Uniformly Valid Asymptotics for Carrier ’ s Mathematical Model of String Oscillations

In the paper, an asymptotic analysis of G.F. Carrier’s mathematical model of string oscillation is presented. The model consists of a system of two nonlinear second order partial differential equations and periodic initial conditions. The longitudinal and transversal string oscillations are analyzed together when at the initial moment of time the system’s solutions have amplitudes proportional to a small parameter. The problem is reduced to a system of two weakly nonlinear wave equations. The resonant interaction of periodic waves is analyzed. An uniformly valid asymptotic approximation in the long time interval, which is inversely proportional to the small parameter, is constructed. This asymptotic approximation is a solution of averaged along characteristics integro-differential system. Conditions of appearance of combinatoric resonances in the system have been established. The results of numerical experiments are presented.


Equation (1.1) is obtained by introducing certain mathematical and physical
preconditions limiting the applicability of the model. For example, mathematical preconditions regarding the smallness of the function v(x, t): |v(x, t)| 1, analyticity or at least smoothing v(x, t) ∈ C 2 ([0, l] × [0, ∞)), physical preconditions regarding string weightlessness, nontensility and absolute tensility. In the opposite case, we should account for the effect of gravitational forces and for the fact that the string's length in the process of its motions may change, therefore the Hooke tensility law does not work.
If we reject at least one above-mentioned or other "obvious" precondition, we shall obtain various generalizations of the linear wave equation (1.1) and of the related model.
For example, upon rejecting the conditions of the small deviation and given that v(x, t) ∼ 1, the equation of motion begins to depend on the first-order derivative v x (x, t) and becomes nonlinear. We can show that in this case from the Lagrangian function, corresponding to such an oscillating string with the usual boundary conditions v(0, t) = v(l, t) = 0, there follows a more complicated equation of motion Employing mechanical reasonings, this equation was obtained [22]. A broader context of modelling is presented in the monographs [12,31]. The monograph [31] deals with the general properties of the mathematical modelling whereas the monograph [12] helps us to find asymptotic solutions of the strongly nonlinear systems of differential equations. The monograph is dedicated to a very important fundamental problem: the inversion of Lagranges theorem on the stability of equilibrium not a trivial task, over which researchers had struggled for more than half a century.
The oscillation equations of the both mentioned models have their regions of validity. Equation (1.2) is nonlinear, valid also in the case when the value of the transversal deviations |v(x, t)| 1. However, in the case transversal deviations are too large, equation (1.2) becomes misleading: in case of large deviations, the deflection of a string element may be not only transversal, but also longitudinal. This means that large deviations are characterized by two functions: transversal v(x, t) and longitudinal u(x, t). It is the idea that has stimulated a number of authors to formulate the corresponding nonlinear models of a fluctuating string not for one but for two functions -v(x, t) and u(x, t) (see, e. g., [1,2,7,21]).
For example, this attracted G.F.Carrier's attention in [5,6] (see also [8,25]); he has formulated a model of the nonlinear motion of a string, in which he analyzes both transversal and longitudinal string oscillations, describing them by a system of two nonlinear second-order partial differential equations: here ρ is the substance density, A -the transection area, E -the Young modulus of string, T -the initial tension. Differently from the classical rectilinear motion, an element of a non-stimulated string is characterized by the spatial axial coordinate x and the time moment t. Deviations of the stimulated string element are characterized by the longitudinal u(x, t) and transverse v(x, t) displacements. The string moves in the (u, v) plane. Such a note may be regarded as a parametrical curve which depends not only on the spatial but also on the time coordinate for which the radius-vector r = (u(x, t), v(x, t)).
In case when the condition The second term will be much larger than the third one if its numerator will be significantly larger than the third term numerator, i.e.
and it is from here that the condition (1.5) follows.
It is well known (see, e. g., [4]) that T EA is a deformation or, in our case, the relative lengthening of the string ∆l l . Condition (1.5) has a simple geometrical interpretation: (1.8) Inequality (1.8) and expansion (1.6) will be satisfied if ∆l < 3 2 l. Let us summarize: when a swaying string lengthens more than 3/2 times, the Carrier's model (1.3) should be used. When the lengthening does not exceed more than 3/2 of the string length but is rather large, e.g., ∆l ∼ l, model (1.2) may be applied. Last but not least, when the condition ∆l l and correspondingly v 2 x 1, equation (1.2) turns into the classical rectilinear swaying equation (1.1). Note that the presented limitations of the application of the models are valid in the bounded region of changing the independent variables (x, t) ∈ [0, l] × 0, t 0 , when the constants l, t 0 do not depend on the asymptotic relations (1.5), (1.7), etc. The more subtle asymptotic analysis performed in this paper shows that when a large (x, t) region of variable changes is analyzed, transition to a simpler model may be groundless even at small oscillation amplitudes. Coming back to system (1.3), let us note that when at the initial moment of time t = 0 the system solutions have small amplitudes (proportional to the small parameter ε), the problem is reduced to a system of two nonlinear wave equations. In this paper, we construct an asymptotic approximation uniformly valid in the long time interval t ∼ ε −1 and determine the appearance conditions of the resonant wave interactions.

Analysis of the model by the small parameter method
Suppose that at the initial moment of time t = 0 the following conditions are satisfied: We shall search for the system (1.3) solution in the form u =εũ, v =εṽ. Substituting u, v in (1.3) and selecting the terms ofε power-series expansions, we obtain:ε Let us note that By omitting members of the O ε 3 order and neglecting the sign (˜), we receive the asymptotic integration problem with the initial conditions corresponding to (2.1). When in system (2.3) ε = 0, we obtain a couple of two independent travelling waves (see, e. g., [30]): Functions (2.4) are obtained also from the linear model (1.1). However, when ε > 0, function (2.4) will be close to the solution of the system (2.3), in the general case only in a short time interval t O ε −1 . The system (2.1)-(2.3) has a classical (continuously differentiable with respect to the variables t ir x) solution u(t, x; ε), v(t, x; ε) in the time interval We are constructing an asymptotic solution At a certain moment of time t = τ 1 ε , there may appear the discontinuous derivatives u t , u x , v t , v x (gradient catastrophe) is constructed and its asymptotic approximation does not describe the solution. This means that the time interval under analysis t ∈ 0, τ 0 ε and τ 0 < τ 1 . Estimation (2.5) is an analog of the well known in asymptotic analysis N.N. Bogoliubov's theorem [24].
The aspects of mathematical substantiation of asymptotic approximations constructed in the paper have been analyzed [13,14].
3 Construction of the uniformly valid asymptotics Let us rewrite system (2.3) in the Riemann invariants r j of the linear part of the system. Note that the asymptotic method does not use the quasi-linear system invariants which may be constructed by the methods of [9] [10].
Then we obtain (3.1) We shall analyze system (3.1) when functions (2.1) are 2π-periodic: If to looking for the asymptotic solution of (3.1) as a regular small parameter ε power series expansion we shall see that in the general case (3.3) we will have the secular terms εt (see, e. g., [26]). Therefore, approximation (3.3) may be used only when the values εt are small, i. e. t ε −1 , or in a short time t interval. On the other hand, equations (3.1) have a classical (diferentiated) solution when t ∈ 0, O ε −1 , i. e. in a long time interval.
Let us note that in papers [5,6] concerning the model (1.3) the author analyzed the asymptotics, but they had secular terms.
It is a non-trivial problem to construct an asymptotic solution uniformly valid in a long time interval t ∈ 0, O ε −1 . Such asymptotics is found as a solution of the following system averaged along the characteristics [15,16]: .

When all numbers
λi−λj λ k −λj are irrational, the right sides of system (3.4) will be equal to zero and the solutions are expressed by formulas (2.4), i. e. travelling by independent waves at the velocity λ j . In the opposite case, the right sides of system (3.4) may not equal zero, and in this case the solution of this system depends on the slow time τ . In this paper, we shall limit ourselves to the case when a and b are integer numbers.
We shall obtain an integral differential system which we integrate from 0 to 2π. Sometimes such averaging is called internal [15], and this is its essential difference as compared with various partial derivatives averaging schemes [23,24] which may be called the external averaging. The systems obtained by internal averaging often remain without a subsequent study and left in the literature as a certain theoretical result of the asymptotic analysis [3,28]. Let us note that system (3.4) does not directly depend on the small parameter ε and thus has no problems of asymptotic integration. Therefore, the problem may be successfully solved by numerical methods [15,16]. Let us note some papers [11,20,27,29] in which, like in our work, the method of two scales and averaging methods are applied, however, the Fourier analysis is performed without directly constructing the averaging system. In our paper, for the definite equations (1.3), we have applied the general method of averaging along characteristics the weakly nonlinear hyperbolic systems. The constructed averaging system allows finding uniformly valid asymptotic approximations of a polynomial form by applying the methods of our earlier work [17]. On the other hand, the theoretical analysis of the obtained averaged system (see [18,19]) allows determining the conditions of appearing combinatoric resonances.

Analysis of the averaged system
We shall analyze the solution of system (3.4) by the Fourier series: (4.1) The functions R jkj (τ ) (3.2) satisfy the initial conditions R jkj (0) = r (0) jkj . When the right sides of system (3.4) are equal to zero, correspondingly the functions R jkj (τ ) ≡ r (0) jkj , i. e. do not depend on τ . Otherwise we shall have the resonant wave interaction and the functions R jkj (τ ) will describe the slowly changing wave amplitude.
Let us denote the sets of resonant harmonics: The resonant wave interaction in the system does not appear if R 1 = R 2 = R 3 = R 4 = ∅. The sets R j are interdependent and are empty only in case when the parameters of the problem a = E ρ and b = T ρA meet certain requirements. Suppose that the sets R 3 and R 4 have no such harmonics k 3 , k 4 for whom is valid. In this case, we shall have R 1 = R 2 = ∅. On the other hand, the sets R 3,4 will be empty, if they and the sets R 1,2 have no such harmonics for which is valid. Note also that for resonances to appear the following relations among all harmonics depending to the set R j are necessary: Let us note that the relations (4.3), (4.4) imply that the corresponding coefficients of the Fourier R jkj (τ ) should not be equal to zero.

Numerical experiments
In practice, we limit ourselves to the first Fourier lines (4.1) harmonics |k j | ≤ K and look for the functions R jkj (τ ) by the following approximations: We find the polynomial coefficients by the method of undetermined coefficients. The same results we obtain by successive iterations.
We shall present the calculated data using the computer algebra system Maple coefficients (5.1): We see that the sets R   Figure 1 presents graphs of the functions R j (τ, y), j = 1, 2, 3, 4 when τ = 0 and τ = 0.9, and y 1 = y 2 = y 3 = y 4 = y are taken. At the initial time moment τ = 0, the functions R 1 and R 2 are absent, and at τ = 0.9 there appear waves, their amplitudes depending on τ . The resonant interactions of waves cause also the dependence of the amplitudes of waves R 3 , R 4 on the slow time τ . The amplitudes slowly change, however, at τ = 0.9 in the graph the new waves R 3 , R 4 are clearly seen.
The calculated coefficients of polynoms (5.1) are presented in Table 1. We see that the sets R j in this case have more elements: This means that not only the transverse oscillations induced the longitudinal ones as in example 1, but also the longitudinal oscillations induced a resonance (not only amplitude) effect on the transverse ones. This effect caused the first harmonic -the basic tone -of the transverse oscillations. The constructed approximations of the Riemann invariants allow to write down the approximations of the required functions u, v, i.e. wave profile approximations: u(x, t; ε) ≈ 1 2a (R 2 (εt, x + at) − R 1 (εt, x + at)) dx = 1.3333 + 0.0394 (εt) 2 − 0.0033 (εt) 4 sin 3(x + t) If at the initial moment of time the transverse oscillations contain no harmonics satisfying the relation (4.2), no resonance appears in the system. This means that the longitudinal oscillations alone cannot arouse the transversal ones, but the transverse oscillations may induce the longitudinal ones in the presence of (4.2).
In this case, the longitudinal oscillations, together with the transversal, ones may induce new harmonics of the transverse vibrations (overtones) of (4.2). Note that all conclusions are valid for the first asymptotic approximations (2.2). By presenting in the system (2.2) terms of the higher order of the small parameter ε, we may construct the second asymptotic approximation and analyze the harmonics which appear not only because of combinatoric resonances (4.2),(4.3),(4.4) but also because of the system's nonlinearities. It is obvious that these effects will be of a higher ε order, i. e. they will be weaker.