Green's Law for the propagation of a linear nondispersive wave in a gradually varying medium conserves energy, which is a quadratic integral property. Green's Law does not conserve linear integral properties, and other waves must arise. This paper treats some general properties that these additional waves must possess for a class that includes long linear water waves in a channel of gradually varying section, The major finding is that both a reflected wave and a re-reflected tail are required to satisfy all of the conservation laws; integral properties of these additional waves are derived.