The complex dispersion relation which describes transverse plasma waves propagating in a cold gyrotropic ambient plasma parallel to the background magnetic field as they interact with charged particle streams is derived by solving the linearized collisionless Boltzmann equation simultaneously with Maxwell's equations using the Fourier-Laplace transform method. The wave frequency is allowed to be complex with a positive imaginary part corresponding to a growing instability. The real and imaginary parts of the dispersion relation yield two separate equations. Under several assumptions, the equations can be simplified to yield an expression for the imaginary part of the frequency (the growth rate) and an equation relating the real wave frequency and the wave number. The theory is then applied to the magnetosphere by choosing a dipole model for the earth's magnetic field and a suitable distribution function for the particles. The specific case of waves of the ion-resonance mode interacting with mono-energetic, contra-streaming protons is considered in detail, and the results of this calculation are used in explaining hydro-magnetic (hm) emissions. In particular, it is suggested that the high frequency cutoff is a result of the pitch angle distribution of the particle stream. Computer calculations are done in order to display the general results of the theory. Specifically, when low energy protons (10 - 20 kev), trapped on a field line with an L value of 5.6 are considered, it is found that the region of instability occurs near the geomagnetic equator, and that the growth rate is a sharply peaked function of the frequency. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate