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

The uniqueness in the inverse problem for transmission eigenvalues for the spherically-symmetric variable-speed wave equation