Traveling Waves

More precisely - traveling waves in a model of a nonlinearly supported beam (like a suspension bridge) or a plate. The motivation for the study of this problem comes from an event that occurred on February 9, 1938 on the Golden Gate Bridge in San Francisco. The bridge engineer Russell G. Cone described in his report that the suspended part of the bridge displayed a wave-like motion (a wave moving along the bridge) resembling cracking a whip. Although it is more or less clear that the cause of this was a strong gust wind blowing through the bridge on that day, it has not been explained exactly how the waves originated.

A mathematical model of such a bridge or a suspended beam goes back to [2]. In its simplest form the model describes the bridge by the equation

,

where is the deflection of the roadbed, the -axis points in the direction along the bridge, is time, the term represents forces of the cables that apply only in case the cables are tight , but not when they are loose. When looking for traveling waves one searches for solutions in the special form , where is a given constant (the speed of the traveling wave).

In [1] we looked at a similar problem but made several generalizations: the cables can apply some force even if they are loose and the force is given by some nonlinear function , we work in more space dimensions than just one - also in 2D and 3D. The equation then has the following form

and we make the traveling wave ansatz (in the direction of the -axis).

Fig. 1		Fig. 2

Movie (1.7 MB)

Fig. 3

The existence of traveling wave solutions of the model was proved under some assumptions on the nonlinear function . Also, many numerical computations were performed. They brought an evidence of existence of multiple solutions with the same speed . Two of them are shown in Figures 1 and 2. It was also observed that these waves have very interesting interaction and stability properties. An example of a collision of two waves is shown in the movie in Figure 3.

References

[1]  J. Horak, P. J. McKenna, Traveling waves in nonlinearly supported beams and plates, Nonlinear Equations: Methods, Models and Applications, Progress in Nonlin. Diff. Eq. Appl., vol. 54, Birkhäuser, Basel 2003, 197-215. [2]  P. J. McKenna, W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math. 50 (1990) 703-715. [3]  B. Breuer, J. Horak, P. J. McKenna, M. Plum, A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam, J. of Differential Equations 224 (2006), no. 1, 60-97.