Measurement of Nonlinear Ultrasonic Parameters from Higher Harmonics

  • Kyung-Young JhangEmail author
  • Sungho Choi
  • Jongbeom Kim
Part of the Springer Series in Measurement Science and Technology book series (SSMST)


Higher harmonic generation is a classic phenomenon of nonlinear ultrasonic interaction, which can occur in both material elastic and contact acoustic nonlinearities. However, this chapter covers only material elastic nonlinearity. The nonlinear ultrasonic parameters quantitatively represent the nonlinear elastic properties of a material and can be measured from the higher harmonics generated during ultrasonic wave propagation in the material. In this chapter, we describe the measurement process of nonlinear ultrasonic parameters. Furthermore, we introduce the configuration of the experimental apparatus and the signal processing techniques in detail. In particular, we focused on the error-inducing factors. Methods that use longitudinal waves are mainly considered; however, a method that uses surface waves is also introduced in brief.



\(A\left(\upomega \right)\)

displacement amplitude


displacement amplitude of the fundamental frequency component


detected signal amplitude of the fundamental frequency component


detected signal amplitude of the fundamental frequency component for the reference material


displacement amplitude of the second-order harmonic frequency component

\(A_{2}^{\prime }\)

detected signal amplitude of the second-order frequency component

\(A_{2,0}^{\prime }\)

detected signal amplitude of the second-order frequency component for the reference material


amplitude of a detected voltage signal


response function of couplant layer function at position i


second-order elastic constant


third-order elastic constant


amplitude composition of an electrical signal sent to a transmitting transducer


response function of an amplifier

\(H\left(\upomega \right)\)

calibration function

\(I^{\prime}_{in} \left( \omega \right)\)

input current in the calibration

\(I_{out} \left( \omega \right)\)

current measured at the receiving part in the nonlinear measurement

\(I_{out}^{\prime } \left( \omega \right)\)

output current in the calibration

\(K\left( \omega \right)\)

transfer function


second-order Huang coefficient


third-order Huang coefficient


window length


number density of precipitates

\(P_{A,in} \left( \omega \right)\)

monochromatic acoustic power at the transmitting transducer

\(P_{A,out} \left( \omega \right)\)

acoustic power at the receiving transducer

\(P_{{{\text{E}},{\text{cal}} - {\text{in}}}} \left(\upomega \right)\)

generated signal in the calibration

\(P_{E,cal - out} \left(\upomega \right)\)

received signal in the calibration

\(P_{E,in} \left( \omega \right)\)

monochromatic electrical power at the transmitting transducer

\(P_{E,out} \left( \omega \right)\)

electrical power at the receiving transducer


response function of a receiving transducer


response function of transmitting transducer

\(V_{in}^{\prime } \left(\upomega \right)\)

input voltage in the calibration

\(V_{\text{out}}^{\prime } \left(\upomega \right)\)

output voltage in the calibration


propagation direction

\(Z\left(\upomega \right)\)

electrical impedance


area of the receiving transducer


line beam spacing


fundamental wave frequency determined by selection of line beam spacing


wave number


wave number compensation factor


wave number for the reference material


data number


quantum voltage


average radius of the precipitates






first-order perturbation solution


second-order perturbation solution

\(u_{{\left( {0^{^\circ } } \right)}}\)

displacement of ultrasonic wave when the phase of the incident ultrasound is 0˚

\(u_{{\left( {180^{^\circ } } \right)}}\)

displacement of ultrasonic wave when the phase of the incident ultrasound is 180˚


particle velocity


line beam width,


propagation distance

\(\Delta \beta\)

variation in the nonlinear ultrasonic parameter


dislocation density


displacement proportionality compensation factor


displacement proportionality coefficient for the fundamental frequency


displacement proportionality coefficient for the fundamental frequency for the reference material


displacement proportionality coefficient for the second-order harmonic frequency


displacement proportionality coefficient for the second-order harmonic frequency for the reference material


displacement proportionality coefficient


second-order nonlinear ultrasonic parameter


relative nonlinear ultrasonic parameter


nonlinear ultrasonic parameter for the reference material


estimated nonlinear ultrasonic parameter


plate-shaped precipitates

\(\beta_{p} {{^{\prime}}}\)

rod-shaped precipitates

\(\beta _{p}^{{''}}\)

needle-shaped precipitates




strain tensor


strain tensor


quantization error


coefficient that determines the profile of the Tukey window








stress tensor


yield strength


angular frequency


fundamental angular frequency


second-order harmonic angular frequency



A major portion of this work was financially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF-2013M2A2A9043241).

We would like to thank all of the ISNDE Lab (Intelligent Sensing & Nondestructive Evaluation Lab, Department of Mechanical Convergence Engineering, Hanyang University) graduates over the past 27 years for their research efforts, discussions, and contributions to the field of nonlinear ultrasonics. A special tribute to Dr. Kyung-Cho Kim, the first Ph.D. graduate of ISNDE to pioneer nonlinear ultrasonics research in solid materials, Dr. Taehun Lee, developed fundamental technologies for the measurement of nonlinear ultrasonic parameters, and Dr. Chung-Seok Kim, who worked as a research Professor at Hanyang University from 2010 to 2012 (currently a Professor at Chosun University) on various applications of nonlinear ultrasonics for material characterization.

Many thanks to Dr. Yob Ha, Dr. Hyunmook Kim, and Dr. Hongjoon Kim, who developed miscellaneous methods for the measurement of nonlinear ultrasonic characteristics, including laser ultrasonic techniques, and Dr. Hogeon Seo who pioneered the measurement of contact acoustic nonlinearity and its synthetic imaging.

Special thanks to the students currently working in the ISNDE Lab: Ph.D. students Dong-Gi Song, Jihyun Jun, Seong-Hyun Park, and Master’s degree students Juyoung Ryu, Youngchang Lee, Seunghoon Lee, Jungyean Hong, who have improved the measurement systems, developed various application techniques, and contributed to the editing and data organization of this chapter.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Hanyang UniversitySeongdong-gu, SeoulRepublic of Korea
  2. 2.Korea Atomic Energy Research InstituteDaejeonRepublic of Korea

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