The dipolar (magnetostatic) interaction dominates the behavior of spin wavesin magnetic films in the long-wavelength regime. In an in-plane magnetizedfilm, volume modes exist with a negative group velocity (backward volumemagnetostatic spin waves), in addition to the forward surface-localized mode(Damon-Eshbach). Inside the film of finite thickness $L$, the volume modes havea nontrivial spatial dependence, and their two-dimensional dispersion relations$\omega(\mathbf{k})$ can be calculated only numerically. We present explicitperturbative expressions for the profiles and frequencies of the volume modes,taking into account an in-plane applied field and uniaxial anisotropy, for theregimes $\lVert \mathbf{k}L \rVert \gg 1$ and $\lVert \mathbf{k}L \rVert \ll1$, which together provide a good indication of the behavior of the modes forarbitrary wavevector $\mathbf{k}$. Moreover, we derive a very accuratesemianalytical expression for the dispersion relation $\omega(\mathbf{k})$ ofthe lowest-frequency mode that is straightforward to evaluate using standardnumerical routines. Our results are useful to quickly interpret and control theexcitation and propagation of spin waves in (opto-)magnetic experiments.
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