THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

2.8 Derivation of the Acoustic Equations 25following format:σ xx = σ yy = σ zz =−pand, neglecting the gravitational body force ρg i (where i = x, y, z), we recastEquations (2.18a)–(2.18c) as− ∂p∂x = ρ DuDt− ∂p∂y = ρ DvDt− ∂p∂z = ρ DwDt2.7 Conservation of EnergyThe energy content W of a fluid is the sum of the macroscopic kinetic energyρ|V | 2 /2 and the internal energy ρE of the fluid. In a gas, the microscopic kineticenergy (i.e., the thermal energy of the molecules) comprises the major portionof the internal energy, so the potential energy between molecules is negligible incomparison. Denoting the energy flux by S we write equation for the conservationof energy as∂W+ ∂ S∂t ∂x = 0 (2.20)The internal energy of a volumetric element can be increased through heat flowfrom the surrounding fluid or from external sources and by the work of compression− ∫ pdV by the surrounding fluid pressure. This energy balance and the fact thatthe internal energy is a thermodynamic state that can be fully specified by twoindependent thermodynamic variables constitute the first law of thermodynamics.With the conservation equations discussed above and the equation of state, wehave all the necessary equations to obtain solutions for the three components ofvelocity V, ρ, p and absolute temperature T . Because the fluid equations are nonlinear,solutions are not easy to come by, even with the aid of supercomputers tomap the complex motions of atmospheric eddies, turbulent jet flows, capillary flow,and so on. Exact solutions exist principally for a few simple problems. Nevertheless,through the derivation of these equations, we have established the foundationfor the derivation of acoustic field equations for fluids.2.8 Derivation of the Acoustic EquationsWe begin with the following assumptions:(1) the unperturbed fluid has definite values of pressure, density, temperature, andvelocity, all of which are assumed to be time independent and denoted by thesubscript 0.