  Faraday waves are a classic example of a system in which an extended pattern
emerges under spatially uniform forcing. Motivated by systems in which uniform
excitation is not plausible, we study both experimentally and theoretically the
effect of heterogeneous forcing on Faraday waves. Our experiments show that
vibrations restricted to finite regions lead to the formation of localized
subharmonic wave patterns and change the onset of the instability. The
prototype model used for the theoretical calculations is the parametrically
driven and damped nonlinear Schr\"odinger equation, which is known to describe
well Faraday-instability regimes. For an energy injection with a Gaussian
spatial profile, we show that the evolution of the envelope of the wave pattern
can be reduced to a Weber-equation eigenvalue problem. Our theoretical results
provide very good predictions of our experimental observations provided that
the decay length scale of the Gaussian profile is much larger than the pattern
wavelength.
