VRIJE UNIVERSITEIT BRUSSEL Acoustics - the Dept. of ...

8 CHAPTER 1. FUNDAMENTAL CONCEPTS OF ACOUSTICS1.2.3 The one-dimensional wave equationFollowing parameters are considered :x coordinate of elementary particle in equilibrium.u particle displacement with respect to equilibrium.v particle velocity v = ∂u∂t .ρ the instantaneous value of the fluid density.ρ 0 fluid density in equilibrium (considered constant)s condensationinapoint(defacto: relativechangeindensity). Thisvariableis defined as :s . = ρ−ρ 0ρ 0or ρ = ρ 0 (1+s) (1.9)p the sound pressure P = P 0 +pc wave propagation speedGravitation is not considered and thus ρ 0 and P 0 areconstant. The gas orfluid is assumed to behomogeneous isotropic elastic : there are no dissipativeforces due to viscosity or heat conduction. We limit this study to waves withsmall amplitude such that the condensation s can be considered to be small :ρ−ρ 0 ≪ ρ 0 . While the wave propagates along the x-axis through the fluid,the adjacent fluid layers are also disturbed from their equilibrium position.This displacement u is function of x and t.In order to derive the wave equation we will use three physical laws:1. The mass conservation principle.2. Thermodynamic change of state.3. Newtons equation.Firstly, the mass conservation principle is applied on a volume between xand x+dx and a deformed volume :ρ 0 Sdx = ρSdx(1+ ∂u∂x ) (1.10)