Transparent Artificial Boundary Conditions for Waves Leaving a Highly-Stretched Cylindrical Computational Domain

We propose artificial boundary conditions for the wave equation considered outside cylindrical computational domains of a high aspect ratio. Such domains can arise in many practical problems, e.g. - as to logging problems - while one develops logging tools, interprets field logs etc. An exact condition on the side boundary is obtained analytically from the representation of a general solution to the wave equation in Fourier series. The discrete counterpart to this non-local condition is generated so that it has recurrence formulae with respect to time: only two sequential temporal levels are required for computations. Instead of conditions on top and bottom boundaries of the domain we use a larger domain where a deformed wave equation is considered. All the formulae have the second order accuracy with respect to both time and space. Test calculations show that proposed conditions do provide a high accuracy without noticeable computational costs.