The author has performed a review of the literature pertinent to wave scattering from rough surfaces. The early techniques developed for this problem are discussed, including the perturbation technique and the Kirchhoff approximation. The limitations of such models are highlighted. The developments in the theory, since the introduction of these models, are also given. These include: the effects of shadowing; multiple scattering effects; integral equation techniques and variational methods. Results from experimental investigations of rough surface scattering are also reviewed. These results are compared with the theoretical predictions. A brief review is also given of the methods by which the profiles of rough surfaces are measured and the problems inherent in making these measurements. A comparison is made between the assumed profiles, used in the theory, and the measured profiles.

https://iopscience.iop.org/article/10.1088/0034-4885/50/12/001/pdf

Wave scattering from rough surfaces

Abstracf-Energy storage at a groove edge, or similar discontinuity in the substrate surface, produces a phase shift in a surface acoustic wave incident on the discontinuitg. Such phase shifts are a major source of phase errors in surface-wave devices employing etched grooves or deposited stripes on the substrate surface. An improved equivalent network, in which this stored energy is represented as a simple shunt susceptence, is presented for a Rayleigh wave normally incident on a step discontinuity on the substrate surface. With this network second-order effects observed at the fundamental frequency of a uniform array of shallow grooves can now be predicted. The presence of stored energy also exerts a profound influence on the wideband harmonic response of the array. In particular, a strong response is predicted at even harmonics of a grating the groove width of which is half the grating period, in contradiction with simpler theories. The validitg of this prediction is verified by measurements at the second harmonic, with excellent agreement between theory and experiment.

The Influence of Stored Energy at Step Discontinuities on the Behavior of Surface-Wave Gratings

In 1887 Lord Rayleigh(1) showed that a semi-infinite, isotropic, elastic medium, bounded by a single, stress-free, planar surface, can support surface vibration modes that are wavelike in directions parallel to the surface of the solid, but whose amplitudes decay exponentially with increasing distance into the solid from the surface, with a decay length that is of the order of the wavelength of the wave along the surface. The displacement vector of these waves lies in the sagittal plane, i.e. in the plane defined by the direction of propagation of the wave and the normal to the surface. These waves are acoustic waves in that their frequencies are linear in the magnitude of the two-dimensional wave vector characterizing their propagation along the surface. They are consequently non dispersive, i.e. their speed of propagation is independent of their wavelength parallel to the surface, which is due to the absence of a characteristic length in the system under consideration. Their frequencies also lie below the continuum of frequencies allowed the normal vibration modes of an infinite elastic medium for the same value of the two-dimensional wave vector. Such surface acoustic waves are now known as Rayleigh Waves.

Surface Acoustic Waves

The use of reflective arrays in surface-wave devices has yielded new devices such as resonators in addition to an enlarged parameter range for devices such as radar pulse expanders and compressors. A number of different grating geometries, substrates, and types of reflectors have been employed in resonators, band-pass filters, filter banks, oscillators, and dispersive delay lines. Simple, yet accurate, models have been developed for design and analysis of reflection gratings in these applications. The major application of reflective arrays has been in radar pulse compressors wherein low spurious levels and precise phase and amplitude response have been achieved in devices with time-bandwidth products as large as 104. Other signal-processing applications such as chirp-z transforms can exploit the large time-bandwidth products available in reflective-array devices.

Properties and applications of reflective-array devices

Finite element analysis of Rayleigh wave interaction with finite-size, surface-breaking cracks.

In an important new class of signal‐processing devices using elastic surface waves, the reflection of a Rayleigh wave from grooves is a key feature. In the present paper, a boundary‐perturbation technique is employed to determine the reflection of a Rayleigh wave from either a shallow groove or a low step in the surface of an isotropic elastic half‐space. Comparison with available experimental data for the case of the rectangular groove and step yields excellent agreement.

Robert C. M. Li

Papers

Rayleigh‐wave reflection from groove and step discontinuities