We have studied the structural elements, including shock ramps and precursor wave trains, of a series of oblique low-Mach number terrestrial bow shocks. We used magnetic field data from the dual ISEE 1 and 2 spacecraft to determine the scale lengths of various elements of shock structure as well as wavelengths and wave polarizations. Bow shocks structure under these conditions is esstentially that of a large-amplitude damped whistler mode wave which extends upstream in the form of a precursor wave train. Shock thicknesses, which are determined by the dispersive properties of the ambient plasma, are too broad to support current-driven electrostatic waves, ruling out such turbulence as the source of dissipation in these shocks. Dissipative processes are reflected in the damping of the precursors, and dissipative scale lengths are ~200--800 km (several times greater than shock thicknesses). Precursor damping is not related to shock normal angle or Mach number, but is correlated with Te/Tt. The source of the dissipation in the shocks does not appear to be wave-wave decay of the whistlers, for which no evidence is found. We cannot rule out the possibility of contribution to the dissipation from ion acoustic and, or lower hybrid mode turbulence, but interaction of the whistler itself with upstream electrons offers a simpler and more self-consistent explanation for the observed wave train damping.