6 Waves in Plates
7.4 Subsurface Longitudinal Waves
kL = ′ + ″k L ik L,kT = ′ + ″kT ik T, and kR = ′ + ″k R ik R.
It can be shown that the damping of Rayleigh waves is similar to that of cylindrical waves, as follows: AR 1/ k rR is compared to bulk waves, where AL T. 1/k rL T. (L for longitudinal and T for transverse). Rayleigh waves are less attenuated than bulk waves and are thus, for example, responsible for greater damage from earthquakes.
Propagation of Rayleigh waves can be studied on curved surfaces. As an example, consider the wave equation in a system using cylindrical coordinates.
A noteworthy analysis of surface waves on anisotropic media may be found in Rose, Pilarski, and Huang (1990). These conditions give rise to variations in surface wave velocity with angle and subsequent differences in phase and in group or energy velocity, as well as to the presence of a skew angle. An inverse problem to evaluate certain composite material properties as a function of skew angle could be carried out.
7.4 Subsurface Longitudinal Waves
Subsurface elastic waves, mostly geoacoustical, are reported under many names: head waves, lateral waves, creeping longitudinal waves, or fast surface waves. Here, the term subsurface waves or subsurface longitudinal (SSL) waves is used to describe the field of longitudinal waves excited in a solid half-space by an angle beam transducer with an angle of incidence close to the first critical angle.
Subsurface longitudinal waves have been extensively investigated, not only theoretically but also experimentally to detect defects in subsurface layers of an isotropic material.
It has been established that – at or near the first critical angle, for longitudinal waves incident onto an interface (liquid–solid or solid–solid) from the medium with a smaller velocity of longitudinal waves – there coexist two waves: SSL and head waves. This is shown in part (a) of Figure 7.10. The two wave types cooperatively fulfill the boundary conditions on the free surface of the solid, where all stresses are supposed to equal zero. Any disturbance on the free surface moves with a velocity equal to the velocity of longitudinal waves in the solid. The amplitude of this displacement decreases as distance increases, according to the 1/rn law, where n ranges from 1.5 to 2.0. This means that the SSL waves close to the free surface are strongly attenuated compared with the bulk waves, since the former are proportional
Normal beam longitudinal wave transducer
Mediator Wedge
Test specimen
Figure 7.9. Mediator technique of generating surface waves in a test specimen.
to 1/rn with n ranging from 0.5 to 1.0. The SSL waves can be detected at some other spot on the same surface, but the receiving transducer must be inclined at an angle equal to the first critical angle.
One characteristic of subsurface waves is the distribution of the amplitude of acoustic pressure in the plane of incidence; this is shown in part (b) of Figure 7.10.
The shape of the pressure field distribution reveals that the maximum sensitivity of the ray occurs at an angle of 10 to 20 degrees from the free surface. Hence, the name chosen for these waves is appropriate, since they can be utilized for the detection of subsurface defects – and especially since the SSL waves show a relatively small sensitivity to surface roughness.
The definition given here of SSL waves is related to the first critical angle, which (for isotropic media) is given by Snell’s law as αcr = sin−1(c1/cL), where c1 and cL are longitudinal wave velocities for the upper (shoe of angle beam probe) and lower media, respectively. For an anisotropic medium the critical angle must be redefined, since the phase and group velocity vectors are generally of different orientations.
7.5 Exercises
1. Show that the equation r2 − 4sq = 0 becomes
η6 − 8η4 + 8(3 − 2ζ2)η2 − 16(1 − ζ2) = 0:
let η = c/cT and ζ = cT/cL.
2. Solve the Rayleigh surface wave velocity equation as a function of Poisson’s ratio ν. Plot graphs. Compare with Viktorov’s approximate solution, η = (0.87 + 1.12ν)/(1 + ν).
3. Where does the particle velocity projection onto an ellipse reverse direction?
4. Study the damping of Rayleigh waves, comparing such media as steel, aluminum, and epoxy. Use Viktorov (1967) or other references.
Longitudinal wave (a)
CSSL = CL Head wave
α1 = αCrl α2 = α1
(b) 30 dB
10° to 20°
Figure 7.10. Subsurface longitudinal waves at first critical angle: (a) coexistence of head and longitudinal waves; (b) pressure field pattern.
119 References
5. For a wave incident on a Plexiglas specimen, assume that the longitudinal velocity is 2.71 mm/μs and the shear wave velocity is 1.38 mm/μs. Calculate the theoretical Rayleigh surface wave velocity and compare with the measured values.
6. Discuss or explain why the Rayleigh equation is a function of only Poisson’s ratio for isotropic material.
7. For t = 0, plot the particle displacement on a 12 × 12 grid as a function of x and z in order to demonstrate concepts of particle motion.
8. Design a comb-type transducer to generate a surface wave in a structure.
9. For a Poisson’s ratio of ν = 0.25, find the surface wave velocity. Which root is correct, and why?
10. Show the development of equation (7.4) from (7.1).
11. How can one be assured of displacement being zero at a depth z of infinity?
12. An SSL wave can create secondary waves, head waves, that can coexist with SSLs. Make a sketch showing head wave formation and the proper angle of propagation into a test material.
13. How do you select the correct root from the cubic equation in a Rayleigh surface wave problem for isotropic material?
14. What is a typical particle motion profile for a surface wave as we consider particles further away from the surface in question?
15. What spacing is required in a comb transducer to produce a surface wave of specified frequency?
16. Show that attenuation for surface waves is less than that of bulk waves.
17. Compare the attenuation coefficients of bulk longitudinal waves with subsurface longitudinal waves.
18. Using an angle beam transducer to generate a surface wave in a material, how would you achieve the best result with maximum energy into the surface wave?
What conditions would be necessary to generate the surface wave with the angle beam transducer?
19. Show on a sketch the SSL wave, longitudinal head wave, SST wave, transverse head wave, and surface wave due to a point source loading on a half-space.
20. Make a plot of surface wave particle displacement for a given time t.
21. How would you proceed to calculate surface wave velocities for excitation on an anisotropic material?
22. Using surface waves, discuss a procedure to identify certain anisotropic material constants as a function of measured skew angles.
REfEREncES
1. Lord Rayleigh Biography. School of Mathematics and Statistics, Univ. of St. Andrews, Scotland, 2003. Web. 6 Dec. 2009. http://www-history.mcs.standrews.ac.uk/Biographies/
Rayleigh.html
2. Lewis, Meirion F. “Rayleigh Waves – a progress report.” European Journal of Physics 16 (1995): 1–7.
3. Viktorov, I.A. Rayleigh and Lamb Waves: physical theory and applications. New York, NY: Plenum Press (1967).
120