In figures a and b we have a representation of the fundamental vibration of such a rod. Here we have a node in the middle-that is to say, a rod free at both ends vibrates as two rods of half the length fixed at one end and free at the other.

In figures c and d we have two nodes each one-fourth of the whole length from the end. Finally, in figures e and ƒ we have three nodes by which the whole rod is divided into 1 + 2 + 2 + I 6 parts, the rods free at one end being only half the size of the rods fixed at both ends, by which means the vibrations of all the various sections take place in the same time.

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It has already been remarked that a column of air vibrates after the same laws as those which regulate the longitudinal vibrations of rods.

Thus an organ-pipe shut at one end is similar to a rod fixed at one end, is in fact a rod of air, and its time of vibration (Art. 156) is the time that a pulse takes to travel twice up and down the organ-pipe, just as the time of vibration of the rod is that occupied by the pulse in travelling twice backwards and forwards along the rod.

In like manner an organ-pipe open at both ends will vibrate like a rod free at both ends, and will therefore have a node in the middle, so that its fundamental note will be the same as that of a shut pipe of half the size.

In fine, the analogy between such rods and organ-pipes is complete. An organ-pipe open at one end will therefore be capable of breaking up into nodes precisely in the same manner as a rod fixed at one end and free at the other, while an organ-pipe open at both ends will split up into nodes precisely after the manner of a rod free at both ends.

159. Vibrations of Plates.-A plate may be made to vibrate by drawing a bow across its edge.

The following law governs the vibrations of such bodies; In plates alike in other respects, the number of vibrations per second varies directly as the thickness of the plate, and inversely as its area.

Gongs, cymbals, and bells are instruments in which the

sound is produced by vibrating plates or masses, while in a drum the sound is produced by a vibratory membrane.

160. Communication of Vibrations.—If a musical note is traversing the air in the presence of an instrument capable of sounding the same note, this instrument seems to take up the note and give it out of its own accord. This may be frequently observed in the case of a piano, which rings to a sound; or, again, the string of a violin may be made to vibrate by sounding a tuning-fork which gives out the same

note.

Now since an undulation of sound is a kind of energy, and since energy cannot be created, it follows that this undulatory energy must be absorbed by the string in order that it may be given out again by the string on its own account. In fact, when a string takes up a note in this manner, there is a communication of the energy of the sound-wave from the air to the string; but when we strike the same string, there is a communication of the energy of the sound wave from the string to the air. We thus see that a string when at rest absorbs that particular kind of undulation which it gives out when struck. It will be afterwards seen that there is a similar law in the case of radiant light and heat.

161. Determination of Number of Vibrations.-One of the simplest machines for measuring the number of vibrations corresponding to a given sound is that of Savart. In this machine a toothed wheel B is made to revolve very fast, and there is a card E at one end, which is so fixed as to be struck by each of the teeth of the wheel B as it revolves with great velocity. Thus, if the wheel B revolves three times in a second, and has 100 teeth, the card will be struck 300 times in a second, and will therefore emit a musical note.

At the side of the wheel there is an indicator which shows how many revolutions have been made by the wheel, from which we can calculate the number of vibrations in a given time. What we have to do, therefore, is to increase the velocity of revolution until we get the required sound. We should then keep the speed of the apparatus constant for a

given time, say 30 seconds, and meanwhile note on the indicator how many revolutions have been made; we shall thus obtain the number of vibrations in one second corresponding to the sound.

The instrument now described affords us an easy method of determining the velocity of sound in air, which can be practised in an ordinary sized room.

First of all let us take a tuning-fork and find the number of vibrations which it makes in one second by means of Savart's machine, or any similar instrument; next let us take a long cylindrical vessel and fill it with water to such a height that when the tuning-fork is held over its mouth the column

of air between the water and the fork vibrates in unison with the fork. A little practice will enable us to decide the height very exactly, because it is accompanied by an unmistakable exaltation of sound-we have in fact made an organ pipe.

Now we know that in this case the blow given by the fork to the air will have travelled twice down to the water and back again during the time of a complete vibration of the fork (Art. 156). If therefore we measure the length of the column and multiply it by four we find the length which the blow has travelled in air during one vibration of the fork, and hence if

we have previously ascertained how many such blows are delivered in one second we can find the velocity of sound.

Suppose, for instance, that a given tuning-fork is found by means of Savart's machine to vibrate 550 times in one second, and that it vibrates in unison with a column of air six inches long. The sound-blow will therefore travel over four times six inches, or two feet of air during one vibration, and as there are 550 of these in one second, it will travel 1,100 feet in one second.

162. Graphical representation of Vibrations.-One of the best methods of making vibrations apparent is that of M. Lissajous, which is represented in Fig. 52. The essentials

of the arrangements are, in the first place, a tuning-fork with a small mirror attached to one arm, and a small counterpoise, in order to balance the mirror, to the other.

A ray of light from a hole in a darkened chimney of a lamp is made to strike this mirror, and is reflected from it towards another mirror m; it then strikes a lens 7, which is so arranged as to throw upon a screen a small luminous point, being the image of the hole in the dark chimney of the lamp from which the light originally proceeds. Thus, if the tuningfork be at rest, we shall simply have a luminous point on the

screen; but if it has been put into vibration, the mirror will of course, move along with it, and the result will be that the luminous dot of light will oscillate on the screen up and down with each vibration of the mirror. But these oscillations will be so rapid that the eye will merely perceive on the screen a luminous line of light, on the same principle that when a burning brand is twirled rapidly round we see a continuous circle of light. Now if, while the tuning-fork is in a state of vibration, we make it rotate, a curved or sinuous bright line will appear on the screen instead of the straight line we have mentioned, the amount of sinuosity depending on the relation between the rapidity with which the tuning-fork vibrates, and that with which it is made to rotate. We are thus furnished with a visible representation of the vibrations of the tuningfork.