Nonlinear Dynamics 30: 179–191, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
On Boundary Damping for a Weakly Nonlinear Wave Equation DARMAWIJOYO and W. T. VAN HORSSEN Department of Applied Mathematical Analysis, ITS, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands (Received: 31 July 2001; accepted: 27 March 2002) Abstract. In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a spring-mass-dashpot system, where the damping generated by the dashpot is assumed to be small. This problem can be regarded as a rather simple model describing oscillations of flexible structures such as suspension bridges or overhead transmission lines in a windfield. A multiple-timescales perturbation method will be used to construct formal asymptotic approximations of the solution. It will also be shown that all solutions tend to zero for a sufficiently large value of the damping parameter. For smaller values of the damping parameter it will be shown how the string-system eventually will oscillate. Keywords: Wave equation, boundary damping, asymptotics, two-timescales perturbation method.
1. Introduction There are a number of examples of flexible structures (such as suspension bridges, overhead transmission lines, dynamically loaded helical springs) that are subjected to oscillations due to different causes. Simple models which describe these oscillations can be expressed in initialboundary value problems for wave equations like in [1–6, 19] or for beam equations like in [7–10, 18]. Simple models which describe these oscillations can involve linear or nonlinear second and fourth order partial differential equations with classical or non-classical boundary conditions. These problems have been studied in [1–8] using a two-timescales perturbation method or a Galerkin-averaging method to construct approximations. In some flexible structures (such as an overhead transmission line or a cable of a suspension bridge) various types of wind-induced mechanical vibrations can occur. Vortex shedding for instance causes usually high frequency oscillations with small amplitudes, whereas low frequency vibrations with large amplitudes can be caused by flow-induced oscillations (galloping) of cables on which ice or snow has accreted. These vibrations can give rise to material fatigue. To suppress these oscillations various types of dampers have been applied in practice [18, 19]. In most cases simple, classical boundary conditions are applied (such as in [1, 4–9]) to construct approximations of the oscillations. More complicated, non-classical boundary conditions (see, for instance, [2, 3, 10–14]) have been considered only for linear partial differential equations. For nonlinear wave equations with boundary damping the approximations have been obtained numerically (see, for instance, [19]). In [19] it has been shown that for large values of the damping parameter the solutions tend to zero. It is, however, not clear how and for what values of the damping parameter the solutions will tend to zero (or not). In this paper On leave from State University of Sriwijaya, Indonesia.
180 Darmawijoyo and W. T. van Horssen
Figure 1. A simple model of an aero-elastic oscillator.
we will study an initial-boundary value problem for a weakly nonlinear partial differential equation for which one of the boundary conditions is of non-classical type. It will also be shown in this paper that the use of boundary damping can be used effectively to suppress the oscillation-amplitudes. Asymptotic approximations of the solution will be constructed. In fact, we will consider the vibrations of a string which is fixed at x = 0 and is attached to a spring-mass-dashpot system at x = π (see also Figure 1). This problem can be considered as a rather simple model to describe wind-induced vibrations of an overhead transmission line or a bridge (see [6, 7]). To our knowledge the use of boundary damping and the explicit construction of approximations of oscillations (which are described by a nonlinear PDE) have not been investigated previously. It is assumed that ρ (the mass-density of the string per unit length), T (the tension in the string), l (the length of the string), m ˜ (the mass in the spring-mass-dashpot system), γ˜ (the stiffness of the spring), α˜ (the damping coefficient of the dashpot), and k (for instance, the stiffness of the stays of the bridge) are all positive constants. Moreover, we only consider the vertical displacement u(x, ˜ t) of the string, where x is the place along the string, and t is time. We neglect internal damping and bending stiffness of the string and consider the weight W of the string per unit length to be constant (W = ρg, g is the gravitational acceleration). We consider a uniform windflow, which causes nonlinear drag and lift forces (FD , FL ) to act on the structure per unit length. The influence of geometrical nonlinearities of the string are assumed to be small (compared to the windforces) and will be neglected in this paper (see [6]). The equation describing the vertical displacement of the string is ρ u˜ t t − T u˜ xx + k u˜ = −ρg + FD + FL ,
(1)
with boundary conditions u(0, ˜ t) = 0 and
˜ u˜ t t (l, t) + γ˜ u(l, ˜ t) + α˜ u˜ t (l, t) = 0, T u˜ x (l, t) + m
In [6] it has been shown that FD + FL can be approximated by 2 a1 a2 2 a3 3 ρa dv∞ a0 + u˜ t + 2 u˜ t + 3 u˜ t , 2 v∞ v∞ v∞
t ≥ 0.
(2)
(3)
On Boundary Damping for a Weakly Nonlinear Wave Equation 181 where ρa is the density of air, d is the diameter of the cross-section of the string, v∞ is the uniform windflow velocity, and the coefficients a0 , a1 , a2 , a3 are given explicitly in [6] and depend on certain drag and lift coefficients. To simplify the initial-boundary value problem for u(x, ˜ t) we introduce the following transformation: u(x, ˜ t) = u(x, ¯ t) + us (x),
(4)
where us (x) is the stationary (that is, time-independent) solution of the initial-boundary value problem, and √ γ˜ − γ˜ cosh( (k/T )l) − T k sinh( (k/T )l) sinh( (k/T )x) us (x) = ρg , + C cosh( (k/T )x) − 1 kC with
√ C = γ˜ sinh( (k/T )l) + T k cosh( (k/T )l).
Then we also introduce the following dimensionless variables x¯ = (π/ l)x, t¯ = ct, v(x, ¯ t¯) = T ¯ t), with c = (π/ l) ρ . In this way Equation (1) becomes (c/(v∞ )u(x, vt¯ t¯ − vx¯ x¯
k + T
2 l ρa d v∞ a0 + a1 vt¯ + a2 vt2¯ + a3 vt3¯ . v= π 2ρ c
(5)
Now we assume that the wind velocity v∞ is small with respect to the wave speed c, that is, ˜ = v∞ /c is a small parameter. Following the analysis as given in [6] it can be shown that the right-hand side of Equation (5) up to order ˜ is equal to ((ρa d)/(2ρ))˜ (avt¯ − bvt3¯ ), where a and b are positive constants which depend on the drag and lift coefficients and which are also given explicitly in [6]. Using the transformation 3b v(x, ¯ t¯), u(x, ¯ t¯) = a putting p 2 = (k/T )(l/π )2 and = ((ρa d)/(2ρ))a ˜ it follows that (5) becomes 1 ut¯t¯ − ux¯ x¯ + p 2 u = ut¯ − u3t¯ , 3 where is a small dimensionless parameter. Finally it is assumed that√m, ˜ γ˜ , and α˜ are small, that is, we assume that m ˜ = mρ(l/π ) , γ˜ = γ T (π/ l) , and α˜ = α T ρ . In this paper we will study the following initial-boundary value problem for u(x, t) (for convenience, we will drop all the bars): 1 3 2 (6) ut t − uxx + p u = ut − ut , 0 < x < π, t > 0, 3 u(0, t) = 0,
t ≥ 0,
ux (π, t) = − (mut t (π, t) + γ u(π, t) + αut (π, t)) , u(x, 0) = φ(x),
0 < x < π,
(7) t ≥ 0,
(8) (9)
182 Darmawijoyo and W. T. van Horssen ut (x, 0) = ψ(x),
0 < x < π,
(10)
where φ and ψ are the initial displacement and the initial velocity of the string respectively, and where p 2 , m, γ , and α are positive constants, and where 0 < 1. In this paper formal approximations (that is, functions that satisfy the differential equation and the initial and boundary values up to some order in ) will be constructed for the solution of the initialboundary value problem (6–10). The outline of this paper is as follows. In Section 2 we apply a two-timescales perturbation method to construct formal approximations for the solution of the initial-boundary value problem (6–10) and we analyze this solution. Also in Section 2 we show that for all values of p 2 mode interactions occur only between modes with non-zero initial energy (up to O( )). Moreover, it will be shown in Section 2 that for α ≥ π/2 all solutions tend to zero (up to O( )). In Section 3 we make some remarks and draw some conclusions. 2. The Construction of Asymptotic Approximations To construct formal asymptotic approximations for the solution of the initial-boundary value problem (6–10) a two-timescales perturbation method [15–17] will be used in this section. Since an approximation in the form of an infinite series will be constructed we will impose some additional conditions on the initial values in order to get a convergent series representation for which summation and differentiation may be interchanged. The additional conditions on the initial values are: φ(0) = φ (π ) = φ
(0) = φ
(π ) = ψ(0) = ψ (π ) = ψ
(0) = 0, φ ∈ C 4 ([0, π ], ) , ψ ∈ C 3 ([0, π ], ) . By using a two-timescales perturbation method the function u(x, t) is supposed to be a function of x, t, and τ , where τ = t. We put u(x, t) = v(x, t, τ ; ).
(11)
By substituting (11) into the initial-boundary value problem (6–10) we obtain vt t − vxx + p 2 v + 2 vt τ + 2 vτ τ 1 3 = vt + vτ − (vt + vτ ) , 3 v(0, t, τ ; ) = 0,
0 < x < π, t > 0,
(12)
t ≥ 0,
(13)
vx (π, t, τ ; ) = − m(vt t (π, t) + 2 vt τ (π, t) + 2 vτ τ (π, t)) + γ v(π, t) + α(vt (π, t) + vτ (π, t))) , v(x, 0, 0; ) = φ(x),
t ≥ 0,
0 < x < π,
vt (x, 0, 0; ) + vτ (x, 0, 0; ) = ψ(x),
(14) (15)
0 < x < π.
(16)
By expanding v into a power series with respect to around = 0, that is, v(x, t, τ ; ) = vo (x, t, τ ) + v1 (x, t, τ ) + · · · ,
(17)
On Boundary Damping for a Weakly Nonlinear Wave Equation 183 and by substituting (17) into (12–16), and by equating the coefficients of like powers in , it follows from the power 0 and 1 of respectively, that v0 should satisfy vott − voxx + p 2 vo = 0, vo (0, t, τ ) = 0,
0 < x < π, t > 0,
(18)
t ≥ 0,
vox (π, t, τ ) = 0,
(19)
t ≥ 0,
vo (x, 0, 0) = φ(x),
(20)
0 < x < π,
vot (x, 0, 0) = ψ(x),
(21)
0 < x < π,
(22)
and that v1 should satisfy 1 v1tt − v1xx + p 2 v1 = vot − 2votτ − vo3t , 3 v1 (0, t, τ ) = 0,
0 < x < π, t > 0,
t ≥ 0,
(24)
v1x (π, t, τ ) = −(mvott (π, t) + γ vo (π, t) + αvot (π, t)), v1 (x, 0, 0) = 0,
(23)
t ≥ 0,
0 < x < π,
v1t (x, 0, 0) = −voτ (x, 0, 0),
(25) (26)
0 < x < π.
(27)
The solution of (18–22) is given by ∞
1 +n x , An cos( λn t) + Bn sin( λn t) sin vo (x, t, τ ) = 2 n=0
(28)
where λn = ((1/2) + n)2 + p 2 , and where An and Bn are still arbitrary functions of τ which can be used to avoid secular terms in v1 . From (21), (22), and (28) it follows that An (0) and Bn (0) have to satisfy 2 An (0) = π
π φ(x) sin
1 + n x dx, 2
(29)
0
2 Bn (0) = √ π λn
π ψ(x) sin
1 + n x dx, 2
(30)
0
for n = 0, 1, 2, . . .. Next, we solve the initial-boundary value problem (23–27). In order to solve this problem we will make the boundary condition (25) homogeneous. For that reason we define the following transformation v1 (x, t, τ ) = w(x, t, τ ) − x(mvott (π, t, τ ) + γ vo (π, t, τ ) + αvot (π, t, τ )).
(31)
184 Darmawijoyo and W. T. van Horssen Substituting (31) into the initial-boundary value problem (23–27) we obtain 1 wt t − wxx + p 2 w = vot − 2votτ − vo3t 3 + x ft t (t, τ ) + p 2 f (t, τ ) , w(0, t, τ ) = 0, wx (π, t, τ ) = 0,
0 < x < π, t > 0,
t ≥ 0,
(33)
t ≥ 0,
w(x, 0, 0) = xf (0, 0),
(32)
(34)
0 < x < π,
wt (x, 0, 0) = −voτ (0, 0) + xft (0, 0),
(35) 0 < x < π,
(36)
where f (t, τ ) = mvott (π, t, τ ) + γ vo (π, t, τ ) + αvot (π, t, τ ). It should be observed that ft t (t, τ ) + p f (t, τ ) = 2
∞
(−1)
n=0
n
1 +n 2
where
A∗n = mλn An − γ An − α λn Bn
and
2 A∗n cos( λn t) + Bn∗ sin( λn t) ,
Bn∗ = mλn Bn − γ Bn + α λn An .
To solve the initial-boundary value problem (32–36) the eigenfunction expansion method will be applied. For that reason, the function w is expanded into the Fourier series ∞
1 +n x . (37) wn (t, τ ) sin w(x, t, τ ) = 2 n=0 The function as defined in (37) satisfies the boundary conditions at x = 0 and x = π . By substituting (37) into (32) the left-hand side of (32) becomes ∞
1 2 +n x . (38) wntt + λn wn sin wt t − wxx + p w = 2 n=0 By multiplying (32) with sin(((1/2) + n)x), and by integrating the so-obtained equation with respect to x from 0 to π , it follows that wn (t, τ ) has to satisfy 2 ∞ 2(−1)n
k 1 ∗ ∗ + n (−1) cos( λ t) + B sin( λ t) A wntt + λn wn = k k k k 2 π(n + 12 )2 k=0 + λn [2A n − An ] sin( λn t) + [Bn − 2Bn ] cos( λn t) ∞ ∞ ∞
1 1
− − (39) − Hk Hl Hm , 4 k,l,m=0 3 k,l,m=0 k,l,m=0 k+l−m=n
k−l−m−1=n
k+l+m+1=n
On Boundary Damping for a Weakly Nonlinear Wave Equation 185 √ √ √ where Hn = λn (−An sin( λn t) + Bn cos( λn t)). The last terms in the right-hand side of (39) (that is, the terms involving products of trigonometric functions. These √ √ the sums) contain products can be equal to sin( λn t) or cos( λn t), which are solutions of the homogeneous equation wntt + λn wn = 0. Obviously these products can give rise to secular terms in w, and so in v1 . To determine the terms in the products of the trigonometric functions that give rise to secular terms we have to solve the following Diophantine-like problems: k + l − m = n,
or
k − l − m − 1 = n,
where √ √ √ √ ± λn = λk + λl − λm ,
or
k + l + m + 1 = n,
(40)
(41)
or √ √ √ √ ± λn = λk − λl + λm ,
(42)
√ √ √ √ ± λn = λk − λl − λm ,
(43)
or
or √
λn =
√
λk +
√
λl +
√
λm ,
(44)
with k, m, l, and n in N, and p 2 > 0. Note that λj = ((1/2) + j )2 + p 2 . To solve these problems (40–44) we use a technique similar to the one used in [5]. By substituting (40) (that is, k + l − m = n, or k − l − m − 1 = n, or k + l + m + 1 = n) into (41), or (42), or (43), or (44), by squaring the so-obtained equation twice, by rearranging terms and by using some elementary algebraic manipulations we find that the Diophantine-like problems (40–44) only have solutions for: √ √ √ √ 1. n = k + l − m and λn = λk + λl − λm . In this case the solution is given by: l = m and n = k, or k = m√and n = √ l. √ √ 2. n = k + l − m and λn = λk − λl + λm . In this case the solution of the equation is given by l = m and √ √ √ n = k. √ 3. n = k + l − m and λn = − λk + λl + λm . In this case the solution of the equation is given by k = m and n = l. We rewrite (39) by taking apart those terms in the right-hand side of (39) that give rise to secular terms in w, yielding 2 wntt + λn wn = 2 λn A n − λn An − (−mλn Bn + γ Bn − α λn An ) π ∞
1 1 2 2 2 2 λn λn An (An + Bn ) − λn An λm (Am + Bm ) sin( λn t) − 4 4 m=0 2 + −2 λn Bn + λn Bn − (−mλn An + γ An + α λn Bn ) π
186 Darmawijoyo and W. T. van Horssen ∞
1 1 2 2 2 2 − λn λn Bn (An + Bn ) + λn Bn λm (Am + Bm ) cos( λn t) − 4 4 m=0 ∞ ∞ ∞
1 1 ∗ − − − Hk Hl Hm 4 k,l,m=0 3 k,l,m=0 k,l,m=0 k+l−m=n
k−l−m−1=n
k+l+m+1=n
∞ 2(−1)n
1 2 ∗ k ∗ + (−1) cos( λ t) + B sin( λ t) , (45) A k + k k k k 2 π(n + 12 )2 k=0 k=n
∗
indicates that terms in this sum giving rise to secular terms are excluded. where the * in √ In order √ to avoid secular terms in w (and in v1 ) we have to take the coefficients of sin( λn t) and cos( λn t) in the right-hand side of (45) to be equal to zero, yielding 2 2 λn A n − λn An = (−mλn Bn + γ Bn − α λn An ) π ∞
1 1 2 2 2 2 λn λn An (An + Bn ) − λn An λm (Am + Bm ) , + 4 4 m=0 2 2 λn Bn − λn Bn = − (−mλn An + γ An + α λn Bn ) π ∞
1 1 λn λn An (A2n + Bn2 ) − λn An λm (A2m + Bm2 ) , + 4 4 m=0
(46)
(47)
√ √ for n = 0, 1, 2, . . .. By taking λn An = A¯ n , λn Bn = B¯ n system (46–47) simplifies to 2 γ
¯ ¯ ¯ ¯ −m λn + √ Bn − α An 2An − An = π λn ∞
1 1 ¯ ¯2 2 2 2 An (An + B¯ n ) − A¯ n (A¯ m + B¯ m ) , (48) + 4 4 m=0 γ ¯ ¯ −m λn + √ An + α Bn λn ∞
1 1 ¯ ¯2 2 2 2 Bn (An + B¯ n ) − B¯ n (A¯ m + B¯ m ) , + 4 4 m=0
2 2B¯ n − B¯ n = − π
(49)
for n = 0, 1, 2, . . . . From (48) and (49) it can easily be seen that if A¯ n (0) = B¯ n (0) = 0 then A¯ n = B¯ n = 0 for τ = t > 0. So if we start with no initial energy in the n-th mode then there will be no energy present up to O( ) on timescales of order −1 . This allows us to truncate the infinite dimensional system (48–49) to those modes which have non-zero initial energy. To study system (48–49) in more detail it is convenient to introduce yn = (1/16)(A¯ 2n + B¯ n2 ). By
On Boundary Damping for a Weakly Nonlinear Wave Equation 187 Table 1. The behaviour of the critical points. α
Critical point
Behaviour
(0,0)
unstable node
0 < α < π2
α> π2
0 , 13 (1 − π2 α)
stable node
1 (1 − 2 α), 0 π 3
1 (1 − 2 α), 1 (1 − 2 α) π π 7 7
(0, 0)
stable node saddle point stable node
multiplying (48) with A¯ n , (49) with B¯ n , adding the so-obtained equations, and by substituting yn into the so-obtained equations it then follows that ∞
α
ym , n = 0, 1, 2, . . . . (50) yn = yn 1 − 2 + yn − 4 π m=0 It is clear from (50) that yn < 0 if the damping parameter α ≥ π/2. So for α ≥ π/2 all solutions of (6–10) will tend to zero for increasing time t. It then follows that, for the damping parameter α ≥ π/2, the equilibrium solution zero of the system of equation (50) is stable. When for instance only energy is initially present in the first two modes (that is, yo (0) = 0, y1 (0) = 0, and yn (0) = 0 for n ≥ 2) a phase-plane analysis can be performed. From (50) it then follows that yo and y1 have to satisfy 2
(51) yo = yo 1 − α − 3yo − 4y1 , π 2
(52) y1 = y1 1 − α − 4yo − 3y1 . π The critical points of (51) and (52) for 0 < α < π/2 are 2 1 2 1 2 1 2 1 1 − α , 0 , 0, 1− α , and 1− α , 1− α . (0, 0) , 3 π 3 π 7 π 7 π For α ≥ π/2 the only critical point is (0, 0). By linearizing (51) and (52) around the critical points for 0 < α < π/2 we find two stable nodes, one unstable node, and one saddle point. For α ≥ π/2 the critical point is a stable node (see Table 1). From the table it can readily be seen that if the damping parameter α is increasing (starting from α = 0) then the two stable nodes and the saddle point are moving to the unstable node. For α = π/2 the four critical points coincide in (0, 0), and for α > π/2 a stable node occurs in (0, 0). The behaviour of the solution of (51–52) can also be seen in Figures 2 and 3. For 0 < α < π/2 it can be seen in Figure 2 that the solution (usually) will finally tend to a single mode vibration up to order for times of order −1 . For α > π/2 it can be seen in Figure 3 that the string vibrations will finally come to rest up to O( ) as t → ∞.
188 Darmawijoyo and W. T. van Horssen
Figure 2. Phase plane for 0 < α < π/2.
Figure 3. Phase plane for α > π/2.
So far our analysis has been restricted to the case where only the first two vibration modes are initially present. The analysis, however, can be generalized to multiple mode initial conditions. We have seen already that for α ≥ π/2 all solutions of (6–10) tend to zero (up to O( )) for increasing time t. For 0 < α < π/2 we define y¯n (s) =
yn (τ ) , 1 − 2 πα
(53)
On Boundary Damping for a Weakly Nonlinear Wave Equation 189 where s = (1 − 2(α/π ))τ . From (50) and (53) it then follows that y¯n (s) satisfies ∞
dy¯n (s) = y¯n (s) 1 + y¯n (s) − 4 y¯m (s) , n = 0, 1, 2, . . . . ds m=0
(54)
Equation (54) has been studied in detail by Keller and Kogelman in [1], and it has been shown in [1] that ⇒ lim yk (τ ) = 0 (55) lim y¯k (s) = 0 τ →∞
s→∞
for those k for which y¯k (0) < maxn y¯n (0), and that 1 − π2 α 1 ⇒ lim yk (τ ) = lim y¯k (s) = s→∞ τ →∞ 4q − 1 4q − 1
(56)
for those k for which y¯k (0) is equal to maxn y¯n (0), and where q is the number of modes for which initially y¯k (0) is equal to maxn y¯n (0). Only for q = 1 the ‘limit’-solution for s → ∞(⇒ τ → ∞) is stable. The limiting values in (55) and (56) are equilibrium solutions of the system of Equations (50). It can readily be seen from (55) and (56) that if the damping coefficient α is increased the equilibrium solution will tend to zero. The interesting thing in this case is that the use of boundary damping can suppress the oscillation-amplitudes efficiently. After removing secular terms in (45) we can finally determine wn from (45). After some lengthy, but elementary calculations we obtain wn (t, τ ) = Fn (t, τ ) + −
1 4
∞
gnk A∗k cos( λk t) + Bk∗ sin( λk t)
k=0 k=n
∞∗
−
k,l,m=0 k+l−m=n
∞
−
k,l,m=0 k−l−m−1=n
1 3
∞
k,l,m=0 k+l+m+1=n
cos(T t + δ ) klm klm , Sklm i 2 λ − (T ) n klm i=1 4
i
where
√ √ Fn (t, τ ) = Cn (τ ) cos( λn t) + Dn (τ ) sin( λn t), A∗n = mλn An − γ An − α λn Bn ,
gnk
Bn∗ = mλn Bn − γ An + α λn An ,
Sklm =
2 = π
k+ n+
1 2 1 2
λk λl λm
2
(−1)k+n , λn − λk
i=k,l,m
λl + λm , = λk + λl − λm ,
1 = Tklm 2 Tklm
λk +
3 2 = Tkml , Tklm
3 2 δklm = δkml ,
1 δklm = αk + αl + αm , 2 δklm = αk + αl − αm ,
A2i + Bi2 ,
i
(57)
190 Darmawijoyo and W. T. van Horssen 4 4 = λk − λl − λm , δklm = αk − αl − αm , Tklm and where αn is defined as follows: for A2n + Bn2 = 0: αn = 0, and for A2n + Bn2 = 0: Bn cos(αn ) = 2 An + Bn2
and
An sin(αn ) = . 2 An + Bn2
It should be observed that wn still contains infinitely many free functions Cn and Dn of τ for n = 0, 1, 2, . . . . These functions can be used to avoid secular terms in the solution of the O( 2 )-problem for v2 . It is, however, our goal to construct a function u¯ that satisfies the partial differential equation, the boundary conditions, and the initial values up to order 2 . For that reason Cn and Dn are taken to be equal to their initial values Cn (0) and Dn (0) respectively. So far we constructed a formal approximation u¯ = vo + v1 for u that satisfies the partial differential equation, the boundary conditions, and the initial values up to order 2 . In [4–6, 8] asymptotic theories have been presented for wave and beam equations with similar nonlinearities. The formal approximations constructed for those problems were shown to be asymptotically valid, i.e., the differences between the approximations and the exact solutions are of order on timescales of order −1 as → 0. It is beyond the scope of this paper to give the asymptotic analysis for the wave equation we discussed. We expect that the asymptotic validity of the constructed approximations can be shown in a way similar to the analysis presented in [4–6, 8]. Finally it should be remarked that from these asymptotic theories it follows that vo + v1 and vo are both (order ) asymptotic approximations of the exact solution on timescales of order −1 . 3. Conclusions In this paper an initial-boundary value problem for a weakly nonlinear wave equation with a non-classical boundary condition has been considered. Formal asymptotic approximations of the exact solution have been constructed using a two-timescales perturbation method. For all values of p 2 > 0 it has been shown that mode-interactions only occur between modes with non-zero initial energy up to O( ). This implies that truncation is allowed to those modes that have non-zero initial energy up to O( ). For the damping parameter α ≥ π/2 it also has been shown that all solutions tend (up to O( )) to zero as t → ∞. For 0 < α < π/2 it can be shown that the string system usually will oscillate in only one mode (up to O( )) as t → ∞. When p 2 = 0 (which models the galloping oscillations of overhead power transmission lines in a windfield (see [6])) truncation is not allowed to those modes that have non-zero initial energy up to O( ). Instead of the Fourier-series method we can use the method of characteristic coordinates (in combination with a multiple-timescales perturbation method). Again it can be shown that for α ≥ π/2 all solutions will tend to zero (up to O( )) as t → ∞. For 0 < α < π/2 it can be shown that the solution will tend to a standing triangular wave as t → ∞ (with a vanishing amplitude as α tends to π/2). The calculations are more complicated than the ones presented in this paper, and will appear in a forthcoming paper. References 1.
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