Short duration, intense pulses of light can experience dramatic spectral broadening when propagating through lengths of optical fibre. This continuum generation process is caused by a combination of nonlinear optical effects including the formation of dispersive waves. Optical analogues of Cherenkov radiation, these waves allow a pulse to radiate power into a distant spectral region. In this work, efficient and coherent dispersive wave generation of visible to ultraviolet light is demonstrated in silica waveguides on a silicon chip. Unlike fibre broadeners, the arrays provide a wide range of emission wavelength choices on a single, compact chip. This new capability is used to simplify offset frequency measurements of a mode-locked frequency comb. The arrays can also enable mode-locked lasers to attain unprecedented tunable spectral reach for spectroscopy, bioimaging, tomography and metrology.