On the degeneration of creeping waves in a vicinity of critical values of the impedance

Creeping waves propagating on a three-dimensional surface with an impedance boundary condition are considered. The standard asymptotic formula for the creeping waves contains the factor l/(ξ + q2) where ξ is the attenuation parameter and q is the Fock parameter q = (kρ/2)1/3Z, where k is the wave number, ρ is the radius of curvature of the geodesics followed by creeping wave and Z is the impedance. This factor diverges when the parameter q takes critical values, which means invalidity of the usual asymptotic formula for creeping wave field. The critical values of the Fock parameter q are found and a new local asymptotics is derived in the supposition that the factor l/(ξ + q2) is infinite on a curve (which we call the degeneration curve) crossed by creeping waves. This new asymptotic decomposition is carried out by powers of the small parameter k−1/9. The effect of creeping wave passing through the degeneration curve is examined.