The paper describes experiments to search for a variable magnetic field close to a rechargeable conductive flat plate and a ball in the air, as well as an experiment looking for a variable electric field near a rotating permanent magnet. It has been found that variable electric and magnetic fields do not induce each other within the measurement error. It means that rotary Maxwell's equations are not applicable in the near-field zone and the classical concept of displacement current in vacuum (air) has no physical meaning. A conclusion is made on the existence of transverse magnetodynamic waves. Statics and dynamics of the magnetic field near the permanent magnet rod are investigated experimentally. The methods to compute magnetodynamic waves from any source are presented. Four types of polarization of these waves are identified: linear, circular, toroidal and mixed. Concentration and deflection of magnetodynamic waves are observed on introducing inhomogeneity in the form of a ferrite rod into their propagation way, which is similar to diffraction in optics. Secondary magnetodynamic waves from the induced magnetic moments in atoms of ferrite are registered near its surface, which is like reflection in optics. Some ideas for observation of effects similar to dispersion and interference are presented for magnetodynamic waves. The structure and properties of electrodynamic, magnetodynamic and electromagnetic waves are discussed. The ideas of experiments to search for their unknown properties are described. In conclusion, technical applications of magnetodynamic waves such as magnetography, magnetic tomography and other are considered.