Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/129

being fixed is necessarily a node, and ${\text{B}}$ being free is the middle of a ventral segment. We have thus a succession of cases in which the rod contains ${\tfrac {1}{2}}$, ${\tfrac {3}{2}}$, ${\tfrac {5}{2}}$, &c. ventral segments. The numbers of vibrations per second are as the squares of these, or, as $1:9:25:$ &c. The reason of this is, that (taking the case of fig. 20, 3) the part ${\text{FB}}$, which may be regarded as an independent rod fixed at the end ${\text{F}}$, is evidently ${\tfrac {1}{5}}$ of the length of ${\text{AB}}$, and consequently, since $n\chi {\tfrac {1}{l^{2}}}$, has a proper note of $5^{2}$ or $25$ times the rapidity of vibration in fig. 20, 1.

By attaching, with a little bees' wax, stiff hog's bristles to one prong of a tuning-fork, or to the edge of a bellglass, or even a common jar, and clipping them on trial to suitable lengths, we shall find that, on drawing a note in the usual way from the tuning-fork or glass, the bristles will divide into one or more separately vibrating segments, as in the above figs.

68.The tuning-fork itself may be regarded as belonging to the class of stiff rods. When emitting its fundamental note, it vibrates, as in fig. 21, with nodes at $b$ and $d$ and extreme positions $abcde$ and $fbgdh$.

69.The transversal vibrations of thin square, circular, and other plates of metal or glass, are interesting, because, if these are kept in a horizontal position, light dry sand or powder sifted over the upper surface, will be thrown off the ventral segments to the nodal lines, which will thus be rendered manifest to the eye, forming what are termed Chlandni's figures. As in the case of a musical string, so here we find that the pitch of the note is higher for a given plate the greater the number of ventral segments into which it is divided; but the converse of this does not hold good, two different notes being obtainable with the same number of such segments, the position of the nodal lines being, however, different.

70.The upper line of annexed figures shows how the sand arranges itself in three cases, when the plates are square.

Fig. 22.

idealised form, and as usually to be found in acoustical works. Fig. 22, 1 corresponds to the lowest possible note of the particular plate used; Fig. 22, 2 to the fifth higher; Fig. 22, 3 to the tenth or octave of the third, the numbers of vibration in the same time being as $2$ to $3$ to $5$.

If the plate be small, it is sufficient, in order to bring out the simpler sand-figures, to hold the plate firmly between two fingers of the same hand placed at any point where at least two nodal lines meet, for instance the centre in (1) and (2), and to draw a violin bow downwards across the edge near the middle of a ventral segment. But with larger plates, which alone will furnish the more complicated figures, a clamp-screw must be used for fixing the plate, and, at the same time, one or more other nodal points ought to be touched with the fingers while the bow is being applied. In this way, any of the possible configurations may be easily produced.

Fig. 23.

71.By similar methods, a circular plate may be made to exhibit nodal lines dividing the surface by diametral lines into four or a greater, but always even, number of sectors, an odd number being incompatible with the general law of stationary waves that the parts of a body adjoining a nodal line on either side must always vibrate oppositely to each other.

Another class of figures consists of circular nodal lines along with diametral (fig. 23). Circular nodal lines unaccompanied by intersecting lines cannot be produced in the manner described; but may be got either by drilling a small hole through the centre, and drawing a horse-hair along its edge to bring out the note, or by attaching a long thin elastic rod to the centre of the plate, at right angles to it, holding the rod by the middle and rubbing it lengthwise with a bit of cloth powdered with resin, till the rod gives a distinct note; the vibrations are communicated to the plate, which consequently vibrates transversely, and causes the sand to heap itself into one or more concentric rings.

72.The theory of the vibrations of plates has not yet been put on a quite satisfactory basis. The following law may, however, be regarded as confirmed by experiment, viz., that when two different plates of the same substance present the same nodal configuration, the numbers of vibrations are to each other directly as the thicknesses, and inversely as the superficial areas.

73.Paper, parchment, or any other thin membrane stretched over a square, circular, &c., frame, when in the vicinity of a sufficiently powerful vibrating body, will, through the medium of the air, be itself made to vibrate in unison, and, by using sand, as in previous instances, the nodal lines will be depicted to the eye, and seen to vary in form, number, and position with the tension of the plate and the pitch of the originating sound. The membrana tympani or drum of the ear has, in like manner and on the same principles, the property of repeating the vibrations of the external air which it communicates to the internal parts of the ear.

74.Rods vibrating longitudinally are, as we have already remarked, subject to the laws of stationary waves. If, for instance, a wooden rod fixed at one end, be rubbed near the top between the finger and thumb previously coated with powdered resin, it will yield a fundamental note when it so vibrates as to have only one node (at the fixed extremity) and half a ventral segment reaching from that extremity to the other, that is, when the length $l$ of the rod is ${\tfrac {1}{4}}\lambda$, or $\lambda =4l$, and therefore $n={\tfrac {\text{V}}{4l}}$. But it may also give overtones corresponding to $2$, $3$, &c. nodes, the free end being always the middle of a ventral segment, and for which therefore the lengths of waves are ${\tfrac {4l}{3}}$, ${\tfrac {4l}{5}}$ &c. (as will be easily seen by referring to figs, in § 67, which may equally represent transversal and longitudinal displacements). Hence, the fundamental and harmonics of a rod such as we are now considering, have vibrations whose rates are as the successive odd numbers.

A series of like rods, each fixed at one end into a block of wood, and of lengths bearing to each other, the ratios $1:{\tfrac {8}{9}}:$ &c. (as in § 61), will give the common scale when rubbed in the manner already mentioned. This follows from the fundamental having $n={\tfrac {\text{V}}{4l}}$, and therefore $n\chi {\tfrac {1}{l}}$.

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