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The goal of this paper is to show that evanescent plane waves are much betterat numerically approximating Helmholtz solutions than classical propagativeplane waves. By generalizing the Jacobi$\unicode{x2013}$Anger identity tocomplex-valued directions, we first prove that any solution of the Helmholtzequation on a three-dimensional ball can be written as a continuoussuperposition of evanescent plane waves in a stable way. We then propose apractical numerical recipe to select discrete approximation sets of evanescentplane waves, which exhibits considerable improvements over standard propagativeplane wave schemes in numerical experiments. We show that all this is notpossible for propagative plane waves: they cannot stably represent generalHelmholtz solutions, and any approximation based on discrete sets ofpropagative plane waves is doomed to have exponentially large coefficients andthus to be numerically unstable. This paper is motivated by applications toTrefftz-type Galerkin schemes and extends the recent results in [Parolin,Huybrechs and Moiola, M2AN, 2023] from two to three space dimensions.

Stable approximation of Helmholtz solutions in the ball using evanescent plane waves