  In this work, we consider a system of two wave equations coupled by
velocities in one-dimensional space, with one boundary fractional damping.
First, we show that the system is strongly asymptotically stable if and only if
the coupling parameter b of the two equations is outside a discrete set of
exceptional real values. Next, we show that our system is not uniformly stable.
Hence, we look for a polynomial decay rate for smooth initial data. Using
frequency domain approach combining with multiplier method, we prove that the
energy decay rate is greatly influenced by the nature of the coupling parameter
b, the arithmetic property of the ratio of the wave propagation speeds a, the
order of the fractional damping. Indeed, under the equal speed propagation
condition, we establish an optimal polynomial energy decay rate. Furthermore,
when the wave propagate with different speeds, under some arithmetic conditions
on the ratio of the wave propagation speeds, we prove that the energy of our
system decays polynomially to zero.

Analysis of PDEs