The dynamic compressive behavior of foams with a non-linear variation in the cross-sectional area profiles is studied. A theoretical model has been developed based on the one-dimensional shock theory and the rigid, perfectly-plastic, and locking (RPPL) material model, to establish the governing equations of motion of an initially stationary foam struck by a rigid projectile. Both decreasing and increasing cross-sectional area profiles with various power-law exponents are considered. The numerical simulations suggest that the decreasing area profiles, referred to as the convergent foams, exhibit a double-shock mode. In contrast, the foams with an increasing area-profile, referred to as the divergent foams, exhibit a single-shock mode. The theoretical predictions of both the double-shock and the single-shock cases are validated against the finite element simulations. Compared to the foam with a uniform cross-sectional area, the convergent foams transmit lower mean forces at the distal end, and the divergent foams transmit higher forces at higher impact velocities. An interesting phenomenon observed is that the mean force at the distal end decreases with an increase in the impact velocity beyond the densification velocity, for the convergent foams. An expression for the plastic energy dissipated by convergent foams has been derived, and we observe that the convergent foams dissipate less energy than the divergent foams. The performance of the foams has been compared through the dissipation performance parameter, defined as the ratio of non-dimensional plastic energy dissipated at the end of the crushing process to the mean non-dimensional force at the distal end. Convergent foams with low gradients and high power-law exponents have the highest dissipation performance. A special case of a decreasing density-gradient combined with an increasing area-gradient is shown (through finite element simulations) to be beneficial for structural protection. © 2021 Elsevier Ltd