On the nonlocality of the state and wave equation of Treeby and Cox

Abstract

In this paper it is shown that the state equation of Treeby and Cox [B. E. Treeby and B. T. Cox, J. Acoust. Soc. Am. \textbf{127} 5, (2010)] is \emph{nonlocal}, more precisely, a \emph{local} density variation causes an \emph{instant global} pressure variation and a \emph{local} pressure variation can only be caused by an \emph{instant global} density variation. This is in contrast to all frequency dependent dissipative state equations known to the author. Moreover, it is shown that the Green function $G$ of the wave equation of Treeby and Cox cannot have a \emph{finite} wave front speed, i.e. there exists no finite $c_F>0$ such that $$ G(\mathbf{x},t) = 0 \qquad\mbox{for}\qquad |\mathbf{x}|/c_F > t $$ holds, where $|\mathbf{x}|/c_F$ corresponds to the \emph{travel time} of a wave propagating with speed $c_F$ from point $\mathbf{0}$ to point $\mathbf{x}$. As a consequence, the density and pressure waves satisfying (i) the state equation of Treeby and Cox, (ii) the equation of motion and (iii) the equation of continuity do not have a \emph{finite} wave front speed.