  Detecting protoplanets during their formation stage is an important but
elusive goal of modern astronomy. Kinematic detections via the spiral wakes in
the gaseous disc are a promising avenue to achieve this goal. We aim to test
the applicability to observations in the low and intermediate planet mass
regimes of a commonly used semi-analytical model for planet induced spiral
waves. In contrast with previous works which proposed to use the
semi-analytical model to interpret observations, in this study we analyse for
the first time both the structure of the velocity and density perturbations. We
run a set of FARGO3D hydrodynamic simulations and compare them with the output
of the semi-analytic model in the code wakeflow, which is obtained by solving
Burgers' equation using the simulations as an initial condition. We find that
the velocity field derived from the analytic theory is discontinuous at the
interface between the linear and nonlinear regions. After 0.2 r$_p$ from the
planet, the behaviour of the velocity field closely follows that of the density
perturbations. In the low mass limit, the analytical model is in qualitative
agreement with the simulations, although it underestimates the azimuthal width
and the amplitude of the perturbations, predicting a stronger decay but a
slower azimuthal advance of the shock fronts. In the intermediate regime, the
discrepancy increases, resulting in a different pitch angle between the spirals
of the simulations and the analytic model. The implementation of a fitting
procedure based on the minimisation of intensity residuals is bound to fail due
to the deviation in pitch angle between the analytic model and the simulations.
In order to apply this model to observations, it needs to be revisited
accounting also for higher planet masses.
