  The steelpan is a pitched percussion instrument that takes the form of a
concave bowl with several localized dimpled regions of varying curvature. Each
of these localized zones, called notes, can vibrate independently when struck,
and produces a sustained tone of a well-defined pitch. While the association of
the localized zones with individual notes has long been known and exploited,
the relationship between the shell geometry and the strength of the mode
confinement remains unclear. Here, we explore the spectral properties of the
steelpan modeled as a vibrating elastic shell. To characterize the resulting
eigenvalue problem, we generalize a recently developed theory of localization
landscapes for scalar elliptic operators to the vector-valued case, and predict
the location of confined eigenmodes by solving a Poisson problem. A finite
element discretization of the shell shows that the localization strength is
determined by the difference in curvature between the note and the surrounding
bowl. In addition to providing an explanation for how a steelpan operates as a
two-dimensional xylophone, our study provides a geometric principle for
designing localized modes in elastic shells.
