Weakly nonlinear waves radiated by pulsations of a cylinder

A cylindrical sound wave is radiated into an unbounded ideal gas by an infinite circular cylinder that executes a sinusoidal pulsation uniformly along its axis with small amplitude and moderate frequency for only one period. In the leading order of approximation, the weakly nonlinear propagation of the wave is studied for the case of sufficiently large acoustic Reynolds number up to the stage that its profile develops into a cylindrical N wave with a tail. In the near field, solving the linear wave equation gives the near-field solution, which has a tail following the body part of the wave. The tail of the velocity profile decreases asymptotically in proportion to t*-⁴ as t* → ∞ at a fixed point in the near field and to t*-5/2 at a fixed point in the far field (t* is the time from the beginning of the pulsation). The tail vanishes in a high frequency limit with an acoustic Mach number being fixed. In the far field, an exact solution, which matches with the near-field solution, is obtained for a far-field equation by the method of strained coordinates. The evolution of the two shocks and the tail are then examined by making use of the equal-areas rule. At large distances from the cylinder, the whole profile of the wave approaches to an N wave with a long tail