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Page 114 ï~~A Physical String Model with a Twist Steven D. Trautmann Center for Computer Research in Music and Acoustics Stanford University (415) 723-4971/wishbon@ccrma.Stanford.edu ABSTRACT: A physical model of a string embedded in a Mobius strip is presented. Due to the nature of this embedding and the global non-orientablity of the strip, interesting properties arise. Given a string at equilibrium, and an initial velocity at any point, a point half way around will function as a node. The fundamental of the string will be the same as its length, but each point in the string will function as if it were the center point of a string with fixed ends. At least one point of the string will be at equilibrium at any time. Taking a circular string and embedding it in a Mobius strip (see fig. 1) does not change the derivation of the wave equation since the string can be orientated locally (for a general derivation see Powers, 1979). Thus the ideal approximation of 82u = 2 still holds locally, though care must be taken due to lack of global orientation. Figure 1: Mobius strips with embedded strings at equilibrium and displaced respectively This lack of global orientation can be compensated for artificially by choosing a point on the Mobius strip where the orientation reverses. In this model, the direction of string motion within the strip is orientated relative to the strip itself, and the twisting of the strip in space does not effect the motion of the string. Choosing a point along the strip where the orientation changes, we can "cut" the strip there and flatten it, maintaining (or producing) the boundary conditions, namely that u (O,t) = -u(2r,t) and (0, t) = -oe(27r, t) and so on for higher derivatives, where the length has been scaled to 27r (see fig. 2) Notice that since the string is continuous in all its derivatives, its position and each derivative (with respect to time) must equal zero at least at one point in the strip, since each derivative changes sign or is zero at the boundaries. Assuming u(x, t) = 4d(x)T(t) (that it is separable) the wave equation implies = T (for 0 < x < 27r). This implies both sides are constant, say equal to -A2 which OW )- cTt) means 0"(x) + A24(x) = 0. The general solution to this equation is 4(x) = a cos(Ax) + b sin(Ax). Given the boundary conditions for the flattened strip, u(0, t) = -u(2ir, t), it must be the case that 6(0) = a and e(2r) = -a which in turn implies a.. = 1, $,"", 2n -1 so that bsin(A)x) = 0 for all LL. Also T(t) = A cos(Act) + B sin(Act) so rÂ~(n 1 / 2~l 2n+l t] u(x, t)= (2n os - xl An cos Â~~~r ct + Bnsin 2~ ti 114 I C MC P ROC EE D I N G S 1995
Page 115 ï~~embedded string +1 equilibrium (0) 0 2n Figure 2: Flattened M6bius strip with boundary conditions which can be compared to the solution to a string of length 2ir bounded on both ends. 00 n - n E n n= 1 Since the wave equation is the usual one, the general solution of the wave equation, u(x, t) = f+ (x + ct) + f-(x - ct) holds in this case. It can be seen that any symmetrical initial condition at x0, say f(xo) will cancel itself when et = ir, due to the relative change in orientation between the left and right going wave. Further, as the waves pass each other the left going wave appears the same as a reflected right going wave due to the difference in direction and orientation, and vice versa. Thus for any symmetrical disturbance, the string in the Mibius strip behaves as a string with fixed ends, the ends being at the node created. Since any initial condition and be broken down into symmetrical components (a countless set of delta-functions if nothing else), the string in the Mibius strip can be viewed as a sum of overlapping strings length 27r with symmetrical initial conditions. A lossless implementation can be made for either position or velocity. Two possible implementations are given (see fig. 3). The first takes the point of view of the directional signals, which see themselves as constant, but the pick-up inverting each time around, which is one-half the total signal length since it takes two trips to be at the same spot with the same orientation. The second implementation is more akin to fig. 2, where the differently oriented parts of the signal are summed (actually differenced) to create directional signals of half the length. The inversion is lumped within the loop, and the number of pick-ups is reduced from four to two. Losses can be lumped and included in either implementation. N-.ampldelay rn * N.eIh delay line NOa. + 0hAs 9 + ou N..e~edelay Un. N-eeipdg delay Urn Figure 3: continuous directional signals and flattened MQbius strip implementations respectively REFER ENCE: Powers, David. Boundary Value Problems, 2nd ed. New York:Academic Press, Inc., 1979. ICMC PROCEEDINGS 1995 115 115