Reservoir computing leverages the dynamic properties of a fixed, randomly connected neural network to facilitate simplified training and enhanced computational efficiency. Many forms of physical reservoir computers have been proposed. In this paper, we use a three-dimensional (3D)-printed reservoir computer as the design environment, develop analytic models to describe its performance, and validate the models through simulations. This approach offers practical insights for designing physical reservoirs with targeted computational capabilities and enables the assessment of the influence of reservoir parameters such as scale or material choice, on performance metrics, including speed and power consumption. Additionally, the proposed approach may be employed to optimally design physical reservoir computers to solve specific problems. This work contributes to the understanding of physical RC systems by providing a detailed analysis of the physical basis that connects computational performance with multidomain physical interactions at the device level. The methods and results from this work not only propel the development of future 3D-printed physical RC systems but also serves as a framework for evaluating and designing diverse physical RC models based on other approaches. Published by the American Physical Society 2024