Green's functions for a source embedded in an isothermal transversely sheared mixing layer are compared with direct numerical simulation (DNS) at various frequencies. Based on the third-order convective wave equation (Lilley 1974), three types of wave responses are analysed. For direct waves, a vortex sheet is used in the low-frequency limit, while in the high-frequency limit the procedure derived by Goldstein (1982) is re-visited. For refracted arrival waves propagating in the zone of silence, the vortex sheet model derived by Friedland & Pierce (1969) is re-visited in the low-frequency limit, while in the high-frequency limit the finite thickness model derived by Suzuki & Lele (2002) is applied. Instability waves excited by a very low-frequency source are formulated in the linear regime using the normal mode decomposition: eigen-functions are normalized using the adjoint convective wave equation, and the receptivity of instability waves is predicted. These theoretical predictions are compared with numerical simulations in two dimensions: DNS are performed based on the full Navier–Stokes equations (the free-stream Mach number is $M_1=0.8$, and the ratios of the acoustic wavelength to the vorticity thickness $\lambda/\delta_V$ are 4.0, 1.0 and 0.25). The DNS results agree fairly well with the high-frequency limit in all three cases for direct waves, although the lowest-frequency case ($\lambda/\delta_V = 4.0$) indicates some features predicted in the low-frequency limit. For refracted arrival waves, the DNS data follow the low- and high-frequency limits to a reasonable degree of accuracy in all cases. Moreover, by setting $\lambda/\delta_V = 16.0$, instability waves are simulated, and a comparison with the theoretical prediction shows that the instability wave response is predicted well when a mixing-layer Reynolds number is high ($Re = 10^5$). They also reveal that the receptivity is fairly sensitive to the Reynolds number and the source position within the mixing layer.