Abstract: A gently sloping perturbation of the rigid wall of a planar acoustic waveguide can cause a Wood anomaly, reflected in an enormously rapid change in the diffraction pattern when the cutoff frequencies are approached from below inside a continuous spectrum. Asymptotic analysis has been used to establish the restrictions imposed on the wall perturbation profile that ensure the appearance, enhancement, or disappearance of the anomaly. The asymptotics of the eigenfrequencies is studied and the complex and threshold resonances are discussed.