The stable analytical solution for the wave fields in the sphere

We investigate the well-known analytical solution to the problem of the wave fields in the sphere. It is shown that the use of the standard asymptotic behavior of the Bessel functions leads to interference in the solution. A new asymptotic expression for the Bessel functions is found which gives a stable analytical solution that allows one to obtain the exact solution. The homogeneous and inhomogeneous waves for the sphere are detected. We present some examples of analytical calculation of the full wave fields and the primary wave for the sphere.