Resonant oscillations set up internal fluid waves within a piezo-drop-on-demand (DoD) print-head channel as a result of actuation drive pulses. Such waves will persist for some time after droplet ejection from the nozzle, and the residual wave amplitude can interfere (constructively or destructively) with all succeeding actuation drive pulses, potentially altering the speed and volume of successive droplets. As uncontrolled interference would worsen printing quality, residual waves are usually reduced by a combination of print-head design and waveform optimization for better performance at continuous (steady state) printing frequencies. However, the residual waves following any changes of printing frequency can influence “first” drops and short bursts of drops. Exact analytic expressions are provided here for the N-pulse burst DoD print-head response function with fixed printing frequency. This article explains the purpose and application of the model predictions to published piezo-driven DoD data. An examination of the effect of fluid properties, the identification of unexpected jetting behavior and some issues with manufacturing prototype quality, tests of assumptions made in the simple model and extensions to the prediction of print-head performance using realistic complex waveforms are also discussed. An earlier shorter article, mainly introducing the multi pulse train modeling approach and some applications within Xaar, was first presented at NIP31/DF2015 [S. D. Hoath, A simple model for DoD inkjet frequency response, Proc. IS&Ts 31st Int'l. Conf. on Digital Printing Technologies (IS&T, Springfield, VA 2015), 8–12]. © 2016 Society for Imaging Science and Technology.