This paper explores the propagation of nonlinear elastic waves in a two-dimensional isotropic medium. The analytical expressions of first-order potentials corresponding to second harmonic acoustic components are obtained and discussed by using the perturbation method. Based on the careful theoretical analysis, it is shown that the first-order P wave always has a resonant term, which is proportional to the propagation distance in the condition of simultaneous excitation of the P wave. On the contrary, the first-order SV wave does not have any cumulative effect. Moreover, the nonlinear interactions between the P wave and SV are also presented.

## 1. Introduction

The degradation and fatigue cracks quickly appear in structures due to various factors such as impact and corrosion, which may cause structural fracture and huge losses if no action is taken in time. Thus, it is essential to find a useful approach to detect cracks early to ensure the safety of facilities. In recent decades, nonlinear elastic wave problems have attracted considerable attention. As opposed to the primary wave, the higher harmonic acoustic components are much more sensitive to damage prior to crack initiation. The theory of nonlinear elastic waves established by Landau and Lishitz^{1} provided the theoretical basis for nonlinear ultrasonic inspection technology.

To date, the research methods used in theoretical analysis and experiments are roughly divided into two kinds. One is the numerical method based on the finite difference method or the finite element method.^{2,3} Wang *et al.*^{2} introduced a general finite-difference time-domain method for numerical simulation of nonlinear acoustic waves. This method is suitable for simulating the time-domain sound field in complex conditions, whereas it is not convenient for physical analysis and has a high demand for computers. The other method is the perturbation method. Zhang and Wang^{4} presented an effective analytical perturbation method for multipole acoustic logging. Eckart^{5} first introduced the perturbation method in the study of nonlinear acoustics in viscous media and made significant progress. After that, the perturbation method has been widely used in the research of nonlinear acoustics.^{6–11} Ginsberg^{6} introduced the perturbation techniques for deriving analytical expression of finite-amplitude waves in both fluids and plates. Only the nonlinear term leading to an accumulated solution for the second harmonic was considered in the research of two-dimensional waves. Thus, one parameter (the acoustic Mach number) was taken for perturbation expansion in both fluid and plates. On the contrary, considering all first-order terms in the wave equation, we introduce multiple perturbation items in this paper. With the development of nonlinear science, nonlinear resonance ultrasound spectroscopy^{12} and other techniques^{8,9} have been developed to determine the acoustic nonlinearity parameter and third-order elastic constant in a one-dimensional model. The advantage of a one-dimensional model is that it is easy to calculate and analyze results. Due to the neglect of some third-order elastic constants, it is difficult to describe all nonlinear properties of elastic waves. Qian^{13} derived the wave equations of the second-order potentials in elastic media without simplifying. Still, the analytical expressions of potentials were not obtained, and the influences of nonlinear waves were not discussed in his work. Later, Guyer and Johnson^{10} briefly explained the interaction of elastic plane waves in special structures, such as bars in the one-dimensional case. Korneev and Demcenko^{11} described the possible nonlinear interaction of elastic plane waves with different frequencies in an isotropic material by complex computation. Also, they gave resonance conditions for strong scattered waves. On the contrary, our work discussed the propagation and interaction of nonlinear elastic waves excited a monofrequency source. These formulas that we obtained are beneficial for analyzing the effect on the second harmonic from different kinds of primary waves.

This paper aims to investigate the interaction between two plane elastic waves in a homogeneous isotropic medium with quadratic nonlinearity. The analytical expressions of zeroth-order and first-order potentials corresponding to the primary wave and the second harmonic are obtained using the perturbation method. Based on the physical analysis of these analytical solutions, the propagation laws of second harmonic acoustics components with different types of excitation in an isotropic solid are presented, which set the basis for the further study of nonlinear ultrasonic inspections.

## 2. Theory

A Cartesian coordinate system *Oxz* is established to solve the two-dimensional problem. In this research, the strain energy density of an isotropic elastic solid proposed by Landau and Lifshitz^{1} is adopted, i.e.,

where *K* = *λ* + 2 *μ*/3 is the bulk modulus, *λ* and *μ* are the second-order elastic constants (Lame constants), *A*, *B*, and *C* represent the third-order elastic constants, and $uik$ is the Lagrangian strain tensor, and *i*, *j*, *k*, *l*, *m* = *x*, *z*.

Then the nonlinear wave equation can be written as (see the supplementary material^{14})

where *ρ* is the density of isotropic media, ** u** =

*u*

_{x}

*e*_{x}+

*u*

_{z}

*e*_{z}represents the particle displacement, and

**=**

*F**F*

_{x}

*e*_{x}+

*F*

_{z}

*e*_{z}is the divergence of quadratic terms of the first Piola-Kirchhoff stress tensor, and

*F*is considered as the functions of

_{i}*u*and

_{x}*u*:

_{z}### 2.1 Perturbation expansion

Since it is quite difficult to solve the nonlinear wave equation directly, the perturbation method is used. First, five dimensionless perturbation items are introduced to describe the nonlinear terms in the equation as follows:

These dimensionless perturbation items are determined by the second-order and third-order elastic constants of material, which represent all nonlinear properties in solids.

The particle displacement is assumed to be the sum of zeroth-order, first-order, second-order, and high-order approximation. In this paper, only the zeroth-order and first-order approximation of the particle displacement are considered, or

After substituting Eq. (6) into Eq. (2), the nonlinear wave equation is divided into two linear wave equations:

where *F*^{1}(*u*^{(0)}) is the driving forces produced by the zeroth-order and first-order approximation due to the nonlinearity of the solid.

### 2.2 Process in solving the linear wave equations

P-wave potential *ϕ*^{(0)}(** r**,

*t*) and S-wave potential

*φ*

^{(0)}(

**,**

*r**t*) are introduced to describe zeroth-order approximation

*u*^{(0)}(

*r*,

*t*):

^{15}

where the vector ** r** =

*x*

*e*_{x}+

*z*

*e*_{z}represents the point in two-dimensional space.

where *c*_{p} and *c*_{s} are the particle velocity of P wave and SV wave, respectively.

where *k*_{p} is the wave vector of P wave and its value is *k*_{p} = *ω*/*c*_{p}, *k*_{s} is the wave vector of SV wave and its value is *k*_{s} = *ω*/*c*_{s}, and *ω* denotes frequency.

In the same way, the first-order approximation **u**^{1}(** r**,

*t*) is written as

where *Q*_{1}, *Q*_{2}, *Q*_{3}, and *Q*_{4}, are linear functions of five perturbation items, and independent of coordinates and time,

and

*α* represents the angle between *k*_{p} and the *x* axis, and *β* is the angle between *k*_{s} and the *x* axis. It is important to note that the nonhomogeneous terms on the right of Eqs. (13a) and (13b) are not symmetric. The observed asymmetry indicates that P wave and SV wave travel in different ways, which is discussed below.

Since Eq. (13a) is a linear nonhomogeneous differential equation, the superposition principle is valid. Finally, the specific solutions of Eqs. (13a) and (13b) are

## 3. Analysis and discussion

Weak nonlinearity is assumed in this paper, which means the first-order approximation is small compared to the zeroth-order approximation. In practice, the second harmonic acoustic components in the received signal of ordinary transducers are weak, and other higher harmonic components are hard to obtain. Therefore, although the perturbation method is an approximate analysis method for elastic waves, the obtained analytical solutions are of reference significance in practical application.

### 3.1 Case I

In this case, only P-wave excitation is set in the solid, i.e., *ϕ* ^{(0)} ≠ 0, *φ* ^{(0)} = 0. Then the first-order potentials are

Equations (17a) and (17b) show that only the first-order P wave is generated by the self-action of zeroth-order P wave, and its amplitude is proportional to the distance travelled by the wave. In this paper, the attenuation in the material is ignored. With the increase of the wave distance passed, the effect of the first-order P wave on the total wave increases. However, due to the assumption of weak nonlinearity, this solution is valid only up to a certain distance travelled by the wave. This conclusion is in agreement with the result obtained in liquids and a one-dimensional case in solids.^{7,16} Besides, there is no first-order SV wave potential, which suggests that P wave does not generate waves of other types in this situation.

### 3.2 Case II

When SV-wave source is excitation in the solid, i.e., *ϕ* ^{(0)} = 0, *φ*^{(0)} ≠ 0, Eqs. (16a) and (16b) are rewritten as

The first-order SV-wave potential is zero, which means that the total SV wave propagates in the form of the primary wave due to the symmetry of the elastic material. When only the second-order and third-order elastic constants of material are considered, the nonlinearity parameter is identically zero for all S waves associated with a material symmetry plane in the media.^{17} At the same time, SV wave displays another property. Some kind of P wave is generated by the self-action of zeroth-order SV wave, but its phase velocity is *c*_{s,} which is the propagation velocity of SV wave. It is important to note that this first-order P wave is not sensitive to propagation distance.

### 3.3 Case III

In this case, P wave and SV wave are excited at the same time, i.e., *ϕ* ^{(0)} ≠ 0, *φ* ^{(0)} ≠ 0, then the solutions of first-order potentials are shown as Eqs. (16a) and (16b). It can be seen that the first-order P-wave potential has three terms. The first term describes the resonant P wave with a cumulative effect generated by the self-action of zeroth-order P wave. The second term represents the self-action of zeroth-order SV wave. And the third term describes the interaction of zeroth-order P wave and zeroth-order SV wave. These three parts have different phase velocities, which are *c*_{p}, *c*_{s}, and 2*ω*/|*k*_{p}+*k*_{s}|, respectively. Therefore, the first-order P wave will be synthesized by amplitude modulation of these three parts that are asynchronous by their wavenumbers. In addition, Eq. (16b) also suggests that the first-order SV wave is generated only when P wave and SV wave are all excited.

Observing Eq. (16a), we found that the denominator of the third term in first-order P-wave potential may be zero, i.e.,

where *θ* donates the angle between *k*_{p} and *k*_{s}.

*θ*_{q,} called the resonance angle between *k*_{p} and *k*_{s,} is only determined by the second-order elastic constants. When *θ* satisfies Eq. (20), the amplitude of the third term in Eq. (16a) also has a monotonically increasing relationship with distance travelled by the wave.

On the contrary, the denominator of that term in first-order SV-wave potential cannot be zero. These three cases show that the resonant first-order SV wave cannot be generated by a monofrequency source. This conclusion explains that the second harmonic components of transverse waves are not observed in some numerical simulations or experiments. In order to observe and use shear waves in soft solids, Hamilton *et al*.^{18} presented the strong cubically nonlinear shear waves in solids, and separated the effects due to compressibility and shear deformation. After that, there has been a stream of works on nonlinear shear waves in soft tissues,^{19–21} which emphasized the applications of nonlinear S wave and ways to reduce or eliminate the influence of P wave on the S wave. However, one crucial part of our work is to obtain the full waveforms with P wave and S wave through the nonlinear wave equation directly, and discuss the nonlinear interaction of P wave and S wave.

In addition, it can be seen from Eqs. (16a) and (16b) that the amplitudes of first-order potentials depend on the amplitudes of zeroth-order potentials, distance travelled by the wave, and wavenumbers. And wave numbers only rely on frequency when material and the propagation direction of waves are known.^{14}

## 4. Conclusion

The analytical solutions of zeroth-order and first-order potentials in isotropic media are derived in this paper. Theoretical analysis suggests that the first-order P wave always has a cumulative effect if the P wave source is used. Moreover, in the first-order P wave, the part describing the interaction of a zeroth-order P wave and the zeroth-order SV wave can become a resonant term for a given condition. On the contrary, the first-order SV wave is generated only when the P wave and SV wave are excited at the same time, and its amplitude is not sensitive to the propagation distance. Furthermore, the nonlinearity degree of the total wave is in a positive relationship with the initial amplitudes and frequency of the wave. These results are important to the development of an experimental process to measure the nonlinearity of material with elastic waves.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China with Grant Nos. 11574343 and 11774377.