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We investigate analytically the linearized water wave radiation problem foran oscillating submerged point source in an inviscid shear flow with a freesurface. A constant depth is taken into account and the shear flow increaseslinearly with depth. The surface velocity relative to the source is taken to bezero, so that Doppler effects are absent. We solve the linearized Eulerequations to calculate the resulting wave field as well as its far-fieldasymptotics. For values of the Froude number $F^2=\omega^2 D/g$ ($\omega$:oscillation frequency, $D$ submergence depth) below a resonant value$F^2_\text{res}$ the wave field splits cleanly into separate contributions fromregular dispersive propagating waves and non-dispersive "critical waves"resulting from a critical layer-like street of flow structures directlydownstream of the source. In the sub-resonant regime the regular waves behavelike sheared ring waves while the critical layer wave forms a street of aconstant width of order $D\sqrt{S/\omega}$ ($S$ is the shear flow vorticity)and is convected downstream at the fluid velocity at the depth of the source.When the Froude number approaches its resonant value, the the downstreamcritical and regular waves resonate, producing a train of waves of linearlyincreasing amplitude contained within a downstream wedge.

Waves from an oscillating point source with a free surface in the presence of a shear current