Using the forced Korteweg‐de Vries equation as a simple model, a perturbation procedure is presented for calculating the amplitude of short‐scale oscillatory tails induced by steady long‐wave disturbances. In the limit of weak dispersion, these tails have exponentially small amplitude that lies beyond all orders of the usual long‐wave expansion. It is demonstrated that by working in the wavenumber domain, the tail amplitude can be determined quite simply, without the need for asymptotic matching in the complex plane. The induced short‐wave tail is sensitive to the details of the long‐wave profile. The proposed technique is applicable to nonlocal solitary waves and to other problems that require the use of exponential