We model and measure the dissipation of water hammer induced by well casing and water elasticity with rapid valve opening at the start of a pneumatic slug test. The higher-frequency water hammer can obscure slower, aquifer-controlled, underdamped oscillations of the rigid water column, so a quantitative description of the elastic motion improves the ability of a slug test to calibrate the aquifer permeability k. Internal friction attenuates the water hammer, subject to a known headspace pressure at the air/water interface and equilibrium pressure at the top of the well screen. An analytical elastic solution is presented and matched to an existing rigid motion analysis, with matching predicated on k exceeding 7 W 10-14 m2 and appreciable water hammer dissipation during the first cycle of the slug test. The model is accurately calibrated with data from underdamped slug tests in a PVC monitoring well in the Plymouth-Carver Aquifer. The calibrated casing elasticity value suggests that effective lateral soil stress appreciably stiffened the casing.