Page:Encyclopædia Britannica, Ninth Edition, v. 24.djvu/443

Rh WAVE Let us express the position of a point of the wire, when undis turbed, by x its distance (say) from one end. Let y represent its transverse displacement at time t. Then, bearing in mind that the disturbance travels with a constant speed v, the nature of the motion, so far as we have yet limited it, will be expressed by saying that the value of y, at x, at time t, will be the value of y, at X + VT, at time t + -r; or, simply, y=f(vt-x). For this expression, whatever be the function /, is unchanged in value, if t + r be put for t, and X+VT for x ; and no other expres sion possesses this special property. If there be a disturbance run ning the opposite way along the wire, we easily see that it will be represented by an expression of the form F(vt+x). These disturbances are superposable, so that y=f(vt-x) + (vt + x) ..... _. (1) expresses the most general state of disturbance which the wire can suffer under the limitations we have imposed. Before going farther we may use this to reproduce the results given above. Nowa; = is one end of the wire, and there yis necessarily always zero. Hence so that the functions/ and F differ only in sign. This condition, inserted in (1), gives us at once the state of matters indicated by the cut above. If x = l be the other end of the wire, we have the new condition The meaning of this is simply that the disturbance is periodic, the period being 2l/v t the other result already obtained. Fourier s theorem (see HARMONIC ANALYSIS, or MECHANICS, 67) now shows that the expression/ may be broken up into one definite series of sines and cosines, whence the usual results as to the various simple sounds which can be produced, together or separately, from a free stretched string. It may be asked, and very naturally, How can this explanation of the nature of all possible transverse motions of a harp or piano forte wire, as the result of sets of equal disturbances running along it with constant speed, be consistent with the appearance which it often presents of vibrating as a whole, or as a number of equal parts separated from one another by nodes which remain apparently at rest in spite of the disturbances to which, if the explanation be correct, they are constantly subjected? The answer is given at once by a consideration of the expression (where i is any integer), which is a particular case of the general expression (1), limited to the circumstances of a wire of length I. This indicates, as we have seen, two exactly similar and equal sets of simple harmonic waves running simultaneously, with equal speed, in opposite directions along the wire. By elementary trigonometry we can put the expression in the form . iirx iirvt y= - 2 sin - cos r-. L L This indicates -first, that the points of the string where x has the values 0, l/i, 2l/i,. . ., I remain constantly at rest (these are the ends, and the i-l equidistant nodes by which the wire is divided into i practically independent parts) ; second, that the form of the wire, at any instant, is a curve of sines, and that the ordinates of this curve increase and diminish simultaneously with a simple harmonic motion, the wire resuming its undisturbed form at intervals of time l/iv. This discussion has been entered into for the purpose of showing, from as simple a point of view as possible, the production of a stationary or standing wave. The same principle applies to more complex cases, so that we need not revert to the question. Recurring to the general expression for y in (1), it is clear that if we differentiate it twice with respect to t, and again twice with respect to x, the results will differ only by the factor v which occurs in each term of the first. Thus r^y ~j 9 &quot;7 o M~ ax* /j. ax is merely another way of writing (1). But in this new form it admits of the immediate interpretation given above in italics. d?y. (pu dt^~ 1S ^ e acce l e ration, and j^ 2 the curvature. (2) Longitudinal Waves in a Wire or Rod.li the displacements of the various parts of the wire be longi tudinal instead of transverse, we may still suppose them to be represented graphically by the figure above by laying off the longitudinal displacement of each point in a line through that point, in a definite plane, and perpendicular to the wire. In the figure a displacement to the right is represented by an upward line, and a dis placement to the left by a downward line. The ex tremities of these lines will, in general, form a curve of continued curvature. And it is easy to see that the tangent of the inclination of the curve to the axis (i.e., its steepness at any point) represents the elongation of unit length of the wire at that point, while the curvature measures the rate at which this elongation increases per unit of length. The force required to produce the elon gation bears to the elongation itself the ratio E, viz., Young s modulus. The acceleration of unit length is the change of this force per unit length, divided by p.. Hence, by the italicized statement in (1), we have v= ^/E, //, (MECHANICS, 270). All the investigations above given apply to this case also, and their interpretations, with the necessary change of a word or two, remain as before. Thus, according to our new interpretation of the figure, the front part of A indicates a wave of compression, its hind part one of elongation, of the wire, the displacement of every point, however, being to the right. B is an equal and similar wave, its front being also a compression, and its rear an elongation, but in it the displacement of each point of the wire is to the left. Hence the displacements of continually compensate one another ; and thus a wave of compression is reflected from the fixed end of the wire as a wave of compression, but positive displacements are reflected as negative. If we now consider a free rod, set into longitudinal vibration by friction, we are led to a particular case of reflection of a wave from a free end. The condition is obviously that, at such a point, there can be neither compression nor elongation. To represent the reflected wave we must therefore take B of such a form that each part of it, when it meets at O the corresponding part of A, shall just annul the compression. On account of the smallness of the displacements, this amounts to saying that the successive parts of B must be equally inclined to the axis with the corresponding parts of A, but they must slope the other ivay. Thus the proper figure for this case is and the interpretation is that a wave of compression is reflected from a free end as an equal and similar wave of elongation ; but the disturbance at each point of the wire in the reflected wave is to the same side of its equilibrium position as in the incident wave. This enables us to understand the nature of reflexion of a wave of sound from the end of an open organ pipe, as the former illustration suited the corresponding pheno menon in a closed one. (3) Waves in a Linear System of Discrete Masses. Suppose the wire above spoken of to be massless, or at least so thin, and of such materials, that the whole mass of it may be neglected in comparison with the masses of a system of equal pellets, which we now suppose to be at tached to the wire at equal distances from one another. The weights of these pellets may be supported by a set of very long vertical strings, one attached to each, so that the arrangement is unaffected by gravity. The wire may be supposed to be stretched, as before, with a definite tension which is not affected by small trans verse disturbances. We will take the case of transverse disturbances only, but it is easy to see that results of precisely the same farm will be obtained for longitudinal XXIV- - 53