The Principle of Closed Wavetrains, Resonance and Efficiency: Past, Present and Future

The principle of closed wavetrains asserts the equivalence of the condition of a traveling wave closing onto itself in phase to the occurrence of a mode. This principle provides a direct conceptual link between spectral descriptions of dynamic responses and a path-based dynamic description. In this paper we present the history and development of the idea since d’Alembert first proposed traveling functional forms to solve the string equation. The subsequent argument between Daniel Bernoulli, Euler and him which led to the development of Fourier analysis and contemporary theories of partial differential equations. Other related developments include the development of chaos theory connected to Poincaré and others, asymptotic solutions associated with Rayleigh, Wenzel, Kramers, Brillouin, and Keller and Kac’s famous isospectral problem. Then we discuss how the traveling functions have been utilized in the numerical simulation of musical instruments through work by Julius Smith, Karplus, Strong and other. This work has recently been extended to additional instruments types, in particular idiophones, which not only are more efficient than finite element based simulations but have the desirable property of stability and ease of interpretation under perturbations. We conclude with outlining possible research based on the advantages and drawbacks of the method. This principle provides a direct conceptual link between modal synthesis and waveguide-style physical models. In addition it suggests the importance of understanding the link between geometry and modes for efficient dynamical simulation. This paper explores these connections with regards to banded waveguide models.