A system for monitoring containers or conduits in which at least two phases or at least two media of a phase in the form of a liquid, a gas and/or a solid are jointly stored or transported, wherein the system is formed with a plurality of test heads, which are arranged along an axis in a series arrangement or in a plane around a container or a conduit at known distances from one another, which test heads are fastenable to the respective container or the respective conduit, and which test heads are connected to an electronic evaluation and control unit via a data and power supply line for the transmission of measurement and control signals and for power supply, and at the same time the test heads are each formed with at least one elastic wave emitting element and at least one elastic wave detecting element or at least one elastic wave emitting and detecting element, wherein the elastic wave emitting elements are arranged and configured to couple guided waves into a wall of the respective container or the respective conduit, and the electronic evaluation and control unit is configured to use measurement signals from the wave-detecting elements to perform damage and crack detection, corrosion and wall thickness determination, fill level measurement and/or a functional check of the test heads and to actuate wave emitting elements, and the test heads are fastenable by means of holding elements and a clamping device to the respective container or the respective conduit on an outer wall, characterized in that the electronic evaluation and control unit is configured to carry out an electromechanical impedance spectroscopy and, in order to determine an impedance, to carry out a sequential transmission and evaluation of the electrical voltage and the electrical current determined with the elastic wave-detecting elements. the system is formed with a plurality of test heads, which are arranged along an axis in a series arrangement or in a plane around a container or a conduit at known distances from one another, which test heads are fastenable to the respective container or the respective conduit, and which test heads are connected to an electronic evaluation and control unit via a data and power supply line for the transmission of measurement and control signals and for power supply, and at the same time the test heads are each formed with at least one elastic wave emitting element and at least one elastic wave detecting element or at least one elastic wave emitting and detecting element, wherein the elastic wave emitting elements are arranged and configured to couple guided waves into a wall of the respective container or the respective conduit, and the electronic evaluation and control unit is configured to use measurement signals from the wave-detecting elements to perform damage and crack detection, corrosion and wall thickness determination, fill level measurement and/or a functional check of the test heads and to actuate wave emitting elements, and the test heads are fastenable by means of holding elements and a clamping device to the respective container or the respective conduit on an outer wall, characterized in that the electronic evaluation and control unit is configured to carry out an electromechanical impedance spectroscopy and, in order to determine an impedance, to carry out a sequential transmission and evaluation of the electrical voltage and the electrical current determined with the elastic wave-detecting elements.