General hydrodynamics ofHe3−Ain finite magnetic fields

We have determined the hydrodynamic equations, including nonlinear terms, for superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}-A$ in finite magnetic fields. The propagating and diffusive normal modes for a uniform texture have been studied, both for bulk and superleak geometries. In bulk, the longitudinal magnetization and the temperature produce two coupled modes; in a superleak, the longitudinal magnetization and the density produce two coupled modes. For both geometries, the coupled modes show crossing effects. (An appendix is devoted to considerations of an experiment designed to observe the coupled longitudinal modes, at "mode crossing.") The transverse spin waves have also been studied, but they do not show mode-crossing effects. For the bulk $A_1$ phase, the relationship of the bulk diffusive mode to the "magnetothermal effect," is discussed. For the $A_1$ phase in a superleak the relationship of the two diffusive longitudinal modes to the "magnetothermomechanical effects," and of the propagating longitudinal mode to the "magnetic fountain effect," are discussed. It is suggested that the longitudinal spin wave in the $A$ phase, at any field, can be studied through its characteristic decay length, when the generator frequency lies below the gap frequency.