In recent years many significant applications have been found for mechanical vibrations in analysis and testing, the study of molecular structure, and the processing of various types of materials. These applications are not limited by the frequency range of hearing of the human ear but extend from infrasonic through ultrasonic frequencies. The concepts and techniques associated with these applications are generally referred to by the term sonies and are the subject of the new book of this same title by Hueter and Bolt.

The symmetry properties of the third-rank tensor β which comprises the set of coefficients of quadratic terms in the expansion of the induced-dipole moment in the electric field are investigated, and the various linear combinations belonging to the irreducible representations are tabulated for the important molecular symmetry groups. Depolarization ratios are calculated for a sample composed of randomly oriented molecules (liquid phase). Examples of the selection rules in both liquid and crystalline phases are discussed, and these selection rules are contrasted with the ones appropriate for infrared absorption and the ordinary Raman effect. Molecular modes allowed in the infrared are always allowed in the hyper-Raman effect and have a depolarization ratio (for linearly polarized incident radiation) ≤⅔, while allowed hyper-Raman transitions forbidden in the infrared have a depolarization ratio of ⅔.

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It is shown that treatment of the molecular vibration problem in the form | F—1G—1—λ—1E | =0 has certain advantages when the data from vibrational spectroscopy are to be combined with information from mean amplitudes of vibration or centrifugal distortion toward the end of determining a complete quadratic potential function. The matrix F—1=C is a compliance matrix; its elements presumably have as much physical significance as those of F. Some new isotopic sum rules are formulated, appropriate to the compliance scheme, but they are not nearly so general as those involving the λ's. Linear molecules are discussed, and some numerical examples are given for nonlinear XY2 and XY4.

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