Continuous models for 1D discrete media valid for higher-frequency domain

Abstract

We deal with a 1D differential-difference equation governing the behavior of a n-mass oscillator. It is known that a string-type approximation is justified for low part of frequency spectra of a continuous model, but for free and forced oscillations a solution of a discrete model and of a wave equation can be quite different. The difference operator makes analysis difficult due to its non-local form. Approximate equations can be gained by replacing the difference operators via a local derivative operator. Although the application of a model with derivative of more than second order improves the continuous model, a higher order of approximated differential equation seriously complicates a solution to the stated problem. It is known that accuracy of the approximation can dramatically increase using one-point Padé approximation. In this report, we show that better results may be obtained when a two-point Padé approximation is applied.