Fluid displacement and velocity

Our next task is to work out an approximate equation of motion for small-amplitude sound waves in a fluid. But before we do this, we must develop a notation which will adequately describe the motion. As a matter of fact, we shall develop two notations, one of which will be generally employed; but the other will be occasionally useful. The most straightforward way, the extension of the notation we have used for the chain of masses and springs, is to label each portion of the fluid  by giving its position at t = 0, and then to give the subsequent position of each portion as a function of time and of its initial position. In this notation, called the Lagrange description, we follow the fluid in its motion.

In one-dimensional motion, the portion of fluid initially at xo has position X at time t, where X is a function of t and of xo. All other quantities similarly follow the motion of the fluid. The density ρ(xo,t) is the density of the portion of the fluid which was initially at Xo and which is at X at time t. This notation gives us a good insight into what is going on inside the fluid as it moves about. For example, if the time rate of change dρ/dt of this density is less than zero, we can be sure the fluid is expanding.

The derivative du/dt, where u = dX/dt is the fluid velocity, is the acceleration of the element of fluid which was originally at xo, and is thus equal to the force acting on the element, divided by its mass. However, the notation also has its disadvantages, for the coordinate system moves with the fluid, and it is not easy to determine what the fluid is doing at some specified point in space at a given time.

So the alternative notation, the Euler description, sets up a coordinate system (x, in one dimension) fixed in space and describes the properties of the fluid which happens to be at a given point at a given time. Therefore, in this notation, the various quantities are functions of x and of t, for one dimension. These are quantities, such as density, velocity, etc., measured at a fixed point in space at a given time, and therefore corresponding to different portions of the fluid, as the fluid streams past the point.

This is the description we shall be using most of the time, with p and u given as explicit functions of x and t; by it we can tell what is happening where, without having to go back to find which part of the fluid happens to be there at time t. In particular, the Euler description enables us to compute easily the spatial variation of a given quantity at time t, the partial derivative ∂p/∂x being the rate of change of density at x, as might be measured from an instantaneous photograph of the density at time t.