This paper is concerned with the semilinear time fractional wave equations on the whole Euclidean space, also known as the super-diffusive equations. Considering the initial data in the fractional Sobolev spaces, we prove the local/global well-posedness results of $L^2$-solutions for linear and semilinear problems. The methods of this paper rely upon the relevant wave operators estimates, Sobolev embedding and fixed point arguments.