In this study the propagation and growth of whistler mode waves generated by electron beams within the earth's bow shock are investigated using a planar model for the bow shock and a model electron distribution function. Within the shock, the model electron distribution function possesses a field-aligned T⊥ >T beam that is directed toward the magnetosheath. Waves with frequencies between about 1 and 100 Hz with a wide range of wave normal angles are generated by the beam via Landau and anomalous cyclotron resonances. However, because the growth rate is small and because the wave packets traverse the shock quickly, these waves do not attain large amplitudes. Waves with frequencies between about 30 and 150 Hz with a wide range of wave normal angles are generated by the beam via the normal cyclotron resonance. The ray paths for most of these waves are directed toward the solar wind, although some wave packets, because of plasma convection, travel transverse to the shock normal. These wave packets grow to large amplitudes, because they spend a long time in the growth region. The results suggest that whistler mode noise within the shock should increase in amplitude with increasing upstream &thgr;Bn. The study provides an explanation for the origin of much of the whistler mode turbulence observed at the bow shock.