Motion of a vibrating string in the presence of a convex obstacle: A free boundary problem

Here we study the motion of a vibrating string in the presence of an arbitrary obstacle. We show that if the string always rebounds on the concave parts of the obstacle, it can either rebound or roll on the convex parts. The latter is the case if the velocity of the string is null at the contact point just before contact, or if the contact point propagates at a characteristic speed. Four examples are given. The three first correspond to the same obstacle, a sinusoidal arc, but with different initial conditions. In the first case, the string rebounds on the whole of the obstacle and the motion is explicitly determined when it is periodic. In the second case, the string rolls on the convex part of the obstacle up to the inflexion point and then rebounds on the concave part and unwinds on the convex part. In the third case, the string is initially at rest on the obstacle; then it instantaneously leaves the concave part while it unwinds progressively on the convex part. The fourth case is similar to the third but with a different obstacle; the motion, which is periodic, is determined explicitly.


Introduction
Problems related to the study of the motion of a vibrating string in presence of an obstacle were first considered by Amerio and Prouse in 1975 [3]. Since then about fifty papers devoted to the subject have been published. The cases most often considered concerne straight-line or point-mass obstacles [5], [lo]. In the case of curvilinear obstacles a fundamental difference appears between concave obstacles (for an observer on the string) and convex obstacles. If the obstacle is concave, the string rebounds instantaneously, while if the obstacle is convex the string can either rebound or can roll up while remaining in contact with the obstacle for some time. This last behaviour also appears when the string vibrates in the presence of a point-mass obstacle [ll], [S). There are only a few papers related to the cases of curvilinear obstacles, in particular one by Schatzman [12] on concave obstacles, while Bumdge et al. [4], and Ameno [2] considered convex obstacles. The reference [4] treats an explicit solution of a vibrating string against a parabolic obstacle with an inelastic reflection law; this solution represents the behaviour of the strings of an Indian musical instrument. In reference [2] Ameno studied in great detail and generality persistent contacts of the string with obstacle, contacts which exist only for convex obstacles.
The purpose of the present paper is to contribute to the study of the motion of a vibrating string in presence of convex obstacles. In the first part, $$ 2,3,4, we study the conditions of rebound and the conditions of wrapping of the string on the obstacle. A priori one can think that the string wraps around when it comes in contact with the obstacle with a null velocity. We will show that even if its velocity is not zero, there are cases in which the string wraps around rather that rebounds because rebound is not possible. If the obstacle is convex in one region and concave in another, these regions being separated by an inflexion point, then when the end of the part of the string wrapped on the obstacle crosses the inflexion point and reaches the concave part, there is a rebound on the concave part while the string unwraps on the convex part. In the second part, $$ 5, 6, 7 and 8, examples are given. The obstacle is either a sinusoidal arc corresponding to one period ($$ 5, 6 and 7), or a portion of a sinusoid corresponding to one and a half period (5 8). The first obstacle possesses a centre of symmetry, which is an inflexion point; half of the obstacle is convex, the other half is concave. The second obstacle possesses an axis of symmetry with two inflexion points. In all cases the reflection of the string on the obstacle takes place with conservation of energy. For both obstacles, periodic motions have been obtained, the string being initially at rest with an appropriate shape ($$ 5 and 8); if the initial conditions are different, the periodic character of the motion disappears. In 4 6, we have obtained a solution for which the string wraps on the convex part of the sinusoid, and then unwraps when the extremity of the arc in contact reaches the inflexion point. Finally, in $$ 7 and 8, the string is at rest at a given time on the obstacles; it then leaves the concave parts instantaneously and the convex parts progressively. The determination of the limits of the arcs of the string in contact with the obstacle during the wrapping and during the unwrapping is a free boundary problem, as Amerio has already shown [2].

Statement of the problem
The motion of a string is defined by the position, at time t, of a point M with curvilinear abscissa s. The string being assumed inextensible, s is the curvilinear abscissa for the initial state, and for any arbitrary state; we then have M = M(s,t) and Iail?/asI = 1. We assume that there is no torque, but that external forces of densityfare applied on the string. If the string is also assumed perfectly flexible, the constraint at each point is a vector tangent to the string; F = T(aM/as). The equations of motion are where e(s) is the density of the string.
These equations constitute a system of four equations, with four unknowns. The tension is assumed positive, which corresponds to a strained string.
During the motion the string can strike against a fixed obstacle. The only force applied on the string is, b assumption, the reaction of the obstacle, so that in the absence of cont act , != 0, whereas during a persistent contact a a@ a2@ a m @ as as as It follows from relation (1') that the reaction is in the osculator plane, and in the case of two dimensional motion it is in the plane of the motion. From now on, we will only consider this last case. With respect to the tangent to the string the reaction is on the side opposed to (a2i@/as2), that is on the side opposite to the concavity (figures 1). To satisfy the laws of action of contact, the reaction must be directed from the obstacle to the string; this will be the case, by relation (l'), only if the string is on the side opposed to  We consider in this paper small motions of a vibrating string. The string, assumed homogeneous, oscillates in a plane related to an orthonormal reference system Oxu. In its equilibrium position the string is on the segment A B of the x-axis (all the variables are non-dimensional). The points A and B, of abscissa x = k0. 5  which it can hit during the motion. In the next § § we will introduce the characteristic coordinates = t + x and q = tx.
In the linear theory the tension is constant, and from equation (1') the density of the reacting forces during a persistent contact is equal to the opposite of the second derivative of the function @ ( x ) : f = -W(x). At an angular point of the obstacle the reaction is a concentrated force

3
Condition for the string to rebound on the obstacle For an arbitrary obstacle (concave or convex), we assume that the string in motion u = u ( x , t ) encounters the obstacle at a point Q of abscissa x = a(t). We denote by T(X) the inverse function of the function Q: Q o t = identity. We have of course (5) u(x,t(x)) = @(x).
At the point Q the string rebounds on the obstacle according to a certain law of reflection. The velocity of the string at point Q, negative before the contact, becomes positive: The reflection coeficient A(x) is given, positive or null. In the x,t plane (figure 2), the curve x = ~( t ) , called the contact curve, is a part of the line of influence (cf. [l], [4]) on which (dr/dx( ,< 1, that is Icf(t)l 2 1; in other words the point Q moves at a supersonic speed, or exceptionally at sonic speed. The contact curve is part of the boundary of the domain u(x,t) 6 @(x) and the line of influence is formed by arcs of contact and their characteristic tangents. On the characteristic segment CD of the contact curve one always has u ( x , t ) = @(x), but relation (6) cannot be satisfied. In fact, the arc CD being characteristic, the motion of the string past C D is determined by the values of the function u ( x , t ) on the two characteristics issuing from the point C , C D and C E. We deduce in particular the value of the velocity au/at, and (by applying relation (6)), the value of the reflection coefficient A(x), which cannot be given, but depends on the motion of the string.
On the non-characteristic arcs of contact, we will adopt the function A(x) = 1 which corresponds to a perfect elastic reflection; and to include the two cases Ici(t)l > 1 and Icf(t)l= 1, we will write the law of reflection of the string in the form We deduce from this law the following result.

Theorem. During the motion, the total energy of the string (kinetic energy and potential energy) remains constant.
This theorem is expressed by the equation We assume first that the string encounters the obstacle in a finite numbers of points Ql, Q2, ...en of abscissae ol(t), az(t), ... with -0.5 < ol < c2 ...
Then and, by virtue of equation (3), one has so that the last line of the right hand side of equation (9) can also be written where the discontinuities are assumed at the points x = a,(t). The function being continuous, in particular for x = oi(t), we have u(o, + 0,t) E u(oi -0,t) and therefore and by virtue of relation (7) (dE/dt) is zero. When the string is in contact with the obstacle not only on isolated points, but also on some arcs, the result remains valid. We have in fact This formula is exactly formula (9) if we note that on a contact arc, cri < x < we have (au/at) = 0 and (&/ax) = @'(x). 4 Wrapping and unwrapping of the string on the obstacle When the string hits a concave obstacle, it always rebounds; when it hits a convex obstacle, it can either rebound or wrap up and remain in contact with the obstacle for some time. We assume that the motion of the string above the obstacle u = @(x) is the free oscillation u ( x , t ) = w ( x , t ) . If On the arcs A B and CD, on which there is a contact between the string and the obstacle, one has Iz'(x)l < 1; the contact point of the string with the obstacle propagates on the string with supersonic speed; exceptionally the boundary of the domain (9) can be composed of characteristic segments.
We assume now that one of the extremities of the string is on the obstacle, @( -0.5) = 0. The boundary of the domain (9) can also meet the axis x = -0.5, and we assume that this is the case at the point A(x = -0.5,    except in isolated cases, the condition u ( -0 . 5 ,~) = 0 is not satisfied. We deduce from this impossibility that the free oscillation ends, not on the boundary of the domain (g), but on the characteristic issuing from the point This is a free boundary problem, as showed by Amerio [lo]; the solution is simple. The system (18) is a system of two equations with two unknowns: the functions ~( x ) and g(q). Taking the derivative of the first equation, and eliminating g'(tx), we obtain the equation which determines the function t(x), the free boundary, that is to say the trajectory of the extremity of the arc of the string rolled round the obstacle. When the function 7(x) is known, the first equation (18) gives the value of the function g(q). The unwrapping of the string appears in similar conditions, which will be examined in the examples of $ § 6, 7 and 8.

First example
We consider an obstacle formed by a sinusoidal arc:    with i= tanxt -l < ( < l .

2
One then determines the function g(q), to obtain g(q) = f ( -q ) , whence

u,(x,t) = f(x + t) + f ( xt):
sin xx + sin 8. cos xt The function ul ( x , t ) represents the motion of the string in the region bounded by the arc A 3 and by the characteristics q = q A , < = cB . Since ul( k 0 . 5~) = 0, it represents the motion of the string after the first rebound, for -0.5 The motion of the string is defined by the function u = ul, up to the second line of influence. To define this curve we must first solve the equation
There is in some cases a third line CDE, 1 I -2cose sin8 * This third line exists only when cosxtc is less than 1, that is if tan(8/2) is greater than 1/3. In this last case the line of influence is formed by the arc CD, the characteristic tangent DG, and the arc GH. The determination of the motion past this second line of influence cannot be made explicitly.
On the other hand, for tan(8/2) < 1/3 the function ul(x,t) represents the motion of the string for ~( x ) < t < 2r ( x ) , and following the second rebound we have again u(x,t) = w(x,t); therefore the motion of the string is periodic with period 2.  figure 5, which corresponds to the limit case tan(8/2) = 1/3. At time t = 1, the string is at rest in the position u = ul(x,l). One can explain the results by considering that the obstacle is the sinusoidal arc u = @(x) = -sin xx * cosxx, while the string is initially at rest in the position u(x,O) = cotan (8/2) . cosscx. For cotan(8/2) 2 3 the motion is periodic with period 2; the periodic character of the motion disappears for -0 < cotan(8/2) < 3.
In a general way, we can express the following supposition: Conjecture. rjthe string, initially at rest in the position u(x,O) = cosxx, oscillates in the presence oythe obstacle u = @(x) = Asin2nrcx (n integer), the motion is periodic with period 2, when IAl is small enough').
') The parameter I, has no relation with the function I(x) introduced in Section 3.

12
In fact if lil is small enough, on all the contact curve cosnx * C O S~ = Lsin2nnx, one has Idt/dxl < 1 and the computation of the function u(x,t) = ul(x,t) following the first rebound can be made by a quadrature; for n = 1 we have obtained this function explicitly, formula (30). For an arbitrary value of n, one has ul(x,t) = f( u ( x , t ) and, always for Ill small enough, there are probably no contacts between t = tl = ~( x ) and t = t2 = 2 -~( x ) , and also the curve of the second contact is t = 2 At time t = 1, the string is at rest, and following the second rebound one again has u ( x , t ) = cosnx . cosnt.

6
Second example We consider the first example but in the limit case 8 = x/2. The contact curve A By which forms the first line of influence, is then the characteristic q = 0.5. On the convex part of the obstacle ( x c 0) the string cannot rebound and wraps on the obstacle, and thereafter we have u ( x , t ) = @(x), whereas on the concave part of the string rebounds, but as the contact curve is a characteristic the coefficient of reflection is not equal to 1, but is a function A(x), which will be determined. Following the first rebound we The velocity of the string on the obstacle (for 5 = 0.5) would be negative, which is not possible beyond the contact. Therefore the unwrapping of the string after t = 0.5 ( figure 6) cannot occur with a characteristic speed. It then happens that on a point of abscissa x = o(t), and on the curve x = a ( t ) or t = 7(x), we must have u~(x,T(x)) = @(x) and, from relation (7), (au2/at)(x,7(x)) = 0. This is a free boundary problem. The functions 7 ( x ) and g ( q ) must consequently satisfy the two conditions 1 ,cosx(r + x) + g ( 7x ) = -sinxx.cosxx x .

Third example
We again consider the same obstacle u = d(x) = -sinx:xcosnx, but we assume now that the string is initially at rest on the obstacle. Immediately after the initial time the string leaves the concave part and the motion in the region t 2 0, x < 0.5, tx < 0 ( figure 9) is defined by the solution u(x,t) = u~( x ,~) of the equation (3) Past the characteristic t + x = 1, we then have u(x,t) = u6(x.r) with 4uS(x,t) = -sin2x(t + x ) + 3 sin2x-. u6(x,t) end at the first line of influence formed (figure 9) by the arc CD (which corresponds to a rebound of the string on the obstacle) and by the characteristic tangent D E .

Fourth example
We consider now a new obstacle, symmetrical with respect to the Ou axis: At the initial time, the string is at rest in the position u(x,O) = cosxx. The free oscillation w (x, t ) = cos 7tx * cos 7tt takes place in all the triangular region 0 4 t < + x + 0.5; for all the region one has w(x,t) 2 @(x).
Beyond the characteristics t k x = 0.5, the equation w(x,t) = @(x), that is to say 3coslct = 1 -2cos2nx, possesses a solution t = z(x) (dotted line in figure 11) for which Idz/dxI 2 1, if t is small enough, xt < Arc cos (-1/9). O n the corresponding arcs, rebound is not possible, as we have shown in 5 4, because the solution determined by the reflection conditions does not satisfy the boundary conditions u(0.5,t) = 0. The motion of the string beyond the characteristic t + x = 0.5 is therefore no longer free oscillation, but is defined by a function u(x,t) = u,(x,t): in the x,t plane; and the conditions which determine the functionsf(5) and z(x) are u7(x,z(x)) = @(x), (au7/at)(x,r(x)) = 0, that is f ' ( z + x ) --sinn(zx) = 0. The motion u ( x , t ) is valid in the triangular region 0.5x < t $ inf (0.5 + x, 1 -2 x ) . In the symmetrical region, with respect to the t-axis u ( x , t ) = u,(x, t ) and therefore beyond the characteristics t x = 0.5, u(x, t ) = U8 (X, t),  At time t = 4/3 we again find the initial conditions. We can then express the following result: If; at some time, the string is at rest on the obstacle 3 u + cos 3 xx = 0, the motion is periodic with the period T = 413.
The positions of the string at different times are illustrated in figure 12.

Conclusion
The present results do not give a complete and definitive solution for the problem of a string vibrating in the presence of obstacles. Nevertheless, they contain new solutions for cases involving convex obstacles, cases which are much more involved than those involving concave obstacles (those involving straight or point-mass obstacles of course being simpler still). These new solutions should help provide a deeper understanding of the physics of this problem: rebounds, and wrapping and unwrapping of the string. We hope they will also help those who will eventually succeed in conceiving and writing a general computer program to obtain the motion of the string for arbitrary obstacles and arbitrary initial conditions.