ИСТИНА

We solve a non-stationary problem of wave propagation in a waveguide excited by a δ-shaped pulse in time and space. The waveguide is assumed to have complex multi-component structure, but be homogeneous in some direction. As it is well-known, solution of this problem can be represented as a double integral over frequency ω and wavenumber k. There is a classical problem of asymptotical estimations of this integral to describe physically conditioned wave pulses propagating in the waveguide. If there are no energy losses in the waveguide, all special points of the formal solution integrand are poles, which give residues. The result of the residue theorem application is the solution in a form of an integral of an analytical function f(k) (or f′(ω)) with a complex integration contour laying on several sheets of the Riemann surface of f(k) [3,4]. We assume that |f(k)| is defined mainly by an exponential factor. This allows one to apply the saddle point method. In the current work we apply a muti-contour saddle point method [5] to describe a forerunner (a fast exponentially decaying wave) in a car tire. To describe non-stationary processes in a tire, it is useful to consider it as a waveguide homogeneous along the azimuthal angle θ [6]. Since a tire is a complex structure composed of rubber and cords, a set of its saddle points is not trivial. We build this set and classify the saddle points into contributing and non-contributing to the integral asymptotic. The tire pulse response was both numerically calculated by the multi-contour saddle point method and measured experimentally. A good agreement between the experimentally observed forerunner and its theoretical prediction is shown.