Back in 1990, D. S. Stewart and the first author contributed significantly to understanding the one-dimensional stability of detonation waves [1]. For this purpose, the reactive Euler’s equation with the one-component reaction term was linearized around the steady state of the well-known ZND (Zeldovich-Doering-von Neumann) model. The key aspect of this paper was to derive the linearized radiation condition (named after A. Sommerfeld). They numerically found multiple eigenvalues for pairs of the temporal frequency and temporal attenuation rate (TAR). Of course, the propagating-wave mode having the least value of the TAR (in the sense of its absolute value) was selected. The successful numerical implementation of the far-field radiation condition is a must when it comes to incorporating a large surrounding space into a problem of finite extent. To one of the sure examples in this category belong the problems involving detonation waves, where high-energy-rate processes take place in spatially confined spaces while the surrounding space should be taken into account for reasons of energy loss (or leaky waves in the language of optics). In another fascinating area of science is nano-photonics, where energy transport should be handled in highly confined regions of space, yet being surrounded by unbounded (dielectric) media. The total energy release in nano-photonics is certainly much smaller than that involved in detonation. However, the energy per unit nanometer-scale volume is not negligibly small in nano-photonics. Over the years, the first author has been successful in implementing both theory and numerical methods to find a multitude of eigenvalues in optics [2]. In this case, the governing Maxwell’s equations are already in a linearized form, being in a sense similar to the linearized Euler equations. In addition, the noble metals such as gold and silver are instrumental in enhancing local electric-field intensities, for which the science of plasmonics is being vigorously investigated in nano-photonics. Even the Bloch’s hydrodynamic equation describing the collective motion of the electrons is akin to the Navier-Stokes equations [3]. Meanwhile, assuming a real-valued frequency has been an old tradition in optics, partly because the real-valued photon’s energy is proportional to frequency and normally the continuous-wave (cw) approximation holds true. In a radical departure from this optical scientists’ tradition, we have recently attempted to deal with complex-valued frequencies in examining the wave propagations around nanoparticles [4, 5]. In consequence, the stability of multiple propagating waves was successfully determined for selecting most realizable wave mode. Further interesting points of the interplay between the two seemingly disparate branches of science (fluid dynamics and photonics) will be expounded in this talk.
Nonlinear oscillations of gas in an open tube near the resonance frequency in the shock-free mode