§ 8 Wheatstone’s Analysis of the Vibrations of Square Plates
And now let us look a little more closely into the mechanism of these vibrations. The manner in which a bar free at both ends divides itself when it vibrates transversely has been already explained. Rectangular pieces of glass or of sheet metal—the glass strips of Fig. 69. the harmonica, for example—also obey the laws of free rods and bars. In Fig. 69 is drawn a rectangle a, with the nodes corresponding to its first division marked upon it, and underneath it is placed a figure showing the manner in which the rectangle, looked at edgewise, bends up and down when it is set in vibration.42 For the sake of plainness the bending is greatly exaggerated. The figures b and c indicate that the vibrating parts of the plate alternately rise above and fall below the average level of the plate. At one moment, for example, the centre of the plate is above the level and its ends below it, as at b; while at the next moment its centre is below and its two ends above the average level, as at c. The vibrations of the plate consist in the quick successive assumption of these two positions. Similar remarks apply to all other modes of division.
Now suppose the rectangle gradually to widen, till it becomes a square. There then would be no reason why the nodal lines should form parallel to one pair of sides rather than to the other. Let us now examine what would be the effect of the coalescence of two such systems of vibrations.
To keep your conceptions clear, take two squares of glass and draw upon each of them the nodal lines belonging to a rectangle. Draw the lines on one plate in white, and on the other in black; this will help you to keep the plates distinct in your mind as you look at them. Now lay one square upon the other so that their nodal lines shall coincide, and then realize with perfect mental clearness both plates in a state of vibration. Let us assume, in the first instance, that the vibrations of the two plates are concurrent; that the middle segment and the end segments of each rise and fall together; and now suppose the vibrations of one plate transferred to the other. What would be the result? Evidently vibrations of a double amplitude on the part of the plate which has received this accession. But suppose the vibrations of the two plates, instead of being concurrent, to be in exact opposition to each other—that when the middle segment of the one rises the middle segment of the other falls—what would be the consequence of adding them together? Evidently a neutralization of all vibration.
Instead of placing the plates so that their nodal lines coincide, set these lines at right angles to each other. That is to say, push A over A′, Fig. 70. In these figures the letter P means positive, indicating, in the section where it occurs, a motion of the plate upward; while N means negative, indicating, where it occurs, a motion downward. You have now before you a kind of check pattern, as shown in the third square, consisting of a square s in the middle, a smaller square b at each corner, and four rectangles at the middle portions of the four sides. Let the plates vibrate, and let the vibrations of their corresponding sections be concurrent, as indicated by the letters P and N; and then suppose the vibrations of one of them transferred to the other. What must result? A moment’s reflection will show you that the big middle square s will vibrate with augmented energy; the same is true of the four smaller squares b, b, b, b, at the four corners; but you will at once convince yourselves that the vibrations in the four rectangles are in opposition, and that where their amplitudes are equal they will destroy each other. The middle point of each side of the plate of glass would therefore be a point of rest; the points where the nodal lines of the two plates cross each other would also be points of rest. Draw a line through every three of these points and you will obtain a second square inscribed in the first. The sides of this square are lines of no motion.
We have thus far been theorizing. Let us now clip a square plate of glass at a point near the centre of one of its edges, and draw the bow across the adjacent corner of the plate. When the glass is homogeneous, a close approximation to this inscribed square is obtained. The reason is that when the plate is agitated in this manner the two sets of vibrations which we have been considering actually coexist in the plate, and produce the figure due to their combination.
Again, place the squares of glass one upon the other exactly as in the last case; but now, instead of supposing them to concur in their vibrations, let their corresponding sections oppose each other: that is, let A cover A′, Fig. 71. Then it is manifest that on superposing the vibrations the middle point of our middle square must be a point of rest; for here the vibrations are equal and opposite. The intersections of the nodal lines are also points of rest, and so also is every corner of the plate itself, for here the added vibrations are also equal and opposite. We have thus fixed four points of rest on each diagonal of the square. Draw the diagonals, and they will represent the nodal lines consequent on the superposition of the two vibrations.
These two systems actually coexist in the same plate when the centre is clamped and one of the corners touched, while the fiddle-bow is drawn across the middle of one of the sides. In this case the sand which marks the lines of rest arranges itself along the diagonals. This, in its simplest possible form, is Sir C. Wheatstone’s analysis of these superposed vibrations.