  In this study, a linear stability analysis is performed for different Weakly
Compressible Smooth Particle Hydrodynamics (WCSPH) methods on a 1D periodic
domain describing an incompressible base flow. The perturbation equation can be
vectorized and written as an ordinary differential equation where the
coefficients are circulant matrices. The diagonalization of the system is
equivalent to apply a spatial discrete Fourier transform. This leads to
stability conditions expressed by the discrete Fourier transform of the first
and second derivatives of the kernel. Although spurious modes are highlighted,
no tensile nor pairing instabilities are found in the present study, suggesting
that the perturbations of the stresses are always damped if the base flow is
incompressible. The perturbations equation is solved in the Laplace domain,
allowing to derive an analytical solution of the transient state. Also, it is
demonstrated analytically that a positive background pressure combined with the
uncorrected gradient operator leads to a reordering of the particle lattice. It
is also shown that above a critical value, the background pressure leads to
instabilities. Finally, the dispersion curves for inviscid and viscous flows
are plotted for different WCSPH methods and compared to the continuum solution.
It is observed that a background pressure equal to $\rho c^2$ gives the best
fidelity to predict the propagation of a sound wave. When viscosity effects are
taken into account, the damping of pressure fluctuations show the best
agreement with the continuum for $p_{back} \sim \rho c^2/2$.
