The laminar boundary‐layer flow resulting from a wave potential flow of formU= &agr;csin(kx — &ohgr;t), with &agr; a parameter andcthe wave velocity, is considered. It is shown, after suitable transformation, that there is an exact solution of the unsteady boundary‐layer equations which is of the form of a power series in the phasekx — &ohgr;t. The coefficients &phgr;nare functions of a similarity variable, and are the solutions of an infinite set of linear third‐order differential equations with nonlinear forcing terms. The forcing term in the equation for &phgr;nis a function of &phgr;0, &phgr;1, …, &phgr;n−1and their derivatives. Solutions for &phgr;0, &phgr;1, and &phgr;2have been computed and are presented. The theory is applied to the laminar boundary layer under a progressive shallow‐water wave, where &agr; =a/h, and compared to a linearized theory. It is concluded that if &agr; ≪ 1, or for any &agr; in a sufficiently small region nearkx — &ohgr;t= 0, the linearized theory is valid. Otherwise, the linear theory does not provide an adequate description of the flow.