Page:Encyclopædia Britannica, Ninth Edition, v. 1.djvu/127

in their progress towards each other, they are then coincident, It is obvious that the particles of the medium will at the moment in question be displaced to double the extent of the displacement producible by either wave alone, so that the resultant wave may be represented by the dotted curve.

Fig. 14.

In fig. 14, 2, the two interfering waves, represented by the full and dotted curves respectively, have each passed over a distance $={\tfrac {1}{4}}\lambda$, the one to the right, the other to the left, and it is manifest that any disturbance of the medium, producible by the one wave, is completely neutralised by the equal and opposite action of the other. Hence, the particles of the medium are now in their undisturbed positions. In fig. 14, 3, a further advance of the two waves, each in its own direction, over a space $={\tfrac {1}{4}}\lambda$, has again brought them into coincidence, and the result is the wave represented by the dotted line, which, it will be re marked, has its crests, where, in fig. 1, are found troughs. In fig. 14, 4, after a further advance $={\tfrac {1}{4}}\lambda$, we have a repetition of the case of fig. 14, 2, the particles are now again unaffected by the waves. A still further advance of ${\tfrac {1}{4}}\lambda$, or of $\lambda$ reckoned from the commencement, brings us back to the same state of things as subsisted in fig. 14, 1. An inspection and inter-comparison of the dotted lines in these figures are now sufficient to establish the accuracy of the laws, before mentioned, of stationary waves.

Part VI.
Musical Strings.

55.We have in musical strings an instance of the occurrence of stationary waves.

Fig. 15.

Let ${\text{AB}}$ (fig. 15) be a wire or string, supposed meanwhile to be fixed only at one extremity ${\text{B}}$, and let the wire be, at any part, excited (whether by passing a violin bow across or by friction along it), so that a wave (whether of transversal or longitudinal vibrations) is propagated thence towards ${\text{B}}$. On reaching this point, which is fixed, reflexion will occur, in consequence of which the particles there will suffer a complete reversal of velocity, just as when a perfectly elastic ball strikes against a smooth surface perpendicularly, it rebounds with a velocity equal and opposite to that it previously had. Hence, the displacement due to the incident wave being ${\text{BM}}$, the displacement after reflexion will be ${\text{BN}}$ equal and opposite to ${\text{BM}}$, and a reflected wave will result, represented by the faint line in the fig., which will travel with the same velocity, but in the opposite direction to the incident wave fully lined in the fig. The interference of these two oppositely progressing waves will consequently give rise to a stationary wave (fig. 16), and if we take on the wire distances ${\text{BC}}$, ${\text{CD}}$, ${\text{DE}}$, &c. $={\tfrac {1}{2}}\lambda$, the points ${\text{B}}$, ${\text{C}}$, ${\text{D}}$, ${\text{E}}$, $\ldots$ will be nodes, each of which separate portions of the wire vibrating in opposite directions, i.e., ventral segments.

56.Now, it is obvious that, inasmuch as a node is a point which remains always at rest while other parts of the medium to which it belongs are vibrating, such point may be absolutely fixed without thereby interfering with the oscillatory motion of the medium. If, therefore, a length ${\text{AB}}$ of wire be taken equal to any multiple of ${\tfrac {\lambda }{2}}$, ${\text{A}}$ may be fixed as well as ${\text{B}}$, the motion remaining the same as before, and thus we shall have the usual case of a musical string. The two extremities being now both fixed, there will be repeated reflexions at both, and a consequent persistence of two progressive waves advancing in opposite directions and producing together the stationary wave above figured.

57.We learn from this that a musical string is susceptible of an infinite variety of modes of vibration corresponding to different numbers of subdivision into ventral segments.

Thus, it may have but one ventral segment (fig. 17), or but two nodes formed by its fixed extremities. In this case, the note emitted by it is the lowest which can possibly be obtained from it, or, as it is called, its fundamental note. If $l$ denote the length of the wire, by what has been already proved, $l={\tfrac {\lambda }{2}}$, and therefore the length of the wave $\lambda =2l$. Hence, ${\text{V}}$ being the velocity of propagation of the wave through the wire, the number $n_{1}$ of vibrations performed in the unit of time with the fundamental note is ${\tfrac {\text{V}}{2l}}$.

The next possible sub-division of the wire is into two ventral segments, the three nodes being the two fixed ends ${\text{A}}$, ${\text{B}}$, and the middle point ${\text{C}}$ (fig. 18). Hence, $l=\lambda$, and the number of vibrations $n_{2}={\tfrac {\text{V}}{l}}$ or double of those of the fundamental. The note, therefore, now is an 8^ve higher.

Reasoning in a like manner for the cases of three, four, &c., ventral segments, we obtain the following general law, which is applicable alike to transversely and to longitudinally vibrating wires:

A wire or string fixed at both ends is capable of yielding, in addition to its fundamental note, any one of a series of notes corresponding to $2$, $3$, $4$ times, &c., the number of vibrations per second of the fundamental, viz., he octave, twelfth, double octave, &c.

These higher notes are termed the harmonics or (by the Germans) the overtones of the string.

It is to be remarked that the overtones are in general fainter the higher they are in the series, because, as the number of ventral segments or independently vibrating parts of the string increases, the extent or amplitude of the vibrations diminishes.

58.Not only may the fundamental and its harmonics