Two velocities

You may wonder where the two velocities v(x,z) and $\bar v ( z )$ came from. The first arises in the wave equation, and it must be x-variable if the model is x-variable. The second arises in a mathematical transformation, namely, equation (15), so it is purely a matter of definition. Experience shows that complications will abound if we try to make the defined velocity $\bar v$ depend on x. But it is nice if the two velocities can be equal so the term containing their difference drops out of the analysis. Thus ordinarily, $\bar v ( z )$ is chosen to be some kind of horizontal average of v (x,z).