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We study the Penrose limit of type-IIB duals of softly broken N = 1 SU(N ) gauge theories in four dimensions, obtained as deformations of the Maldacena-Nùñez and Klebanov-Strassler backgrounds. We extract the string spectrum on the resulting pp-wave backgrounds and discuss some properties of the conjectured dual gauge theory hadrons, the so called "Annulons". The string zero-point energy on the light-cone is nontrivial, due to the loss of linearly realized worldsheet supersymmetry, and negative, even in the unbroken supersymmetric case. This causes the appearance of non-perturbative corrections to the hadronic mass spectrum. We briefly discuss the thermodynamic behavior of these string models, calculating the corresponding Hagedorn temperatures.