Reflection and transmission of waves incident on time-space modulated media

This paper studies the reflection and transmission properties of longitudinal waves incident on one-dimensional, finite-length time-space modulated beam. The Young's modulus of the so-called time-space modulated beam is modulated in both time and space in the form of a traveling wave. The objectives of this paper are (I) to provide more insights into the nonreciprocal transmission through modulated media, (II) to study the influence of the modulation wave on the nonreciprocal transmission, and (III) to quantify the one-way energy isolation effect. Our results show that the strong nonreciprocal transmission observed within the stop bands of the two fundamental Bloch modes of the modulated beam is caused by the fact that, within these frequency bands, incident waves from opposite directions will excite different Bloch modes in the modulated beam. At frequencies where strong reflection is observed, the incident waves excite not only the zero-order evanescent mode but also some higher-order propagative modes in the modulated beam, leading to unavoidable wave transmission when isolation is what we desired. This undesired transmission becomes more significant as the velocity of modulation wave increases when the waves are incident on the modulated beam in the same direction as the modulation wave. The nonreciprocal property of modulated media may be explored to realize one-way energy isolation. Energy balance resulting from the reflection and transmission is studied to quantify this application.