This Research Note presents a method for the numerical estimation of eigenvalues which is easy to understand, practical to implement and effective in its results. It is developed with complete details of how to calculate eigenvalues associated with vibrating membranes and plates, waveguides, sloshing fluids and Stekloff problems; however, this flexible technique can be applied to a variety of eigenvalue problems. The text is illustrated by a number of simple examples, many of which are worked out in full. By discussing both theoretical and computational aspects, this book is of use to electrical and mechanical engineers as well as applied mathematicians.