Leaky Rayleigh and Scholte waves at the fluid–solid interface subjected to transient point loading Jinying Zhu and John S. Popovicsa) Department of Civil and Environmental Engineering, University of Illinois at Urbana, Illinois 61801 Frank Schubert Lab EADQ, Fraunhofer Institute for Nondestructive Testing, D-01326 Dresden, Germany ͑Received 6 February 2004; revised 9 July 2004; accepted 18 July 2004͒ The analysis of acoustic waves generated by a transient normal point load applied on a fluid–solid interface is presented. The closed-form exact solution of the wave motion is obtained by using integral transform techniques. The obtained analytical solution provides necessary theoretical background for optimization of fluid-coupled ultrasonic and acoustic wave detection in experiments. Numerical simulation ͑elastodynamic finite integration technique͒ is performed to verify the obtained analytical solution. Detailed descriptions of leaky Rayleigh and Scholte wave solutions are presented. A simplified solution to isolate the contributions of leaky Rayleigh and Scholte waves generated by a transient point load is proposed, and closed-form formulations for displacement and stress components are then presented. The simplified solution is compared to the exact solution for two configurations: water/concrete and air/concrete. The excitation effectiveness of leaky Rayleigh waves for the air/concrete configuration is studied, which has practical significance to air-coupled sensing in civil engineering structures. © 2004 Acoustical Society of America. ͓DOI: 10.1121/1.1791718͔ PACS numbers: 43.35.Pt, 43.20.El, 43.35.Zc ͓YHB͔ Pages: 2101–2110 I. INTRODUCTION tally by Glorieux et al. ͑2001͒. The propagation of leaky Rayleigh waves under the influences of viscous damping and The propagating leaky Rayleigh wave that emanates heat conduction at the fluid–solid interface was studied by from a fluid–solid interface has been used as an effective Qi ͑1994͒, who concluded that the effect of viscosity can be means for surface and subsurface defect detection. With re- neglected for fluids with Reynolds number larger than 2500. cent improvements in instrumentation, air-coupled transduc- For common fluids such as water and air at normal driving ers have been used for detection of leaky surface ͑Zhu et al., frequencies ͑Ͻ10 MHz͒, the viscous effect of the fluid can 2001͒ and leaky guided waves ͑Castaings et al., 2001͒. With be neglected because the Reynolds numbers are far above the the advantage of noncontact sensing, air-coupled transducers critical value of 2500. provide an opportunity for quick scanning and imaging of The response of leaky waves owing to transient point large civil engineering structures by detecting the leaky Ray- loading is of great practical interest to nondestructive evalu- leigh wave. Therefore, detailed study of leaky surface waves ation ͑NDE͒ researchers, especially in civil engineering, for this case is needed. where an impulse hammer or a point impactor applied to the Extensive studies and applications of leaky surface surface of the solid is often used as a transient wave source. waves have been reported during the past 40 years. A com- Gusev et al. ͑1996͒ provided detailed theoretical analyses of prehensive study of Rayleigh waves and leaky Rayleigh laser-induced wave motions at the fluid–solid interface, waves has been given by Viktorov ͑1967͒, where leaky Ray- which include Scholte, leaky Rayleigh, and lateral waves. leigh waves at the interfaces of a solid half-space with both a General solutions for interface wave motion were given us- fluid layer and a fluid half-space were investigated in great ing a 2-D formulation. 2-D and 3-D analytical solutions for a detail. Leaky Rayleigh waves propagate with a velocity fluid–solid configuration subjected to implosive line and slightly higher than the ordinary Rayleigh wave, and attenu- point sources in the fluid have been given by de Hoop et al. ate more intensively with distance due to continuous energy ͑1983, 1984͒. However, for the case of the ‘‘Lamb’’ problem radiation into the fluid. It was initially believed that the leaky in fluid–solid configuration, where a normal transient point Rayleigh wave exists when the fluid wave velocity cF is load is applied at the fluid–solid interface, no closed-form smaller than the leaky Rayleigh wave velocity cLR . How- exact solution has been reported so far. ever, Mozhaev and Weihnacht ͑2002͒ showed the actual In this paper, Laplace and Hankel integral transforms are threshold phase velocity for leaky Rayleigh wave existence employed to derive the full analytical solution to the was 1.45% lower than the acoustic wave velocity of the ‘‘Lamb’’ problem in a fluid–solid half-space system, where a fluid. The character and existence conditions of leaky Ray- point load, varying with time as a step function, is applied at leigh and Scholte waves were also investigated experimen- the interface. The obtained step response solution is shown in integral format, which can be calculated numerically. Im- a͒Author to whom correspondence should be addressed. Electronic mail: pulse responses are then obtained by differentiating the step [email protected] responses. Therefore, for any transient impact loading that J. Acoust. Soc. Am. 116 (4), Pt. 1, October 2004 0001-4966/2004/116(4)/2101/10/$20.00 © 2004 Acoustical Society of America 2101 ␺͒ ץ ␸ץ ␺ץ ␸ץ 2 2 1 ͑r u ϭ Ϫ , w ϭ ϩ , ͑2͒ rץ z rץ z 2ץ rץ 2 where the subscripts 1 and 2 represent quantities related to the fluid and solid, respectively. B. Continuity and initial condition A point load is applied at the origin as a Dirac delta function in space and varies as a Heaviside step function in FIG. 1. A transient point load applied at the interface of a fluid–solid half- time, which can be expressed as QH(t)␦(r)/2␲r. Because space system. ideal fluid and shear-free interface conditions are assumed, only the normal stress and vertical component of the dis- has arbitrary temporal variation and spatial distribution, the placement are continuous at the interface. The continuity ϭ responses can be obtained by convolving the impulse re- conditions at z 0 are sponse in time and space domains. The obtained analytical ϭ w1 w2 , solutions are verified by EFIT ͑elastodynamic finite integra- tion technique͒ numerical simulation, which is a powerful ␦͑r͒ ␦͑r͒ ␶ ϭ␶ ϩ͑ϪQ͒H͑t͒ ϭϪPϪQH͑t͒ , ͑3͒ tool for elastodynamic wave field analysis. Then a simplified zz2 zz1 2␲r 2␲r analytical formulation for the pressure field in the fluid is ␶ ϭ␶ ϭ derived and illustrated. zr2 zr1 0, ␶ ␶ where zz and zr are the normal and shear components of stress, and P the pressure in the fluid. II. COMPLETE FORMULATION Assuming the system is at rest prior to tϭ0, we have Consider a fluid–solid half-space system as shown in ␸ ͑r,z,0͒ϭ␸¨ ͑r,z,0͒ Fig. 1. The solid half-space is given by zϾ0, and the fluid by 1 1 Ͻ ´ ϭ␸ ͒ϭ␸ ͒ z 0. The properties of the fluid are given by the Lame con- 2͑r,z,0 ¨ 2͑r,z,0 ␭ ␳ stant 1 and mass density 1 , and those of the solid by the ␭ ␮ ␳ ϭ␺ ͑r,z,0͒ϭ␺¨ ͑r,z,0͒ϭ0. ͑4͒ Lame´ constants 2 and and density 2 . The interface be- 2 2 tween the fluid and solid half-space is subjected to a normal point load of magnitude QH(t), where H(t) is the Heaviside C. Integral transform step function. Because the wave motion in the fluid and solid One-sided Laplace and Hankel transforms are used to generated by the point load is axially symmetric about the z obtain solutions to the equations. The Laplace and nth-order axis, cylindrical coordinates are employed, where the origin Hankel transforms are defined respectively as is located at the load point on the interface. ϱ ϱ Ϫ A. Fluid–solid half-space ¯͑ ͒ϭ ͵ ͑ ͒ pt Hn͑␰͒ϭ ͵ ͑ ͒ ͑␰ ͒ ͑ ͒ f p f t e dt, f f r Jn r rdr, 5 ␸ 0 0 Introducing displacement potential functions 1 in the ␸ ␺ fluid and 2 and in the solid, the governing equations for where p and ␰ are variables of the Laplace and Hankel trans- the fluid and the solid half-spaces are forms, respectively. Applying the Laplace transform to Eq. ͑1͒ with respect 2␸ץ ␸ץ 2␸ץ 1 1 1 1 1 ϩ ϩ ϭ ␸¨ , to time t and the zeroth- and first-order Hankel transform 1 2 2 ץ rץ r 2 ץ r z cF with respect to the radial distance r yields 2␸ 2␸H0ץ ␸ץ 2␸ץ 2 1 2 2 1 d ¯ ϩ ϩ ϭ ␸ ͑ ͒ 1 2 H0 ¨ 2 , 1 Ϫ␣ ¯␸ ϭ0, 1 1 2 2 2 ץ rץ r 2 ץ r z c P dz ͑6͒ 2␺ ␺ 2 Hץ ␺ץ 2␺ץ 1 1 d ¯␸ 0 2␺¯ H1 ϩ ϩ Ϫ ϭ ␺¨ 2 H d , Ϫ␣2¯␸ 0ϭ Ϫ␤2␺¯ H1ϭ ,0 ,0 2 2 2 2 2 ץ rץ r 2 ץ r z r cS dz2 dz2 2 ϭ␭ ␳ where cF 1 / 1 is the acoustic wave velocity in the fluid, where 2 ϭ ␭ ϩ ␮ ␳ 2ϭ␮ ␳ and c P ( 2 2 )/ 2 and cS / 2 are the P- and S-wave ␣2ϭ␰2ϩ 2 2 ␣2ϭ␰2ϩ 2 2 ␤2ϭ␰2ϩ 2 2 ͑ ͒ velocities in the solid. The double dots represent a double 1 sFp , 2 s Pp , sSp , 7 derivative with respect to time. The displacements are related to potential functions by and sF , s P , sS are P- and S-wave slowness in the fluid and the solid. Only choosing the terms which lead to finite re- ␸ץ ␸ץ 1 1 sponses for large values of ͉z͉, we obtain the solutions to Eq.