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For the B-type supergiant $\kappa$ Cassiopeiae (HD 2905) variabilities withperiods between several hours and a few days have been observed bothphotometrically and spectroscopically. A recent study of this star bySim\'on-D\'{\i}az et al. (2018) has revealed variability with a dominant periodof 2.7 days. To understand this variability, we present a linear non-adiabaticstability analysis with respect to radial perturbations for models of $\kappa$Cassiopeiae. Instabilities associated with the fundamental mode and the firstovertone are identified for models with masses between 27 M$_{\odot}$ and 44M$_{\odot}$. For selected models, the instabilities are followed into thenon-linear regime by numerical simulations. As a result, finite amplitudepulsations with periods between 3 and 1.8 days are found. The model with a massof 34.5 M$_{\odot}$ exhibits a pulsation period of 2.7 days consistent with theobservations. In the non-linear regime, the instabilities may cause asubstantial inflation of the envelope.

Instabilities and pulsations in models of the B-type supergiant $\kappa$ Cassiopeiae (HD 2905)