Formulas for Vibrating Strings with Obstacles

Henri Cabannes

June 16, 2001

The string moves in the plane Oxu (orthonormed system of references) ; the extremities are fixed at the points (x = ±1/2, u = 0), and the shape of the string at time t is u(x,t). The string is initially at rest in the position u(x,0) = u₀(x) ³ -h, with 0 < h < 1.

1  Fixed Point-Mass Obstacle⁽²⁾

The obstacle is located at the point x = 0, u = +h. The function u₀(x) is an even function and u₀(0) = 1. We introduce the two functions K(y) and f(y), periodic with period 1, and défined for -(1/2) < y < (1/2) by :

K(y) = ì í î |y|-[|y|] -  1 2 ü ý þ 2   ,

(1)

f(y)+y = sgn(y){ 1-u0(y)} /2.

(2)

[|y|] denotes the highest integer smaller than |y|, absolute value of y ; sgn(y) denotes the sign of y. The function K(y) is even, the function f(y) is odd. Then we introduce the new variables X and T, and the new functions K₁ and K₂.

X(x,t) = x+  f(x+t)+f(x-t) 2 ,

(3)

T(x,t) = t+  f(x+t)-f(x-t) 2 .

(4)

K1 = K ì í î 2  X+T 3-h ü ý þ +K ì í î 2  X-T 3-h ü ý þ ,

(5)

K2 = K ì í î  2(X+T)+1-l 3-h ü ý þ +K ì í î  2(X-T)+1-l 3-h ü ý þ .

(6)

The parameter l is equal to h when 0 < 4x < 1+h, and equal to 3 when 1+h < 4x < 2.  With all those auxiliary functions, the motion of the string is represented by the function :

u(x,t) =  1+h 3-h { 1-2|X|} +  3-h 2 { K1-K2} .

(7)

The motion is an almost periodic function of the time, and a periodic function when h is rational.

2  Fixed Straight Line Obstacle⁽³⁾

The obstacle is located on the line u = -h. The initial position is ''unimodal'', which means that the derivative (du₀/dx) is positive or zero when -(1/2) < x < a < (1/2), and negative or zero when -(1/2) < a < x < (1/2). We assume u₀(a) = 1. We introduce the new function L(y) = {1-(y/2)} ² when 0 < y < 1, and then L(y+1) = L(y). We introduce also the functions F(y), L₁, L₂ and L₃ :

F(y) =  1 2 { 1-u0(y)} sgn(y-a).

(8)

 

L1 = L ì í î  F(x+t) 1+h ü ý þ +L ì í î  F(x-t) 1+h ü ý þ ,

(9)

L2 = L ì í î  F(x+t)-F(x-t) 2(1+h) +  1 2 , ü ý þ ,

(10)

L3 = L ì í î  1-eF(x+t) 1+h ü ý þ -L ì í î  1-eF(x-t) 1+h ü ý þ .

(11)

e = ±1 possesses the sign of F(x+t)+F(x-t).

The function u(x,t) which represents the motion of the string is given by formula (12) for the points which encounter obstacles : |F(x+t)+F(x-t)| < 1-h, and by formula (13) for points which do not encounter obstacles : 1-h < |F(x+t)+F(x-t)| < 1.

u(x,t) =  1-h 2  { F(x+t)+F(x-t)} 2 2(1+h) +(1+h){ L1-2L2} ,

(12)

u(x,t) =  1-h 1+h { 1-|F(x+t)+F(x-t)|}+(1+h){ L1-L3} .

(13)

The motion is an almost periodic function of the time, and a periodic function when h is rational.

3  Rebound on a Curvilinear Obstacle⁽⁴⁾

The obstacle is the curve u = lsin(2px), and the intial position of the string u(x,0) = u₀(x) = cos(px). Before the first contact the motion is the free oscillation u(x,t) = w(x,t) = cos(px)cos(pt) ; the time of the fist contact is

t1 = t(x) = (1/p)Arccos{ 2lcos(px)} .

(14)

For |2l| < 1, the first contact corresponds to a rebound and, after this rebound, the motion of the string is represented by the function

u(x,t) = w(x,t)-  1-4l2 2l G(x,t),

(15)

G(x,t) = px-Arctan  (1+4l2)sin(px)-4lcos(pt) (1-4l2)cos(px) .

(16)

For |2l| < (1/3), the time of the second contact is

t2 = 2-t(x),

(17)

and this second contact corresponds still to a rebound. After this second rebound the motion of the string is again the free oscillation, so that the motion is periodic with period 2.

4  Wrapping on a Curvilinear Obstacle⁽⁴⁾

Always considering an obstacle u = lsin(2px), and the string always in the initial position u(x,0) = u₀(x) = cos(px), one obtains for the peculiar case l = -(1/2), the wrapping of the string on the convex part of the obstacle, and the rebound on the concave part.

For the new peculiar case corresponding to the obstacle u = (1/3)cos(3px) , the motion of the string is periodic with period T = (4/3), with successive wrapping and unwrapping on the convex part of the obstacle.

References

[1]. L. Amerio and G. Prouse, Study of  the motion of a string vibrating against an obstacle, Rendiconti di Matematica, Serie VI, 8, 563-584, (1975).

[2] . H. Cabannes , Motion of a vibrating string against a fixed point-mass obstacle. C.R. Acad. Sci. Paris, Série II 298, 613-616, (1984).

[3] . H. Cabannes, Motion of a string in the presence of a straight rectilinear obstacle. C.R. Acad. Sci. Paris, Série II 295, 637-640, (1982).

[4] . H. Cabannes, Motion of a vibrating string in the presence of a convex obstacle, C.R. Acad. Sci. Paris, Série II 301, 125-129, (1985).

[5] . A. Haraux and H. Cabannes, Almost periodic motion of a string vibrating against a straight fixed obstacle, Nonlinear Analysis, Theory, Methods and Applications 7, 129-141, (1983).

[6] . H. Cabannes, Periodic  motions of a string vibrating against a fixed point-mass obstacle, Mathematical Methods in Applied Sciences 6, 55-67, (1984).

[7] . H. Cabannes, Cordes vibrantes avec obstacles, Acustica 55, 14-20, (1984).

[8] . H. Cabannes and C. Citrini, Editors, Vibrations with unilateral constraints, In Proceedings of the Euromech Colloquium 209, Tecnoprint, Via del Legatore 3, Bologna, Italy, (1986).