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We discuss three long‐wave approximations (the Born, quasistatic, and extended quasistatic) to the scattering of elastic waves from a flaw embedded in an isotropic medium. First, we derive the Born and quasistatic approximations by the technique of infinite‐order perturbation theory. For these approximations this derivation clearly reveals the precise nature of the approximations and the generality of the quasistatic approximation in particular. Next, we give the complete details on how to calculate within the three approximations the long‐wave scattering from ellipsoidal voids and inclusions. Then, we calibrate the approximations by comparison with exact results for the scattering from a sphere and also present computations based on the extended quasistatic approximation for the scattering from various ellipsoidal voids.