the distance between two nodal lines.
Vibrations of a Rectangular Plate in their Simplest Case.
c. Fig. 43 a. shows the situations of the nodal lines, when the ends of the plate are free, and the number of the nodal lines is 2, 3, 4, and 5. Figs. 43 b. and c. are profiles of the preceding, and represent the curvature of each parallel fibre perpendicular to the nodal lines at the two opposite limits of their vibrations. The quantity of motion at each point is indicated by the corresponding ordinate of the curve, and its direction by its situation above or below the horizontal line.
d. It will be convenient to distinguish these states of motion, in which every corresponding point is moving in direct opposition. The first, b. will therefore be called the positive states of vibration; and the second, c. the negative states of vibration. When there is an even number of nodal lines, the positive state of vibration may be considered as that in which the motion at the central part is above the plane of equilibrium, and the negative, that in which it is below it.
127. The subdivision of a plane surface in vibration by its nodal lines may be rendered visible to the eye, by holding it in a horizontal position and strewing it over with sand; for the sand will be thrown away from the vibrating parts and accumulate on those at
In the preceding cases of the rectangular plate, the sand will then be arranged in straight lines across the plate parallel to its edges, and their distances apart and from the ends of the plate may be measured at leisure.
128. If we suppose two similar surfaces with the same number of nodal lines to be superposed, and both to vibrate in concurrence, i. e. both either positively or negatively, they will mutually assist each other's effects; but if they vibrate in opposing directions, they will destroy each other's motions, and the entire surface will be at rest.
Coexistence of Vibrations. Superposition of two Similar Modes of Vibration.
129. When the rectangular surface is a square, it is obvious that it may vibrate in two different rectangular directions, so as to give the same sound, and present the same arrangement of nodal lines. Now, by the principle of the superposition of vibrations, these two modes of vibration may coexist, and produce a compound, in which the position of the nodal lines will be greatly changed; but the number of vibrations, and consequently the pitch, will not materially differ from that of the components.
a. If a plate is excited at a point where the motion of each rectangular mode of vibration is at its maximum, in the same direction, and of equal intensity; there is no reason why one mode of vibration should be produced in preference to the other. On calculating the effect of such coexistence, Mr. Wheatstone found that, the resultants of these combined modes of vibration, similar in every thing but in their direction with regard to the sides of the plate, gave rise to new nodal lines which accurately corresponded with figures described by Chladni.
b. The principal results of the superposition of two similar modes of vibration are these.
First. The points, where the nodal lines of each figure intersect each other, remain nodal points in the resulting figure.
Secondly. The nodal lines of one figure are obliterated, when superposed, by the vibrating parts of the other.
Thirdly. New nodal points or lines are formed wherever the vibrations in opposite directions neutralize each other.
Fourthly. At all other points the motion is as the sum of the concurring, or the difference of the opposing, vibrations.
c. A primary figure, having an even number of nodal lines, may be superposed two ways, and may consequently give rise to two distinct resultant figures; one, when the central vibrating parts concur; and the other, when they are in opposition.
Superposition of two Similar Modes of Vibration.
But if the number of nodal lines in the primary figure is uneven, there can be only one resultant figure.
d. The nodal lines which thus result may be very easily ascertained.
Take, as an example, the first mode of vibration having two parallel nodal lines. This, being superposed in two rectangular directions, and so that the states of vibration are opposing, (fig. 44,) it is obvious that no lines of compensation can exist in the four rectangular segments a a a a, as every point included within them is actuated by concurrent motions. But, in all the other rectangles, they must necessarily be forined, as every point within them is affected by two opposing motions; and if the two modes be of equal intensity, the compensations must occur at every point equally distant from the two rectangular nodal lines, each appertaining to a different mode of vibration. The resultant figure will thus be found to consist of two diagonal lines, perpendicular to each other, and passing through the centre of the plate.
But if the two superpositions vibrate in concurrence, (fig. 45,) the rectangles bbbb will be free from compensating points; but these will occur in the other rectangles, and form a figure which also consists of diagonal lines.
e. In the same manner the resultant of any two similar modes of vibration with nodal lines, parallel to the sides, may be proved to consist of lines parallel to the diagonals.
130. It is not a necessary condition for the vibrations of a square plate, that the primary nodal lines shall be parallel to a side; they may also be parallel to a diagonal, or to any line intermediate between a transverse and a diagonal line; and these vibrations may be distinguished by the term primary vibrations.
131. In these cases the superpositions take place according to the following rule.
Superposition of two Similar Oblique Vibrations of a Square.
The axes of the superposed modes of vibration must make equal angles with a line parallel to the edges of the plate passing through its centre; for, otherwise, the modes of vibration would not be similar.
a. By the axis of a primary mode of vibration is meant a straight line passing through the centre of the plate, and parallel to the nodal lines.
b. Considerations, of the kind already employed, will show that in all these instances the resultant figures consist of lines parallel to the edges of the plate, and that they are always the same in number as the nodal lines of a component mode of vibration, but differently distributed in the two directions, according as the angle of superposition varies.
c. Some of the various primary modes of vibration, transverse, intermediate, and diagonal, and the angles which the nodal lines of two similar figures make with each other when they are superposed, are represented in the first column of Plates 5 and 6. In the second column of these plates are placed the figures resulting from their opposing superpositions; and in the third column those which arise from their concurring superpositions.
132. We obtain by experiment a limited number only of figures, which can be considered the resultants of primary modes of vibration consisting of any given number of oblique lines; but it would seem, that as the various degrees of obliquity are infinite, so there should be an infinite number of resultant figures passing into each other by insensible gradations. By calculation this should be so; but there is a cause of limitation in the circumstance that no resultant figure is maintainable, unless the greatest excursions of the external vibrating parts occur at the edges of the plate.