Treeby和Cox状态方程和波动方程的非定域性
On the nonlocality of the state and wave equation of Treeby and Cox
摘要：
本文证明Treeby和Cox [B. E. Treeby和B. T.考克斯，J. Soc. Am. \textbf{127} 5，（2010）]是\emph{nonlocal}，更精确地，\emph{local}密度变化引起\emph{instant global}压力变化，并且\emph{local}压力变化只能由\emph{instant global}密度变化引起。这与作者所知的所有与频率有关的耗散状态方程相反。此外，还证明了Treeby和Cox波动方程的格林函数$G$不可能具有\emph{finite}波前速度，即不存在有限的$c_F>0$使得$$ G(\mathbf{x},t) = 0 \qquad\mbox{for}\qquad |\mathbf{x}|/c_F > t $$成立，其中$|\mathbf{x}|/c_F$对应于以速度$c_F$从点$\mathbf{0}$传播到点$\mathbf{x}$的波的\emph{travel time}.因此，满足（i）Treeby和Cox的状态方程、（ii）运动方程和（iii）连续性方程的密度波和压力波不具有\emph{finite}波前速度。
英文摘要：
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.
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