Numerical solutions to the guided toroidal Alfv¿n wave equation in a dipole field are presented with symmetric ionospheric Pedersen conductivities ΣPN,S, and using a range of field-aligned plasma density profiles $propto$ 1/rp, where p takes integer values from 0 to 6. We show that weakly damped fundamental guided toroidal Alfv¿n waves can occur when ΣPN,S are both less than or greater than a critical value. Our results show that for low p values the equatorial plasma density inferred from a measurement of a fundamental mode wave frequency using the high ΣPN,S (fixed end) solution can be up to a factor of 4 different from that determined from the low ΣPN,S (free end) solution. These results illustrate that in order to correctly invert wave frequency observations and to obtain accurate estimates of the equatorial plasma mass density in the nighttime sector where ΣPN,S may drop below a critical conductivity, the numerical solution of the guided toroidal Alfv¿n wave equation must be used. In addition, we present results which can be used to scale equatorial plasma densities derived from observed eigenperiods using the analytic WKBJ solutions, with p = 6, into the equatorial plasma densities derived from the solution of the toroidal wave equation for any integer value of p, from 0 to 6, in both the daytime (high ΣPN,S) sectors and nighttime (low ΣPN,S) sectors of the magnetosphere.