We study random waves on smooth, compact, Riemannian manifolds under the spherical ensemble. Our first main result shows that there is a positive universal limit for the critical radius of a specific deterministic embedding, defined via the eigenfunctions of the Laplace-Beltrami operator, of such manifolds into higher dimensional Euclidean spaces. This result enables the application of Weyl's tube formula to derive the tail probabilities for the suprema of random waves. Consequently, the estimate for the expectation of the Euler characteristic of the excursion set follows directly.