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\section{Introduction}  The reservoir-wave hypothesis was introduced in order to resolve a number of difficulties in the interpretation of arterial mechanics. The capacitive nature of the compliant arteries was recognised by \citet{Borelli1680}, Borelli \cite{Borelli1680},  popularised by Hales \cite{Hales1733} and made quantitative by Frank \cite{Frank1899}. The Windkessel model introduced by Frank is an excellent representation of arterial mechanics during diastole when the arterial flow is isolated from the left ventricle by the closed aortic valve. However, despite decades of work, the Windkessel model could not describe the observed pressure and flow in the arteries during systole. As a result, it fell into disfavour. The wave nature of arterial mechanics was obvious to the earliest observers and a great deal of attention was given to the arterial pulse. In the second half of the twentieth century, the wave analysis of arterial mechanics became the dominant interpretation of arterial physiology. The measured pressure and flow were Fourier analysed and state of the cardiovascular system was characterised by its impedance, derived by dividing the pressure coefficient by the flow coefficient for each frequency. This method proved to be a very powerful descriptor of the state of the cardiovascular system \cite{Nichols2005}.