742 MECHANICS Both ends fixed. to x=q, which runs up to the origin. After the lapse of any inter val greater than q/a we have y=f(at-x), for at + x has now become greater than q. This is a wave of exactly the same form as before, but the sign of the disturbance and the direction of its propagation are both reversed. Every portion of a wave is therefore reflected, with simple reversal of the displace ment, as soon as it reaches the fixed end. For we may take the limits > and q as close together as we choose. Now suppose the string to have another fixed point atx=?. Then we have 0-f(at-l)-f(at + l). Thus/ is ( 67) a periodic function, of period llja, and can there fore be expressed as a series of simple harmonic terms of the full period, half period, one-third period, &c. Hence we may write, the coefficient | being put in for convenience, Emsinirml-^at - x) = Sj 00 A TO siinrml - l at sin -n-ml - } x - 2i B, n cosirinl - l at sin irml ~ l x . This expression contains the complete solution of the problem. To adapt it to any particular case, we must know at some definite time (say t = Q) the value of y in terms of x, i.e., the initial disturb ance ; also the corresponding value of y. We have then As y , 2/0 are given in terms of x, we can find, by the process of 67, the values of A OT and B m , and thence the required value of y. Oscilla- 268. As another example, suppose a uniform chain to be sus- tions of pended by one end, and to make small oscillations in a vertical a chain plane. fixed at AA 7 e cannot enter here into details ; so we simply assume that one end. elementary persistent harmonic solutions are possible, or, what comes to the same thing, that there are permanent forms in which the chain can rotate about the vertical from the point of sus pension. If the axis of x be vertical, the equations of motion are d dscls. Avhere /j. is the mass of unit length of the chain. As the oscillations are supposed to be small, we may neglect the change in the ver tical ordinate of any point of the chain, because it must be of the second order of small quantities if the horizontal displacement is of the first order. Hence we may put everywhere x for s, and therefore consider x to be independent of t. Thus the first equa tion becomes dT = whence T*=/j.g(l-x) , where I is the length of the chain. The second equation then becomes dn 1 or, if we measure x from the lower end of the chain upwards, d^i dv 1 /^ _ v. i ___ y_ _ _ ^i dx 2 dx g The complete integral of this equation would be much more general than we require, for it would express every possible small motion of the chain, however apparently irregular." What we seek are the fundamental modes of simple harmonic oscillation, any number of which, as in the case of a musical string, may be superposed. Hence we may write y = rism(nt+a) , where n is a numerical quantity as yet undetermined, but which is confined to one or other of a series of definite values ; rj, on the other hand, is a function of x only. With this value of y the equation becomes 9 By the usual method of undetermined coefficients we easily find the particular integral t nx >j =A 1 - g 2-g 2 2 3 2 </ 3 This series is obviously convergent for all finite values of ?i-x/g. The general integral is of the form (D. gz) + Ar? ; where T?J is a function of x, finite for all values of x, but which we need not determine. For it is clear that, to suit our present purpose, we must put B = ; otherwise we should have rj infinite at o; = 0. Thus (1) is the expression we require under the limita tions above imposed. The quantity A represents the semi-amplitude of oscillation of the lower extremity of the chain. The condition that the upper end is fixed gives rj = for x = l, i.e., (J The roots of this equation (which are all real and positive) give the values of n for the several fundamental modes of vibration. AVe have i7 = for the following values of n^ljg : 1 454, 7 - 62, 1874, 34-79, &c. From these we find for the periods of the various simple dis turbances the following multiples of the period of a simple pen dulum equal in length to the chain, viz., 83, 36, 23, 17, &c. AVhen nHjg has the least of the above values, the chain is always entirely on one side of the vertical, and the time of a complete oscillation is to that of a simple pendulum of the same length as 5 : 6 nearly. 269. AVhen a free chain, at rest, has an impulsive tension applied Impul- at one end, the calculation of the consequent impulsive tension at sive different parts of the chain and the velocities generated is very tension, simple. For, calling the instantaneous speeds along the tangent and along the radius of absolute curvature v s and Vp respectively, we have where /u is the mass of unit length of chain at s. It is obvious that there can be no impulsive speed perpendicular to the osculating plane. The kinematical condition is simply that an elementary arc 8s is not altered in length. But the tangential increment of speed alone would imply an increase of the length of 8s in the ratio 1 + -St: 1 in time St. Also the impulsive speed vp would imply a diminution of its length in the ratio 1 - v P 8t/p : 1 by virtu ally making it an arc of a circle of smaller radius, but subtending the same angle at the centre. Hence, neglecting the square of tit as compared with its first power, we find for the kinematical con dition
- >_ ty =0 _
ds p This gives, by eliminating the impulsive velocities, ds /j. ds j jj. p- If the chain be uniform, this becomes 7O rp m n 79 o V . as- p- The whole kinetic energy generated in the chain by the impulse is and the condition that this shall be a maximum is the differential equation above. This is a particular case of a general theorem due to Sir W. Thomson, viz. : A material system of any kind, given at rest, and subjected to an impulse in any specified direction and of any given magnitude, moves off so as to take the greatest amount of kinetic energy which the specified impulse can give it. The direction in which an element of the chain begins to move is inclined to the tangent at an angle <J> where s els 270. It is to be observed that, in such questions as those just Waves c treated, the possibility of an impact s being propagated instan- extensio taneously along the whole length of a chain depends upon its of a assumed inextensibility. AVhen a wire (such as that employed for string. a distance-signal on railways) is regarded as extensible, there is a definite speed with which a disturbance of the nature of extension is transmitted along it. Thus, recurring to the equations of 267, we see tnat for the motion of a stretched elastic string in the direction of its length we have If there be no applied forces, X-=0. Also, if we use x instead of s to characterize a particular point of the string, we must put x + for x and x for s, being a function of x and t which denotes at any
instant the displacement of that point.