On the coefficient of nonlinearity β in nonlinear acoustics

An analysis valid to second order in the acoustic variables is used to investigate the origin and form of the coefficient of nonlinearity β. Two cases are considered. First, for progressive plane waves, the well-known formula is β=1+B/2A, where B/2A is the coefficient of the quadratic term in the isentropic pressure–density relation. It is shown that the component 1, which is associated with convection, comes solely from nonlinear terms in the equation of continuity. The second component comes from a quadratic term in the equation of state. The momentum equation is linear for this type of wave motion and therefore contributes nothing to the expression for β. Second, for noncollinear interaction of two plane waves, quasilinear analysis shows that the expression for β appropriate for the sum and difference frequency waves has three components. In this case, the momentum equation does contribute to the formula. For most practical problems the three-component formula reduces to β=cos θ +B/2A, where θ is the angle of interaction. The factor cos θ represents the combination of contributions cos2(θ/2) and −sin2(θ/2) from the continuity and momentum equations, respectively.