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Stellar bow shocks, bow waves, and dust waves all result from the action of a star's wind and radiation pressure on a stream of dusty plasma that flows past it. The dust in these bows emits prominently at mid-infrared wavelengths in the range 8 to   60  |$\mu$| m. We propose a novel diagnostic method, the τ–η diagram, for analysing these bows, which is based on comparing the fractions of stellar radiative energy and stellar radiative momentum that is trapped by the bow shell. This diagram allows the discrimination of wind-supported bow shocks, radiation-supported bow waves, and dust waves in which grains decouple from the gas. For the wind-supported bow shocks, it allows the stellar wind mass-loss rate to be determined. We critically compare our method with a previous method that has been proposed for determining wind mass-loss rates from bow shock observations. This comparison points to ways in which both methods can be improved and suggests a downward revision by a factor of two with respect to previously reported mass-loss rates. From a sample of 23 mid-infrared bow-shaped sources, we identify at least four strong candidates for radiation-supported bow waves, which need to be confirmed by more detailed studies, but no strong candidates for dust waves. [ABSTRACT FROM AUTHOR]