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Passage of Oblique Vibrations into Tangential, or into Transverse.
cord, ce, stretched parallel to its length over a bridge, e, and put in vibration by a bow perpendicular to it, FF.
Then, if the plane of the bow and string coincide with the plane of the surface of the lamina, as in fig. 117, a, the latter will execute tangential vibrations across its breadth, and will exhibit on its upper surface a single nodal line, n n' n", as in fig. 117, but on its under none, all the sand being driven off.
Now incline the bow to the surface of the lamina, as represented in fig. 118, a, at an angle of about 20°, still keeping it perpendicular to the string, and the nodal line will assume the curvature represented in fig. 118.
If the bow be still more inclined, the curve breaks up, and at 45° of inclination, as in fig. 119, a, becomes changed into transverse and oblique lines, as in fig. 119; and it is now observed that the sand not only runs in the direction of the arrows, but also begins to leap, indicating an oblique vibration of the surface.
Lastly, when the bow is inclined 90° to the plane of the lamina, as in fig. 120, a, the vibration becomes altogether transverse, the nodal lines are similarly disposed on both sides of the plate, as in fig. 120, and the sand merely leaps up and down, till it is danced off the vibrating parts, without any tendency to creep.
152. The preceding proposition can also be extended to the vibrations of the air; and the motions of the aerial molecules in every part of a spherical wave, propagated from a vibrating body as a centre, instead of diverging like radii in all directions, so as to be always perpendicular to the surface of the wave, are all parallel to each other; in a word, they are disposed, not as in fig. 8, but as in fig. 7, agreeing with the remarks of art. 45.
And the same thing holds good, not only in air, but in liquids, as the experiments hereafter to be related satisfactorily demonstrate.
Nature of the Aerial Motions in a Sound-wave.
a. If a very thin membrane be stretched horizontally over the orifice of a circular bowl, as a drinking-cup, or harmonica-glass, (extremely thin paper wetted and glued to the edges, and then suffered to become tight by drying, answers very well,) and if fine sand be strewed on it, it becomes a most delicate detector of aerial vibrations.
b. Suppose now a circular disc of glass, held concentrically over it, with its plane parallel to that of the membrane, and set in transverse vibration, so as to form any of Chladni's acoustic figures, as for instance fig. 59. Then will this figure be imitated exactly by the sand on the membrane.
Now let the vibrating disc be shifted laterally, so as no longer to have its centre vertically over that of the membrane, but keeping its plane, as well as that of the membrane, horizontal. Still the figures marked out on the latter will be fac-similes of those on the disc, and that, whatever be the extent of lateral removal, till the vibrations become too much enfeebled by distance to have any effect at all.
c. But, in place of shifting the disc laterally, let its plane be inclined to the horizon. Immediately the figures on the membrane will change, though the vibrations of the disc remain unaltered, and the change will be the greater, the greater be the inclination of the plane of the disc to that of the membrane. And when the former plane is perpendicular to the horizon, the nodal figure on the membrane is found to be transformed into a system of straight lines, parallel to the common intersection of the two planes; and the particles of sand, instead of dancing, creep in opposite directions to meet in these lines. One of these always passes through the centre, and the whole system is analogous to what would be produced by attaching a cord to the centre of a disc, and, having stretched it very obliquely, setting it in vibration by a bow drawn parallel to the surface. In a word, the vibrations of the membrane are now tangential, and they preserve this character unchanged, however the disc be now shifted laterally, provided its plane be not turned from the vertical position.
Data on which the Vibrations of Membranes Depend.
If the disc be made to revolve about its vertical diameter, the nodal lines on the membrane will rotate, following exactly the motion of the disc.
153. The vibrations, communicated to a stretched membrane by the air, vary with the pitch, quality, and direction of the original vibrations, and also with the tension of the membrane.
a. If, cæteris immutatis, the pitch of the sound, whose vibrations, communicated through the air to the membrane, excite its motions, be altered, the membrane will still vibrate, differing in this respect from a rigid lamina, which will only vibrate by sympathy with sounds corresponding to its own subdivisions. The membrane, be it observed, will vibrate in sympathy with any sound, but every particular sound will mark out on it its own particular nodal figure, and as the pitch varies the figure varies. Thus if a slow air be played on a flute near it, each note will call up a particular form, which the next will efface, to establish its own.
b. Suppose the exciting cause be the vibration of a disc, or lamina of any form. If its mode of vibration be varied, so as to change its nodal figures, those on the membrane will vary; and if the same note be produced by different subdivisions of different sized discs, the nodal figures on the membrane will be different.
c. The effect of a change in the direction of the primitive vibrations is clearly shown by the change of inclination of the disc in the experiment of the preceding article.
d. If the tension of the membrane be varied ever so little, most material changes will take place in the figures it exhibits. If paper be the substance employed, mere hygrometric changes affect it to such a degree, that, if moistened by breathing on it, and allowed to dry while the exciting sound is continued, the nodal forms will be in a constant state of fluctuation, and will not acquire permanence, till the paper is so far dried as the state of the surrounding atmosphere will permit.
Stretched Membranes employed to Detect Sonorous Vibrations.
Indeed, this fluctuation is so troublesome in experiments of this kind, that to avoid them it is necessary to coat the upper or exposed side of the paper with a thin film of varnish. Of all substances, which can be employed for the exhibition of these beautiful experiments, M. Savart observes, by far the best is such a varnished paper stretched on a frame and moistened on the under side. The moisture diminishes the cohesion of the fibres, and renders them nearly independent of each other, and indifferent to all impulses. As a proof of this, he observes, that he has frequently obtained, on a circular membrane of paper so prepared, a nodal figure composed of no fewer than twenty concentric annuli, which is far beyond what can be obtained in any other way.
154. A very important application of the properties of stretched membranes is, to employ such a one as an instrument for detecting the existence and exploring the extent and limits of contiguous and oppositely vibrating portions of masses of air.
For, since such a membrane is thrown into vibration by all aerial vibrations of a certain force, the fact of the existence, or not, of a vibratory motion in any point of the air, of a chamber, for instance, or a box, or large organ-pipe, may be ascertained by observing whether sand strewed on it is set in motion, and arranged in regular forms, on holding the membrane at that point.
Thus if an organ-pipe be made to sound with a constant force, and the exploring membrane be so far removed from it, that the membrane shall just cease to be visibly agitated, the force of the sound being increased by a quantity not sensible to the ear, the sand will recommence its motion.
Nay, if two such pipes placed close together, be made to beat, the membrane will be seen to be agitated at the coincidences, and at rest in the interferences of their vibrations.
b. Were the membrane to be entirely destitute of tension, its vibrations would exactly coincide with those of the air.
Stretched Membranes employed to Detect Sonorous Vibrations.
it has always some tension or thickness, this modifies in a most complicated manner the effects of the direct aerial action.
c. In order to explore the actual state of the air in different parts of a vibrating mass of determinate figure, as to motion or rest, the sound should be excited and maintained by a constant cause at a high degree of intensity, especially if the mass of the air be large, as in a chamber or gallery; and to give the membrane the greatest possible sensibility, it ought to be stretched so as to be, naturally, in unison with the note sounded, so as to act as a receiver and condenser of the small aerial motions.
d. The greatest purity and intensity of the sounds, to be employed for this purpose, may be obtained by a harmonica glass, or the bell of a clock, maintained in vibration by a bow; and this may be still further augmented by adapting to it a resonant cavity, as, for instance, a large cylindrical vase of considerable diameter, closed at one end, and of such dimensions as separately to vibrate the same note. The tones thus produced, especially when large harmonica glasses are used, as M. Savart remarks, are of such intensity, that no ear can long support them, and at the same time, of such a rich and mellow quality, that all other musical sounds appear poor and harsh in comparison.
e. In order yet more to increase the sensibility of the membrane, the frame on which it is stretched should be fitted over the orifice of a similar resonant cavity. For convenience, and lest the tension of the membrane should vary by hygrometric changes, it is proper to have means of varying this at pleasure.
155. When the vibrations of the air in a chamber are examined, there are found to be nodal lines, which, in the case of rectangular chambers or galleries, wind spirally round the walls as in rectangular rods, and will even extend into the open air through a window.
These nodal lines are rendered evident to the ear by the diminution of sound; and there is also a re