2. Basic Differential Equations
The flux equations can be used to determine the net rate of increase of each quantity within a cube of volume ΔV= ΔxΔyΔz. We implicitly assume the limit where the dimensions approach zero.
For mass, the net flux inflow through the face at x=xo is ρux kg-m-2-s-1. The outflow through the face at x=xo+Δx is
Therefore the net flow across the two faces, each of area ΔyΔz, is
Converting this to density, and adding in the flow through the other four faces,
This is known as the equation of continuity, and basically reflects the law of conservation of mass.
The net flow of x-directed momentum across the same two faces parallel to the y-z plane is
In addition there is a flow of x-directed momentum across the other four faces, given by
Putting this together with the other momentum components, and dividing by ΔV yields
This equation reflects the law of conservation of momentum.
Following the same procedure for the third and final flux yields
For a monatomic molecule, the factor of 5 in the total energy term on the left side of equation (15) would be replaced by 3, and the factor of 7 on the right-hand side of the equation would be replaced by 5. These terms represent the effects that are typically explained in terms of specific heat ratios, which we do not need to be concerned with. This equation reflects the law of conservation of energy, and also implicitly defines adiabatic behavior, since no energy is added to or subtracted from the system as a whole.
Expanding the time derivative in equation (14), substituting the right-hand side of equation (11) for the left-hand side, and using the following vector identity
This is known as Euler's equation [Landau and Lifshitz page 3].
Pressure Differential Equation
Following a similar procedure, equation (15) can be transformed into a differential equation for pressure. The right-hand sides of both equations (11) and (14) are substituted for their left-hand sides, and the following vector identity is also used:
The derivation is a bit messy, but otherwise straightforward. The result is:
This equation probably also has a name, but I haven't run across it. Equations (11), (17) and (19) are the basic equations we need to proceed. The next section makes several approximations based on the assumption that the variation in all of the parameters are small compared to the static values, which finally leads to one primary goal of the analysis - the wave equations.
One-dimensional versions of equations (11), (17), and (19) can be found here.
It is interesting to note that this entire derivation could have been done in terms of expected values, using only the number of particles per unit volume, and the velocity statistics, without ever making a connection to mass, momentum, or energy. Newton's 2nd law, relating force, mass, and acceleration, which is central to the usual derivation of sound waves, is not used at all in the above derivation. The concepts of pressure and force are not necessary. In order to relate the results to the real world a connection to physical quantities is necessary, but the derivation itself is almost entirely independent of Newtonian physics.
To the next section.
To the list of Physics of Sound Subsections.