1.1 Background to wave intensity

Vein Man - Vesalius (1543)

Wave intensity analysis is rooted in the development of gas dynamics during and after the Second World War. The advent of supersonic flight, jet engines and rockets required a new approach to aerodynamics that could explain the 'new' phenomena that were being observed; particularly shock waves. For low Mach number flows [The Mach number is defined as the speed convection divided by the speed of sound, m = U/c.], air could be considered to be incompressible with reasonable accuracy, but this was no longer true as we neared the 'sound barrier' when the Mach number approached and exceeded one. For supersonic and hypersonic flow, it became important to track the propagation of waves through the flow field.

The mathematical tools for solving these problems were provided nearly a century earlier by Riemann who introduced the method of characteristics for the solution of hyperbolic equations [B Riemann, Gesammelte mathematische Werke un wissenschaftlicher Nachlass (1860)]. The import of this solution is impressive; he found a general solution for an entire class of partial differential equations that is valid for linear and nonlinear equations with continuous or discontinuous boundary conditions. It is an achievement that would easily have secured his place in mathematical history, even if he hadn't also laid the foundations for the theory of relativity with his work on non-Euclidean geometry and also made major contributions to the theory of complex numbers.

Although arteries have complex geometries, for many purposes it is sufficient to consider them as long, thin tubes - the 1-D approximation. This approximation ignores the variation of velocity across the cross-section, necessarily abandoning the no-slip condition at the wall. It is, therefore, not suitable for the calculation of the detailed distribution of wall shear stress, for example, but does provide information about the axial distribution of pressure and velocity. Interestingly, current methods of computational fluid dynamics that do provide detailed solutions for the 3-D distribution of velocity, pressure, and shear stress are generally restricted to local regions of an artery such as a single bifurcation or occluded section of artery. These solutions require boundary conditions at the inlet and outlet of the section which are generally unknown. The approximate, simplified boundary conditions that are generally used do not take account of the wave nature of arterial flow. An interesting future direction for theoretical haemodynamics is the combination of 1-D and 3-D theory to provide more realistic local solutions that take account of global conditions.