Trace formulas are derived for the coefficients of a matrix differential equation representing wave propagation in a medium which supports N types of waves. In general these waves are coupled to each other but are assumed to become uncoupled as lzl -> ?o. The two coefficients of the ordinary differential equation, N x N matrices, are assumed known for z 0. A point source is located in the region z 0. The unique features of this work are a derivation of the trace formula in which the Jost functions are not needed, an alternate measurement method based on an impedance concept and a trace equation based on the impedance concept. Introduction. Motivated by the work of Trubowitz (1), trace formula methods were introduced by Deift and Trubowitz (2) to study one-dimensional inverse quantum mechanical scattering problems on the line. The trace formula expresses the potential as a functional of the reflection coefficient. Greene (3) used the same techniques to study the one-dimensional acoustical wave equation with the angular frequency playing the role of the quantum mechanical energy. Stickler and Deift (4) examined the acoustic problem of a point source in an inhomogeneous half space bounded by a pressure release surface. The sound speed was recovered from a measurement of the normal component of the acoustic velocity at the pressure release surface. The role of the quantum mechanical energy was played by a vertical wave number, which does not correspond to either a spatial or temporal derivative. Using again the analogy between vertical wave number (the wave number in the direction of the coefficients' variation) and the quantum mechanical energy, Stickler (5) showed how the trace formula methods could be used to recover the sound speed when it approaches different values at plus and minus infinity. In a quantum mechanical context this means that the potential approaches zero, say at minus infinity, but some positive constant, not necessarily known, at plus infinity. (In the Deift-Trubowitz analysis the potential approached zero in both limits.) Stickler (6) also discussed this same problem in an electromagnetic context. Tomei (7) has used the trace methods to recover the coefficients of a linear, third order equation associated with the Boussinesq equation. In the present paper the trace method is extended to a one-dimensional vector Helmholtz equation which supports N types of waves and for which the waves are coupled to each other except at infinity. The trace method yields a functional relation- ship between the coefficients of the Helmholtz equation, a reflection coefficient described below and assumed known and a specific solution of the vector Helmholtz equation with known and prescribed initial conditions. These relationships are called the trace formulas and are derived here. The coefficients of the vector Helmholtz differential equation can be replaced by these trace formulas and this results in a system of coupled, nonlinear ordinary differential equations with known initial condi- tions. Once this solution is determined, the coefficients can be recovered. The existence of a solution of this coupled system is not given.