  The Kuramoto model is a commonly used mathematical model for studying
synchronized oscillations in biological systems, with its temporal
synchronization properties well studied. However, the properties of spatial
waves have received less attention. This paper investigates the spatial waves
formed by locally coupled oscillators arranged in an $n\times n$ grid.
Numerical simulations show that directional waves can form when the system
exhibits heterogeneity, while spiral waves can arise in a homogeneous system.
Interestingly, both wave patterns remain stable under minor noise disturbances.
To explain the properties of the spatial wave pattern, starting from the
simplest case of a $2\times 2$ grid, we analytically calculate the phase
differences between oscillators to discuss the formation of wave patterns in
the system. We then apply this method to compute the stable and saddle points
and corresponding wave patterns of some $n\times n$ grid cases and discuss
their stability. Furthermore, linear approximation reveals that the wave
pattern under noise is the noiseless wave pattern plus its first-order
approximation, indicating that the wave pattern remains stable within a certain
range of noise. These results suggest that the necessary condition for
directional wave propagation in biological systems is the presence of
heterogeneity that far exceeds noise. In contrast, the disappearance of
heterogeneity may induce spiral waves, often corresponding to disease states.
