波与wave equations服从衰减的因果关系分析
Causality analysis of waves and wave equations obeying attenuation
摘要：
本文证明了衰减波的标准因果关系条件，即与波的衰减规律和相速度有关的Kramers-Kronig关系，对于波的因果关系是必要的，但不是充分的。根据波的因果关系，我们理解波的波前速度是有界的。虽然这种情况并不新鲜，但对波衰减的后果还没有进行充分的分析。我们推导出服从衰减和因果关系的波动方程（对于均匀和各向同性介质），并推广了Paley-Wiener-Schwartz定理（参见。定理7.4.3.在\cite{Ho03}中），我们对服从频率功率衰减规律的波进行因果分析。后来，Szabo波动方程的因果关系行为（参见。\cite{Szabo94}）和热粘性波动方程。最后，我们提出了一个泛化的热粘性波动方程，服从因果关系和频率幂律（功率在$(1,2]$和）的小频率，我们提出的热声层析成像。
英文摘要：
In this paper we show that the standard causality condition for attenuated waves, i.e. the Kramers-Kronig relation that relates the attenuation law and the phase speed of the wave, is necessary but not sufficient for causality of a wave. By causality of a wave we understand the property that its wave front speed is bounded. Although this condition is not new, the consequences for wave attenuation have not been analysed sufficiently well. We derive the wave equation (for a homogeneous and isotropic medium) obeying attenuation and causality and with a generalization of the Paley-Wiener-Schwartz Theorem (cf. Theorem 7.4.3. in \cite{Ho03}), we perform a causality analysis of waves obeying the frequency power attenuation law. Afterwards the causality behaviour of Szabo's wave equation (cf. \cite{Szabo94}) and the thermo-viscous wave equation are investigated. Finally, we present a generalization of the thermo-viscous wave equation that obeys causality and the frequency power law (for powers in $(1,2]$ and) for small frequencies, which we propose for Thermoacoustic Tomography.
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