A note on the formula for the Rayleigh wave speed

In this brief note, we have derived an approximate analytical expression for the Rayleigh wave speed for an isotropic half-space using Lanczos's approximation. This expression is simpler than the exact expressions reported earlier by a number of authors while at the same is very accurate, with an error not exceeding 0.45%. The same approximation has been also used to derive an approximate analytical expression for an incompressible orthotropic half-space. The latter has been shown to be within a maximum error not exceeding 0.05%. In many theoretical and practical applications, such approximate formulas might prove more useful and convenient to manipulate with.