It is well known that the propagation of both bulk and surface acoustic waves (SAWs) is affected by the presence of static stresses, a phenomenon known as the acoustoelastic effect. Ultrasonic measurements of velocity therefore depend on the stresses within the material, as well the elastic constants and the density. Although the effect of stress on the velocity is small, many ultrasonic methods are sufficiently accurate to detect the changes involved. When inverting such measurements to obtain elastic constants, it is desirable to take the effect of stress into account. Similarly, when using ultrasonic methods to measure stress, it is necessary to have accurate values for the elastic constants. In practice, the material parameters of the ‘natural’, unstressed state are often either completely unknown or not known with sufficient accuracy (as is usually the case for residual stresses), or else cannot be assumed to be equal to bulk values (as in the case of layered materials). This is a major distinction between situations involving residual as opposed to applied stress, since a reference state of some description is always available in the latter case.