Fi may establish this, however, in a very simple manner, as follows : Let AB (fig. 65) be a small portion of the cord, and AC, CB the tangents at its ex tremities; and let the (small) exterior angle at C be 0. ^-^" t5"* Then, p being the pressure ^ A per unit length of the string, we have at once ultimately. But A.T> = pO, so that If there be friction, and if the element of the rope be just coiled on about to slip, in consequence of the difference of the ten- roll .sb sions at its ends, we have cylinder. so that Wave- velocity on stretched cord. Reflected wave. This leads to the formula for the growth of a sum at com pound interest at /z per cent, payable every instant. Hence for a, finite angle a W 7 e have f jua-p a~ ; A It is to be remarked here that neither the dimensions nor the form of the curve on which the cord is stretched, pro vided only it be plane, have any influence on this result, which involves only the coefficient of friction and the angle between the two free portions of the cord. KINETICS OF A CHAIN OR PERFECTLY FLEXIBLE CORD. 265. The equations of motion of a chain, under the action of any finite forces, are at once formed from those of equilibrium by introducing the forces of resistance to acceleration according to Newton s principle. Here we enter on a subject of extreme importance, but also (at least in the majority of cases) of great mathematical difficulty. One valuable result, however, can be obtained by very simple means. A uniformly heavy and perfectly flexible cord, placed in the interior of a smooth tube in the form of any curve, and subject to no external forces, will exert no pressure on the tube if it have everywhere the same tension, and move with a certain definite speed. For, as in 264, the statical pressure due to the curva ture of the rope is T$ cr per unit of length (where <r is the length of the arc AB in that figure) directed inwards to the centre of curvature. Now, the element o-, whose mass is ma- (if m be the mass per unit of length), is moving in a curve whose curvature is 0/cr, with speed v (suppose). The requisite force is " -- = mv-d , and for unit of length mv-O a: Hence if T = ??t*r the theorem is true. If we suppose a portion of the tube to be straight, and the whole to be moving with speed v parallel to this line, and against the motion of the cord, we shall have the straight part of the cord reduced to rest, and an undula tion, of any, but unvarying, form and dimensions, running along it with linear speed s/T/w. Suppose the tension of the cord to be equal to the weight of W pounds, and suppose its length I feet and its own mass iv pounds. Then T = Wy, Im = w, and the speed of the un dulation is tJWlff/w feet per second. 266. As will be shown later, when sivih an undulation reaches a fixed point of the cord or chain, it is reflected, and runs back along the cord with the same definite speed. But the reflected form differs from the incident form in being turned about in its own plane through two right angles. When the string is fixed at both ends any disturbance runs along it, backwards and forwards, with this speed, and thus (in a piano or harp) administers periodic shocks to the sounding board, causing it to give out a musical note. The interval between these periodic shocks is of course the time taken by the disturbance in running from end to end of the string. Dividing the length I of the string by the speed above reckoned, we find for this interval the value the reciprocal of which is the number of impulses per second. It is thus seen to be directly as the square-root of the tension of the string, inversely as the square-root of its mass per unit of length, and also inversely as its length. These are well-known facts in Acoustics. It is to be observed that there is no necessity for limiting the pro position above to a plane curve, though we have treated the question as if it were such. The demonstration applies even to a knot of any form. 267. We will now consider more particularly the vibra- Vibrati< tions of a musical string, whose tension is great and its of " lusii own mass small. Forming the equations of motion as above hinted, we have three of the type In the special case of a tightly stretched inextensible string, per forming very small transverse oscillations, we may greatly simplify these by assuming that no external forces act. This practically means that the weight of the string is negligible in comparison with the tension. If the axis of x be taken to coincide with the undisturbed position of the string, we have to the second order of small quantities s = a . With this the equation above written becomes or the tension is the same throughout, equations now become The second and third The y and z disturbances are therefore of the same general character, and perfectly independent of one another. We will therefore con fine our attention to one of them. From the equations we see that T/ju. must be of two linear dimensions, and we will therefore write for it 2 . As this quantity must also be of two negative dimensions in time, a represents a speed. What speed will be seen imme diately. The equation in y is now whose integral is known to be y =f(at -x) + Y(at + x) , where/ and Fare arbitrary functions. As we have already seen ( 53), the first part of the value of y expresses a wave running with speed a along the axis of x in the positive direction ; the second part a wave in the negative direction with the same speed. Thus we see that any small disturbance whatever, of a stretched string, gives rise to two series of waves propagated in opposite directions with equal speeds. Also, as the equation is linear, the sum of any two or more particular integrals is also an integral. If we suppose one extremity of the string to be fixed at the origin, One em we have the condition x = Q, 2/ = 0, and therefore fixed. As this holds for all values of t, the function F is simply the negative of/, so that y-f(at-x)-f(at + x). To investigate what becomes of a disturbance which runs along the cord to the fixed end, let us suppose that f(r) (which, by the remark above, may represent any part of a disturbance of the string) is a function which vanishes for all values of r which do not lie between the positive limits p and q, but which for values of r between these limits takes definite values. Then at time < = we have 2/- -/(*). for by hypothesis, f(r] vanishes for all negative values of r. This
denotes a disturbance of the string originally extending from x-p