This is the first of a series of three papers which examine the excitation of low-frequency plasma waves arising from a current source moving through a cold, magnetized plasma. In particular, the effects of motion of the source upon the radiation pattern are studied. In the present paper the general theory of wave excitation by a moving source is developed and specific approximations are made which are valid for frequencies much below the ion-cyclotron frequency. The amplitude and pattern of shear-Alv¿n waves excited by a current source are studied for sources whose size in the direction of motion is large in comparison to the size in other directions and also for sources whose size is large in the direction of current flow (also in comparison to the size of the source in other directions). It is shown that the radiation pattern of the shear waves arises naturally as a consequence of the motion of the source and that any modulation of the source current has little effect upon the waves, other than to reduce their amplitude.