The electro-mechanical analogies between inductances and capacitances in lumped circuits on one hand, and masses and springs on the other hand, are commonly known, and have been found extremely useful in many different ways. In the paper the analogies are considered for distributed elements, the detailed relations being shown between the electromagnetic fields in cavities, waveguides, etc. (with or without discontinuities) and the mechanical vibrations of elastic sheets having similar boundaries. The conclusions reached are quite general, and apply to plane sections of these electromagnetic fields, having any conducting boundaries, and with either TE or TM modes. The equivalent lumped circuits, whereby such fields may be represented to any degree of accuracy desired, are considered and illustrated by experimental examples. It is shown that these circuits result from applying the method of relaxation to the distributed-field calculation.It is shown that, although two different sets of electro-mechanical analogies are commonly applied to circuits of one dimension (such as transmission lines and filters), only one of these sets is at all applicable to distributed systems, namely that in which velocity corresponds to voltage (or ξ-vector), and force corresponds to current (or magneticH-vector). This latter analogy is strictly true for steady-state or transient conditions. Again, mass corresponds to capacitance (or to κ) and elastic constant to inverse inductance (or to 1/μ).The possible applications of such analogies, apart from their aiding in the understanding of electromagnetic field problems, fall into two classes. In the first, elastic membranes are stretched over frames of suitable shape, and a study of their vibration patterns may assist in the design of microwave components, such as valve anode blocks, resonators, guides with discontinuities such as diaphragms or slots, etc. In the second, the development of the lumped equivalent circuits, which are essentially two-dimensional circuits, is a contribution to the general study of periodic structures. The selectivity and impedance properties of these lumped circuits are examined in relation to the properties of conventional constant-kfilter structures.