Radiation damping in the context of one-dimensional wave propagation: A teaching perspective

This paper has two objectives: to connect directly radiation damping and 1D elastic wave propagation, and to create a simple teaching tool to introduce the subject to students and engineers trained in Structural Dynamics. The first objective is achieved by obtaining the equivalent radiation modal damping using time domain solutions for the fundamental mode in shear of an elastic layer (soil) on flexible rock, for the case in which the rock–soil Impedance Ratio in shear, I=(ρrVr)/(ρsVs)>1, where ρ=mass density and V=shear wave velocity. These time domain solutions are developed for the case of steady-state input sinusoidal shear waves propagating vertically in the rock as well as for horizontal free vibration of the layer. Both derivations result in the same approximate expression for the modal radiation damping in the first mode, ζr1≈2/(πI), which is in turn identical to the approximate equation obtained by Roesset and Whitman [11] using a frequency domain approach. This expression for ζr1 is linked to the fact that, during free vibration, the ratio between two successive positive displacement peaks uso and us1 at the ground surface is us0/us1=[(1+I)/(1−I)]2, associated with two wave reflections at the soil–rock interface. From this ratio, and after applying the standard expression to obtain modal damping from damped free vibration, the same expression for ζr1 is reached again, ζr1≈[1/(2π)] ln(us0/us1)≈[1/(2π)] ln[(1+I)/(1−I)]2≈2/(πI). This finding allows development of the simple teaching tool proposed at the end of the paper. While only a crude approximation lacking in rigor, this teaching tool is physically intuitive, links directly wave propagation and modal damping in a simple way and gives the correct result.

► Soil radiation damping important in many soil and soil-structure systems. ► Radiation damping is connected to 1D elastic wave propagation and Impedance Ratio. ► Fundamental mode radiation damping of layer on flexible rock derived from wave reflections. ► Main purpose is to teach radiation damping concept to students and engineers. ► Crude but simple and intuitive teaching tool is proposed that gives the right answer.