The properties of adjoint operators and the method of composite expansion are used to study the generation of Tollmien–Schlichting (TS) waves in the leading-edge region of an incompressible, flat-plate boundary layer. Following the classical asymptotic approach, the flow field is divided into an initial receptivity region, where the unsteady motion is governed by the linearized unsteady boundary-layer equation (LUBLE), and a downstream linear amplification area, where the evolution of the unstable mode is described by the classical Orr–Sommerfeld equation (OSE). The large $\bar{x}$ behaviour of the LUBLE is analysed using a multiple-scale expansion which leads to a set of composite differential equations uniformly valid in the wall-normal direction. These are solved numerically as an eigenvalue problem to determine the local properties of the Lam and Rott eigensolutions. The receptivity coefficient for an impinging acoustic wave is extracted by projecting the numerical solution of the LUBLE onto the adjoint of the Lam and Rott eigenfunction which, further downstream, turns into an unstable TS wave. In the linear amplification region, the main characteristics of the instability are recovered by using a multiple-scale expansion of the Navier–Stokes equations and solving numerically the derived eigenvalue problems. A new matching procedure, based on the properties of the adjoint Orr–Sommerfeld operator, is then used to check the existence and the extent of an overlapping domain between the two asymptotic regions. Results for different frequencies are discussed.