  The recent development of subwavelength photonic and phononic crystals shows
the possibility of controlling wave propagation at deep subwavelength scales.
Subwavelength bandgap phononic crystals are typically created using a periodic
arrangement of subwavelength resonators, in our case small gas bubbles in a
liquid. In this work, a waveguide is created by modifying the sizes of the
bubbles along a line in a dilute two-dimensional bubbly crystal, thereby
creating a line defect. Our aim is to prove that the line defect indeed acts as
a waveguide; waves of certain frequencies will be localized to, and guided
along, the line defect. The key result is an original formula for the
frequencies of the defect modes. Moreover, these frequencies are numerically
computed using the multipole method, which numerically illustrates our main
results.
