Measurement of Nonlinear Guided Waves

  • Cliff J. LissendenEmail author
  • Mostafa Hasanian
Part of the Springer Series in Measurement Science and Technology book series (SSMST)


Characteristics of nonlinear guided waves provide information about the current state of the material that comprises the waveguide. Notably, the information is related to the material's microstructure, which in turn influences the strength properties. However, the dispersive nature of guided waves makes their nonlinear characteristics more complicated than those of bulk waves. This chapter strives to describe nonlinear features of guided wave propagation including higher harmonic generation and wave mixing. It provides a methodology for selecting wave modes and frequencies that provide the best opportunities for measurements as well as measurement techniques and a glimpse of some recent results.



second rank identity tensor


trace of the tensor


real part of the complex argument


displacement vector


displacement profile through thickness of plate (i.e., wavestructure)


displacement gradient tensor


Lagrange strain tensor


Cauchy stress tensor


First Piola-Kirchhoff stress tensor


Second Piola-Kirchhoff stress tensor


Outward normal unit vector


mass density

λL, μ

Lame’s constants

A, B, C

Landau-Lifshitz 3rd order elastic constants

\(\beta^{\prime} = A_{2} /A_{1}^{2}\)

relative nonlinearity parameter, where A1 and A2 are the amplitudes of the primary and secondary wave fields respectively


strain energy function


half-thickness of plate


plate thickness








position vector


mixing power associated with the interaction of waves a and b, which generates the internally resonant mode r


wave interaction angle between the primary waves


direction of secondary wave vector with respect to wave a


circular frequency


frequency in Hz

cL, cT

longitudinal wave speed, transverse wave speed

cp, cg

phase velocity (speed of an individual wave), group velocity (speed of a group of waves having similar frequencies, i.e., rate at which the envelope travels)


peak-to-peak voltage



The first author acknowledges that our research in nonlinear guided waves was partially supported by projects funded by the Nuclear Energy Universities Program in the U.S. Department of Energy (awards 102,946 and 120,237) and the U.S. National Science Foundation (awards 1,300,562 and 1,727,292). In addition, he wants to acknowledge graduate and postdoc students Yang Liu, Vamshi Chillara, Gloria Choi, Hwanjeong Cho, Baiyang Ren, and Chung Seok Kim for their weighty contributions, as well as encouragement from his colleague Joseph Rose.


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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of Engineering Science and MechanicsThe Pennsylvania State UniversityUniversity ParkUSA

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