Exact solution of the fundamental equation of acoustics for a pressure wave developing in two directions

The authors proceed from the hyperbolic equation for acoustic pressure. Using the integral Fourier transform along the axial coordinate, an equation in partial derivatives for the kernel of this transformation is found. This equation contains only one spatial coordinate and time. Applying the integral Laplace transform in time to the last equation, we obtain an ordinary differential equation with respect to the radial coordinate for the corresponding image. It turns out that the solution of the last equation is the well-known Macdonald function. For this function, it was possible to find the original image according to Laplace. All this made it possible to write an integral formula for the pressure in a sound wave. If the function of the initial pressure distribution along the pipe axis is taken in the form of a Gaussian impulse, then the integrals included in the representation of the desired solution are taken explicitly. As a result, we obtain an explicit compact formula for the acoustic pressure distribution in the axisymmetric case. It is convenient to use this formula to analyze the distribution of sound disturbances both along the pipe axis and in the radial direction. Therefore, the results are presented as isobars in the $(z, r)$ plane corresponding to different times.