  The recovery of a spherically-symmetric wave speed $v$ is considered in a
bounded spherical region of radius $b$ from the set of the corresponding
transmission eigenvalues for which the corresponding eigenfunctions are also
spherically symmetric. If the integral of $1/v$ on the interval $[0,b]$ is less
than $b,$ assuming that there exists at least one $v$ corresponding to the
data, it is shown that $v$ is uniquely determined by the data consisting of
such transmission eigenvalues and their "multiplicities," where the
"multiplicity" is defined as the multiplicity of the transmission eigenvalue as
a zero of a key quantity. When that integral is equal to $b,$ the unique
recovery is obtained when the data contains one additional piece of
information. Some similar results are presented for the unique determination of
the potential from the transmission eigenvalues with "multiplicities" for a
related Schr\"odinger equation.
