Surface wave dispersion beyond cutoff for a fast layer on a slow substrate
O. Lefeuvre, P. Zinin, G. A. D. Briggs, A. Every
1998
Applied Physics Letters
The velocity and attenuation of pseudosurface waves beyond cutoff have been calculated for several layers on a slow substrate and unexpectedly different kinds of behavior can be distinguished. When the elastic properties of the layer and the substrate are not very different, the velocity of the pseudosurface wave beyond cutoff increases up to the Rayleigh wave velocity of the layer. When the elastic properties of the two materials are quite dissimilar, the pseudosurface wave tends towards an
more »
... enuated interfacial mode. In addition, another mode appears evolving into the Rayleigh wave of the layer material. Acoustic microscopy and Brillouin spectroscopy are commonly used to determine the elastic constants of layered materials. 1 The wavelength, the frequency, and/or the thickness of the layer are varied to obtain dispersion curves for the surface acoustic waves ͑SAW͒, and the elastic properties are then inferred from these dispersion curves. The case of slow on fast has been widely studied, 2 but little attention has been given to the case of a fast layer on a slow substrate, 3-5 despite the fact that such hard layers are widely used as protection against erosion and abrasion. The purpose of this letter is to consider some general features of the low lying SAW on such systems. We have performed calculations with and without fluid loading because it is present in acoustic microscopy but not in surface Brillouin spectroscopy. The results for both cases differ slightly but the conclusions are similar. We present the fluid loaded results here. A layer is said to stiffen the substrate when its presence, at whatever thickness, increases the velocity of the surface wave above that of the Rayleigh velocity of the substrate. 2 Farnell and Adler 2 have shown that a sufficient but not necessary condition for this to be the case is when the ratio of the shear velocity of the layer (V t ) to the shear velocity of the substrate (V t ) is larger than &. An example of a stiffening layer for which the ratio is smaller than & has been investigated in Refs. 6 and 7. We consider here the dispersion curves of two different stiffening layers, chromium on steel (V t /V t Ͻ&) and titanium nitride on steel (V t /V t Ͼ&). These materials have been chosen to illustrate two kinds of surface wave propagation which are totally different, one when the properties of the layer and the substrate are close, one when they are dissimilar. The elastic constants used for the calculations are given in Table I . All materials are taken as isotropic and in our calculations we assume water loading so that all surface modes are leaky. Numerical simulations were made using a program written by Lowe. 8 The frequency 225 MHz has been used in the calculations, because it is a common frequency of excitation in quantitative acoustic microscopy. Figure 1 shows the dispersion curve of a chromium layer on steel. As the film thickness is increased, the velocity of the leaky Rayleigh wave increases from the substrate Rayleigh wave velocity until it reaches the substrate shear velocity of 3.26 km s Ϫ1 ͑we refer to this as cutoff͒. The behavior around the cutoff has been described previously. 6,7 Below cutoff, the SAW leaks energy into the liquid only and it would be a true surface wave without water loading. Beyond cutoff, the SAW radiates energy into the substrate as well, this also happens without water loading, and it becomes a pseudo SAW ͑PSAW͒. The velocity of this wave increases until it reaches the velocity of the Rayleigh wave of the layer material. The attenuation is maximum just after the cutoff and then tends to the attenuation of the Rayleigh SAW on the film. The dispersion curve for titanium nitride on steel beyond cutoff is quite different. With increasing layer thickness, the velocity of the PSAW reaches a value that is smaller than that of the Rayleigh wave in the layer ͑Fig. 2͒. The attenuation of this wave is also much higher than for the Rayleigh wave. Calculations show that the displacements of the mode TABLE I. Materials constants used for numerical simulations. (kg m Ϫ3 ) V l (km s Ϫ1 )
doi:10.1063/1.120916
fatcat:7ujesq4yqvhcxe7j6z4z3vxapu