A new theory is developed for the wake far downstream of a cylindrical body of height h, placed with its generators perpendicular to the flow on a surface above which there is a boundary layer of thickness δ. If the streamwise (x) velocity in the wakeis (U + u), then assuming (h/δ) is small enough that the velocity profile in the boundary layer may be regarded as U = αy, and assuming |u| [Lt] U, linear differential equations governing u are derived. It is found that a constant along the wake is\[I=\frac{3}{2}\int_0^{\infty} yUu\,dy.\]This result can be used to find an order of magnitude estimate for u, because I is related to the forces on the body producing the wake by the approximate formula\[I \simeq - C_1/\rho,\]where C1 is that component of the couple on the body produced by pressure and viscous stresses in the x direction. For the particular case of a small hump on the boundary of height h and length b, such that h [Lt] b, the above relation is shown to be exact. The perturbation velocity in the wake is found to have a similarity solution\[u = [I/(xv)]f(y^3/[xv/\alpha]),\]the physical implications of which are discussed in detail. The relevance of the theory to the problem of transition behind a trip wire is also mentioned.