This paper derives new recurrence relations for scalar spherical waves in Cartesian coordinates and compares the performance of these relations with that of traditional recurrence relations. They are numerically stable for travelling waves and some standing waves, and are faster for evaluating scalar spherical waves up to order 40 on a nonspherical surface. Because of their algebraic simplicity, they will be especially useful for symbolic-algebra computer programs, allowing explicit expressions for scalar and vector spherical waves to be obtained using a succinct procedure.