  CeRh$_3$Si$_2$ has been reported to exhibit metamagnetic transitions below
5~K, a giant crystal field splitting, and anisotropic magnetic properties from
single crystal magnetization and heat capacity measurements. Here we report
results of neutron and x-ray scattering studies of the magnetic structure and
crystal-field excitations to further understand the magnetism of this compound.
Inelastic neutron scattering (INS) and resonant inelastic x-ray scattering
(RIXS) reveal a $J_z$\,=\,1/2 groundstate for Ce when considering the
crystallographic $a$ direction as quantization axis, thus explaining the
anisotropy of the static susceptibility. Furthermore, we find a total splitting
of 78\,meV for the $J$\,=\,5/2 multiplet. The neutron diffraction study in zero
field reveals that on cooling from the paramagnetic state, the system first
orders at $T_{\text{N}_1}=4.7$\,K in a longitudinal spin density wave with
ordered Ce moments along the $b$-axis (i.e. the [0 1 0] crystal direction) and
an incommensurate propagation vector $\textbf{k}=(0,0.43,0$). Below the
lower-temperature transition $T_{\text{N}_2}=4.48$\,K, the propagation vector
locks to the commensurate value $\textbf{k}=(0,0.5,0)$, with a so-called
lock-in transition. Our neutron diffraction study in applied magnetic field
$H\parallel b$-axis shows a change in the commensurate propagation vector and
development of a ferromagnetic component at $H=3$\,kOe, followed by a series of
transitions before the fully field-induced ferromagnetic phase is reached at $H
= 7$\,kOe. This explains the nature of the steps previously reported in
field-dependent magnetization measurements. A very similar behaviour is also
observed for the $H\parallel$ [0 1 1] crystal direction.
