To study the effect of SHG of guided waves propagation, Deng first investigated the feature of the second-harmonic field of the Lamb wave propagation in an isotropic plate,^[29] using the technique of nonlinear reflection of acoustic waves at an interface.^[33,34] Primary Lamb wave propagation can be taken as the superposition of the four bulk wave modes (two longitudinal and two transverse waves) propagating in the plate, as described in Fig. 2.

Fig. 2.

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	Fig. 2. Acoustic field of primary Lamb wave propagation in a plate.[29]

The primary wave field of Lamb wave propagation can be expressed as 5 where u₁ satisfies Eq. (3). The stress-free boundary conditions determine the Lamb wave dispersion relations and the amplitudes of the four partial waves (two longitudinal waves u_L1 and u_L2, and two transverse waves u_T1 and u_T2), which require that the normal and tangential stress should be zero at the two surfaces of the plate.

The acoustic field of second harmonic of primary Lamb wave propagation can be obtained by analyzing the self- and cross-interactions of the two longitudinal waves and two transverse waves. Through these complicated self- and cross-interactions, ten driven longitudinal second harmonics and four transverse second harmonics will be generated.^[29,33–35] Generally, only the self-interaction of the primary longitudinal wave (i.e., u_L1 or u_L2 can generate the resonant (or cumulative) longitudinal second harmonic due to the phase velocity matching between the primary wave and the second harmonic generated. For each of the other driven second harmonics generated, due to the phase velocity mismatching, the efficiency of its generation is very inefficient. That is to say, the corresponding second-harmonic solution does not have a cumulative term.

The second-harmonic boundary condition at the two surfaces of the solid plate requires that the components of second-harmonic stress equal zero. Generally, this boundary condition cannot be satisfied if only the driven second harmonics are considered. In fact, these driven second harmonics (generated by f(u₁) in Eq. (4)) are the particular solution to Eq. (4). The full solution to Eq. (4) should include both the particular and general solutions. When f(u₁) in Eq. (4) is set to be zero, the general solution satisfying Eq. (4) can readily be obtained.^[29] The general solution to Eq. (4) is also referred to as the freely-propagating second harmonic since there is no driving force. The boundary condition of second harmonics may be satisfied by taking into account both the particular and general solutions.

Based on the technique of nonlinear reflection of acoustic waves at an interface, the general solution to Eq. (4) can formally be determined, which includes the plane wave term, as well as the cumulative wave term.^[29,35] Through solving the equation of the boundary condition where the second-harmonic stresses arising from the driven and the freely-propagating second harmonics are zero at the two surfaces, the solution to the freely-propagating second harmonics can be derived. Especially for the case of phase velocity matching between the primary and double frequency Lamb wave modes, the solution to the cumulative second harmonic (consisting of the driven and the freely-propagating cumulative second harmonics) of primary Lamb wave propagation can be determined, which satisfies the wave equation, boundary, and initial conditions of excitation.

The above analysis based on the technique of nonlinear reflection of acoustic waves at an interface is straightforward and convenient for understanding the physical process of the nonlinear guided waves propagation in solid media. In addition, the solution derived is relatively easy for numerical computations and can also be readily used to demonstrate the influence of the location of the excitation source on the second-harmonic field. However, the solution derived by this method is relatively complex and cumbersome, and it is not applicable to the case where the phase velocity matching is approximately satisfied, as well as the case where ultrasonic guided waves propagate in anisotropic media or waveguides with arbitrary geometries. To avoid these deficiencies, Deng and de Lima et al. separately adopted the modal expansion analysis approach to investigate the secondary wave field of guided wave propagation within a perturbation approximation.^[31,32]

According to the modal analysis approach, the solution of acoustic field generated by the surface traction s and the volume force f in a plate can be written as a linear combination of normal guided wave modes^[3] 6 where u_m(y) is the displacement field function of the mth guided wave mode, and A_m(z) is the corresponding expansion coefficient, and 1/2 is multiplied to ensure real quantities. A_m(z) is the solution of the ordinary differential equation 7 where 8 9 10 where v = ∂u/∂t, Ω is the waveguide cross-sectional area, and Γ is the curve enclosing Ω. The terms and are defined as the complex external power due to the surface traction s and the volume force f of ultrasonic guided waves, respectively. k_n is the wavenumber of the wave mode that is not orthogonal to the mode with the wavenumber k_m.

Generally, the normal modal analysis approach is based on the reciprocity relation, while the reciprocity theory cannot be used to analyze the nonlinear problem. However, under a second-order perturbation, the acoustic nonlinear response can be considered as a perturbation to primary wave motion. Thus, the SHG of ultrasonic guided wave propagation can be taken as a linear response with the given excited sources due to propagation of primary guided waves.

The acoustic field of the second harmonic generated can be constructed via this modal expansion approach. By using the normal mode expansion approach, the second-harmonic field of Lamb wave propagation can be considered as a linear summation of a series of DFLW modes. However, it is inconvenient for numerical computations by employing the normal modal analysis approach to analyze the second harmonic field of guided waves propagation. Considering the complexity of nonlinear guided waves, combining the nonlinear reflection of acoustic waves at interfaces and normal modal expansion can be used to understand the physical insight and propagation procedure of SHG of guided waves. A straightforward approach for analysis of DFLWs by using the partial wave technique and modal analyses approach was proposed by Deng.^[36] As described in Fig. 2, the Lagrangian coordinates are established for a solid plate with a thickness of 2d. The lth Lamb wave mode consists of the four partial bulk waves, denoted by the four displacement vectors u_(l)Li and u_(l)Ti (i = 1, 2). The field of the lth Lamb wave is given by . There is a bulk driving force with the double frequency 2f, denoted by , in the solid plate due to the elastic nonlinearity of the plate material. Besides, there are second-order stress tensors with the double frequency 2f at the two surfaces of the solid plate, denoted by and obtained through the first Piola–Kirchhoff stress tensor formula.^[36] The total second-harmonic field of the lth Lamb wave can be expressed by the summation of a series of DFLWs, i.e., , where is the field function of the nth DFLW, and A_n(z) is the corresponding expansion coefficient. The equation governing A_n(z) is given by 11 where P_nn is the average power flow per unit width along the x axis for the nth DFLW; and are respectively the oz-axis components of the wave vectors of the bulk waves constituting the lth Lamb wave (namely and the nth DFLW; is the oy-axis unit vector; and are the phase velocities of the lth Lamb wave and the nth DLFW, respectively. The sign ~ on the top of denotes the complex conjugate operation. Combining the initial condition of SHG with Eq. (13) yields the formal solution of A_n(z) 12 It is found that A_n(z) may be linearly proportional to propagation distance z when both the conditions F_(n) ≠ 0 and (namely phase velocity matching are satisfied. It has been shown that F_(n) = 0 (namely, A_n(z)=0) when the nth DFLW is anti-symmetrical. Thus the field of u^(2f) is only symmetrical.^[29–31] Under the condition , the value of the algebraic expression in { } in Eq. (12) is 1, and then the solution of the nth DFLW component that linearly grows with propagation distance can be given by 13

It is known that the particle displacement of shear horizontal (SH) guided wave propagation is in parallel to the surface of the waveguide.^[37] In general, each guided wave mode can be taken as superposition of partial bulk acoustic waves satisfying the corresponding boundary conditions. A SH guided wave mode can be represented by two shear waves that are reflected at the upper and lower boundaries of the solid plate. Modal analysis of waveguide excitation can effectively be employed to investigate the SHG of SH guided wave modes under second-order perturbation.^[38,39]

If the excitation source of the SH wave mode is located at z = 0, and the mth SH wave mode with angular frequency ω is produced in the plate of thickness 2d as shown in Fig. 3, the solution to the mth SH wave mode can formally be expressed as 14 where is the unit vector along the ox axis and the subscript m denotes the ordinal of SH mode. k_(m) = ω/c_(m) in Fig. 3 is the oz axis component of K_(m)T1 and K_(m)T2. c_(m) is the phase velocity of the mth SH mode.

Fig. 3.

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	Fig. 3. Acoustic field of the mth SH wave mode and nth DFLW mode.[38]

The SHG of an SH wave mode can be analyzed by using the modal analysis approach. It is shown that the total second-harmonic field of a SH wave mode can be decomposed into a series of DFLW modes, due to the self- and cross-interactions of primary partial bulks waves. The field of a DFLW will have a cumulative growth effect versus propagation distance if its phase velocity exactly or approximately equals that of the primary SH wave mode.

Recently, Liu et al. used the mathematical approach to predict the cumulative behavior of the third-harmonic generation of SH guided waves with a hyperelastic material model.^[40] In their investigation, reciprocity and normal mode expansion approach were used to find the ordinary differential equation for the modal participation factors. Results indicate that primary SH wave modes are holo-internal-resonant with the third-harmonic fields. Lissenden et al. developed an experimental procedure by using magnetostrictive transducers for measurement of the third harmonic of SH0 wave mode. The proposed setup was used to evaluate the localized microstructural evolution in an aluminum plate.^[41]

Li et al. studied the nonlinear behavior of SH guided wave propagation using partial wave technique and normal mode expansion approach.^[42] In their study, all of the phase matched SH modes in an isotropic plate were provided, as indicated in Fig. 4.^[42]

Fig. 4.

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	Fig. 4. (color online) SH guided wave dispersion curves and phase matched SH wave modes with 1 mm thickness.[42]

Deng et al. studied the SHG of SH wave modes in a layered planar structure with imperfect interface.^[43] The influences of interfacial properties on the effect of SHG by the SH guided wave were discussed in their study. The coordinates system associated to the structure with two solid layers is shown in Fig. 5, where d_i(i = 1,2) denotes the solid layer thickness, and ρ_(i) is the mass density. According to the partial wave analyses approach of guided wave modes, SH wave propagation in the structures shown in the layered structure can be decomposed into four partial plane shear waves that have the same wave vector components along the oz axis.

Fig. 5.

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	Fig. 5. (color online) Layered planar structure with the SH guided wave mode propagation.[43]

The analyses indicate that the second-harmonic field generated by primary SH wave propagation consists of a series of DFLW modes with both longitudinal and transverse wave components for the reason that the cross-interaction of two shear waves can generate double frequency longitudinal wave components. As shown in Fig. 6, at the condition of phase velocity matching between the primary SH mode and the DFLW component, the amplitude of the corresponding second harmonic has a cumulative effect with propagation distance. Meanwhile, the mismatching of phase velocity can remarkably decrease the SHG efficiency of primary SH guided wave propagation.

Fig. 6.

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	Fig. 6. (color online) Phase velocity dispersion curves for primary SH wave modes and the symmetric DFLWs in an aluminum sheet.[43]

Investigation of SHG of Lamb wave propagation in an isotropic plate was firstly studied by perturbation method and partial wave technique.^[27] In addition, a straightforward analyses of SHG of Lamb wave propagation by perturbation method and normal modal analysis technique were separately reported by Deng^[31] and de Lima et al.^[32] Srivastava et al. reported the possibility of the existence of anti-symmetric or symmetric second-harmonic Lamb wave modes.^[44] Recently, this subject was also studied by using displacement gradient-based formation during the procedure of modal expanding analysis.^[45,46]

To analyze the second-order solution of Lamb wave propagation, the normal model analysis approach is used. The total second harmonic field can be written as 15 The solution of A_m(z) in Eq. (15) with the source condition u₂ = 0 at z = 0 is given by 16 where 17 18 Here u₂ is the displacement field function, z is the wave propagation distance, and A_m(z) is the corresponding expansion coefficient, which is the second-order modal amplitude to be determined. The terms and are respectively defined as the complex external power due to the surface traction and the volume force generated by propagation of primary Lamb waves.

Based on the above analyses, it is found that second-harmonic amplitude grows linearly with propagation direction, when (synchronism) and (nonzero power flux). If the selected wave mode satisfies the two conditions, the second-harmonic amplitude will have a cumulative effect versus propagation distance. In practice, interest is focused on the case where the second harmonic generated grows with propagation distance, since the cumulative second harmonic plays a dominant role in the second-harmonics field after primary wave propagates a certain distance.^[47]

The symmetric (S) modes, anti-symmetric (A) modes, and symmetric DFLW (D) modes are shown in Fig. 7, where the phase velocity matching modes can be found in the crossing points. The symmetric properties of DFLW modes will be discussed in Section 6. The frequencies along with corresponding phase velocities of different types of Lamb wave modes that can be used to generate cumulative second-harmonic waves have been discussed by Matsuda and Biwa.^[48] It is important to note that the second-harmonic field of Lamb wave propagation can be considered as superposition of a series of DFLW components. However, most researchers focus on the DFLW component that has the same phase velocity as primary Lamb wave mode since only the DFLW mode with a proper phase velocity matching can remain in the second-harmonic field after propagating some distance, while all the other DFLW modes can be ignored due to destructive interference with each other.

Fig. 7.

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	Fig. 7. (color online) Dispersion curves of primary Lamb waves and the symmetric DFLWs for a given aluminum plate with 1 mm thickness.[47]

SHG of Lamb wave propagation in a two-layered solid waveguide was studied by combining the modal analyses approach and the nonlinear reflection of acoustic waves at interfaces.^[49–51] A straightforward and convenient analytical expression of cumulative second harmonic of Lamb wave propagation in this layered structure was provided. To illustrate the theoretical model, nonlinear Lamb wave propagation in a two-layered waveguide made of two different materials denoted by M_i with thickness of d_i (i = 1,2) was shown in Fig. 8. The oy axis is normal to the interface and the oz axis is along the interface of M₁ and M₂. The partial bulk acoustic waves constituting the primary Lamb wave with frequency f and ordinal l were denoted by the displacement fields and (i = 1, 2; q = 1, 2), where the subscript (i) = (1) or (2) of U corresponds to the layer M₁ and M₂, T or L in U means that the corresponding quantity is associated with the transverse or longitudinal wave, respectively.

Fig. 8.

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	Fig. 8. The wave vectors and displacement fields of partial plane bulk acoustic waves constituting the lth primary Lamb wave.[51]

A modal analysis method was used to analyze the waveguide excitation. The second-harmonic field of Lamb wave propagation in the structure shown in Fig. 8 can be expressed as 19 with 20 where is the displacement field function of the nth DFLW mode. The modal expansion coefficient A_n(z) can be determined as 21 with 22 23 and 24 where or is the forcing function due to the coupling between and , or due to the coupling between and , respectively. P_nn is the average power flow per unit width along the ox axis for the nth DFLW mode, and denotes the unit vector along the oz axis. is the stress tensor associated with . Considering the initial condition of SHG, the formal solution of A_n(z) can be obtained as 25

It shows that A_n(z) is linearly proportional to propagation distance z under the condition of k^(f,l) = k^(2f,n)/2, which represents the phase velocity matching between the lth primary Lamb wave and the nth DFLW component.

Cumulative SHG of Lamb wave propagation in a solid plate contacting a liquid layer was also analyzed by Deng.^[52] In this investigation, a second-order perturbation and approach of nonlinear reflection of acoustic wave at an interface were employed to describe the physical process of SHG of Lamb wave propagation.

Compared with Lamb wave propagation in the isotropic plate, the strain energy function for a nonlinear elastic transversely isotropic material is expressed in terms of the five invariants of the Green–Lagrange strain tensor. The formal solution of second-harmonic field of guided wave propagation in anisotropic media was analyzed by Deng et al.^[53,54] and the analytical results show that the efficiency of SHG of guided wave propagation can be significantly affected by the elastic anisotropy.

As shown in Fig. 9, the Lagrangian coordinates (a₁, a₂, and a₃) are established for an anisotropic solid plate and guided waves are assumed to propagate along the a₃ axis. The formal solution of displacement field of a guided wave mode with the angular frequency ω can be expressed as^[54] 26 where the subscript i = 1, 2, or 3 corresponds to the component of displacement field along the a₁, a₂, or a₃ axis. As illustrated in Fig. 9, the elastic constants of the anisotropic material are originally defined in the , , coordinate system. The corresponding elastic constants of the material in the a₁a₂a₃ coordinate can be obtained via the rotation transformation determined by the rotation angles γ₁, γ₂, and γ₃.

Fig. 9.

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	Fig. 9. Schematic representation of an anisotropic/piezoelectric solid plate.[54,57]

For anisotropic material, the component of the nonlinear term in Eq. (2) is given by 27 where C_ijkl and C_ijklmn are the second- and third-order elastic constants in the a₁a₂a₃ coordinate, respectively. The second-order traction stress tensors at the top and bottom surfaces of the anisotropic plate can be written as 28 It is important to note that there is a factor exp[j(2ka₃ − 2ωt)] in f_i and P_ij.

Similar to the analysis of SHG of primary Lamb waves in an isotropic plate, the perturbation approximation and normal modal expansion approach are used in this problem. The second-harmonic field u₂ of guided wave propagation in anisotropic plate can be expressed as 29 where A_m(a₃) is the expansion coefficient of the mth double-frequency guided wave mode U^(2ω,m)(a₂), which is determined by 30 31 32 where k^(2ω,m) is the wave number of the mth double-frequency guided wave along the a₃ axis and P_mm is the average power flow of the mth double-frequency guided wave per unit width along the a₁ axis. The initial condition for the SHG requires that A_m(a₃) equal zero at a₃ = 0. Thus we have 33 where D = 1 − k^(2ω,m)/2k. It shows that the amplitude of the second harmonic of primary guided wave propagation will grow with propagation distance a₃ under the condition of D = 0 or D ≈ 0. The result of analysis given in Eqs. (29)–(33) is similar to the case where primary Lamb waves propagate in isotropic media.

Recently, Zhao et al. reported a theoretical study on the SHG of Lamb waves in composite.^[55] The analysis shows that the efficiency of SHG can be determined by the propagation direction of Lamb waves in the composites. Li et al. provided an example of Lamb wave modes selection for generation of cumulative second harmonic in co-directional composites structure.^[56]

In piezoelectric material, piezoelectric and dielectric nonlinearities can be other sources for acoustic nonlinearity of primary guided wave propagation. Recently, Deng et al. studied the nonlinear behavior of ultrasonic guided wave propagation in a piezoelectric plate with nonlinear elastic, piezoelectric, and dielectric properties.^[57] Generally, besides dispersion and multi-mode characteristics of guided wave propagation, the inherent coupling between the electric and mechanical fields in piezoelectric materials makes the theoretical analysis of SHG of guided waves much more complex than that in an isotropic elastic plate.

The Lagrangian coordinate system established for a single piezoelectric plate is shown in Fig. 9, where γ₁, γ₂, and γ₃ are the angles of orientation between the two coordinate systems a₁a₂a₃ and , and the elastic, piezoelectric, and dielectric constants of the piezoelectric material are originally defined under the coordinate system . The excitation sources for the generation of a series of double-frequency guided wave modes include bulk driving force components are indicated as with double frequency. The second- and third-order elastic constants, second-order piezoelectric one, as well as the electrostrictive constant are involved into the expression of the bulk driving force .^[58,59] Compared with non-piezoelectric material, there is a bulk charge density with the double frequency in piezoelectric material, denoted by ρ^(2f). All the material constants in and ρ^(2f) should be transformed from the coordinate system to a₁a₂a₃ as illustrated in Fig. 9. Besides the bulk driving force and bulk charge density in the interior of the piezoelectric plate, there are second-order stress tensors with the double frequency at the bottom and top surfaces of the piezoelectric plate, denoted as 34 and second-order surface charge density, written as 35 where φ is the electrical potential of the fundamental frequency, and M_ijklmn is defined by c_ijklmn + δ_kmc_ijln + δ_imc_jnkl + δ_ikc_jlmn (i, j, k, l, m, n = 1, 2, 3). The constant e_kmj is a second-order piezoelectric one. It should be noted that the electrostrictive constant d_klij is related both to elasto-optic and electrostrictive effects, and that the third-order piezoelectric constant f_kijlm is related to the electroacoustic effect.

The second-order terms, , , , and , can provide a complete and accurate description for the SHG effect.^[57] According to the modal analysis approach for waveguide excitation, these second-order terms can be regarded as excitation sources for generation of a series of double frequency guided wave modes. Thus, the field of the SHG by primary guided wave propagation (U_i, φ) can formally be expressed by the summation of these double frequency guided waves, i.e., 36 where (i = 1, 2, 3) and φ^{(2f, m)}(a₂) are respectively the a_i-axis displacement component and the electrical potential of the guided wave mode (with the driving frequency 2f and the order m), and A_m (a₃) is the corresponding expansion coefficient. Based on the reciprocity relation and the orthogonality of guided waves, the equation governing A_m(a₃) is given by 37 with^[57] 38 39 where P_mm is the average power flow per unit width along with the a₁ axis for the mth double frequency guided wave and k^(2f,m) is the a₃-axis component of the wave vectors. Considering an initial condition for SHG of primary guided wave propagation, i.e., A_m(a₃) = 0 when a₃ = 0, A_m(a₃) can formally be given by 40 where . When A_m ≠ 0 and D = 0, A_m(a₃) can be written as A_m(a₃) = (A_m/4d)a₃ exp[jk^(2f,m)a₃], which means that the second-harmonic amplitude will increase with propagation distance a₃. When A_m ≠ 0 and D ≠ 0, there will be a beat effect for the amplitude of the mth guided wave with propagation distance and its contribution to and φ⁽²⁾ can be negligible.

For a given set of orientation angles (γ₁, γ₂, γ₃, the primary guided wave propagating in the piezoelectric plate with the specific electrical boundary conditions at a₂ = −d and 0 (see Fig. 9) can be selectively generated to ensure A_m ≠ 0 and D = 0. In this way, the mth double frequency guided wave component will grow with propagation distance, and the obvious second-harmonic signals of primary guided wave propagation can be observed. When changes in the electrical boundary condition take place, the analysis given in Ref. [57] indicates that the efficiency of SHG by primary guided wave propagation will be remarkably influenced. This provides a means to regulate the SHG efficiency by changing the electrical boundary conditions.

de Lima et al. theoretically studied the elastic wave in waveguides with arbitrary cross-section media.^[60] They reported the harmonics generation of ultrasonic guided wave propagation in cylindrical rods and shells, based on the perturbation method and normal mode decomposition technique which was suggested by Auld.^[3] Practically, unlike the case of guided wave propagation in plate-like structures where the diffusion of energy for primary wave propagation makes it difficult for long range detection, the guided wave propagation in tube-like structures provides a more efficient approach to detect material nonlinearity due to its less energy diffusion.

The normal modal expansion analysis shown in Section 3 is applied to a cross-sectional area of any shape. For the tube-like structure as shown in Fig. 10, the particle displacement components of primary wave mode propagating along the z axis would be^[60] 41 42 43 where the circumferential order is for n = 0, 1, 2, …. For axis-symmetric modes, n = 0, and n ≥ 1 for the flexural modes. The flexural modes with n = 1 are called first-order flexural modes. Generally, the terms ( , , and ) depended on the type of waveguide and family of modes.

Fig. 10.

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	Fig. 10. Schematic of cylindrical waveguides (a) rod and (b) pipe.[60]

Accompanying propagation of primary wave mode ( , , , there are second-order bulk driving force f⁽²⁾ (r,θ) within the waveguide and second-order driving traction at the surfaces r = a, b. Within a second-order perturbation, the second-harmonic field u₂ of primary wave mode ( , , can be decomposed into a series of double frequency guided waves generated by f⁽²⁾ (r,θ) and , i.e., , where u_m(r,θ) is the displacement field function of the mth double frequency guided wave and the corresponding traction stress is denoted by s_m(r,θ), and A_m(z) is the modal expansion coefficient. The formal expression of A_m(z) is given by 44 where D = 1 − k^(2ω,m)/2k and k^(2ω,m) is the wave number of the mth double frequency guided wave. For cylindrical rod (see Fig. 10), the expressions of , , and are shown as 45 46 47 For the cylindrical shell, they can be expressed as 48 49 50 Here and are the unit vectors along the radial and axial directions, respectively. If we submit Eqs. (41)–(43) under the condition of n = 0 into Eqs. (45)–(47) or Eqs. (48)–(50), the condition (nonzero power flux) will always be satisfied. It shows that all axisymmetric modes, including longitudinal modes and torsional modes in a tube-like structure, can be used to generate cumulative second harmonics under the condition of synchronism, when the secondary waves have the same phase velocity as that of primary wave mode (i.e., D = 0 in Eq. (44)). Similarly, for the SHG of flexural mode propagation, under the condition of n ≠ 0, the theoretical analyses can be used to choose the suitable double frequency flexural mode with non-zero power flux from primary wave mode to second harmonic ones.

Theoretical studies of the conditions for the existence of longitudinal or flexural waves in nonlinear, isotropic rods were presented by Srrivastava et al.,^[61–63] who applied the analytical solutions in rail tracks for the potential measurement of thermal stress in welded rail. Liu et al. studied the interaction of axisymmetric torsional and longitudinal guided waves in circular cylinders.^[64] Li, et al. investigated the axisymmetric guided waves modes in pipe and applied the phase matched longitudinal guided wave modes to evaluate the thermal damage.^[65,66] Recently, numerical studies of nonlinear guided waves in uniform waveguides with arbitrary cross section were conducted by Zuo et al. and the SHG of guided waves in a steel rectangular bar was studied using the proposed numerical model.^[67]

Circumferential guided wave (CGW) is another kind of fundamental wave mode that propagates along the circumference of the circular tubes. Recently, Gao and Deng et al. established a theoretical model to analyze the SHG of CGW propagation in a circular tube with isotropic, homogeneous, and dispersionless features.^[68–70] The cylindrical coordinates for analyzing CGW propagation were shown in Fig. 11.

Fig. 11.

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	Fig. 11. (color online) Lagrangian cylindrical coordinates (r,θ) for analyzing CGW propagation.[70]

The displacement field of the primary CGW mode propagating along the circumference of the circular tube can be expressed as 51 where n⁽¹⁾ is the dimensionless angular wave number of the CGW mode. The components of bulk driving force in the interior of the circular tube and surface stress are the sources for the generation of a series of double frequency CGW modes under second order perturbation approximation, denoted as and (q = r,θ), which can be expressed in the cylindrical coordinates as^[70] 52 53 and 54 55 56 57 where U_r = r ⋅ u₁, , and and are the unit vectors along the radial and circumferential direction, respectively. The parameters a, b, d, and e in Eqs. (54)–(57) are defined in Ref. [70]. The second-harmonic field of primary CGW propagation generated by and (q = r,θ) can be written as 58 where A_m(θ) is the expansion coefficient of the mth double frequency CGW, determined by 59 where n⁽²⁾ and P_mm are the dimensionless angular wave number of the mth double frequency CGW and the corresponding average power flow per unit width perpendicular to the tube section. Due to the bulk driving force (q = r,θ) and surface stress (q = r,θ), and can be expressed as 60 61 The solution to A_m (θ) in Eq. (59) is given by 62 where Δn = [n⁽¹⁾ − n⁽²⁾/2 is used to describe the dispersion degree between the primary mode and the mth double frequency CGW. It can be found that the amplitude of the mth double frequency CGW will grow linearly along the tube circumference under the condition of Δn = 0 and , and can be written as 63

Obviously, the theoretical results obtained for SHG of primary CGWs are similar to those of ultrasonic guided wave propagation in plate-like structures.

The symmetry of second-harmonic fields of Lamb wave propagation was investigated by Deng,^[29–1] de Lima et al.,^[32] and Chillara et al.^[75] Based on the results of these existing investigations, it can be concluded that a symmetric primary guided wave mode can generate a symmetric secondary wave mode. However, their controversial conclusion is about the symmetry of the secondary wave mode in the case of the primary antisymmetric guided wave mode. De Lima et al. state that a primary mode can generate a secondary mode only with the same symmetry, which means that an antisymmetric primary mode can generate an antisymmetric but not a symmetric secondary mode.^[32] In contrast, the analytical expressions and numerical analyses of the second-harmonic field were provided by Deng, and the results obtained indicate that the cumulative second harmonic is symmetrical regardless of whether primary Lamb wave propagation is symmetrical or antisymmetrical.^[29–31] Müller et al. also investigated the symmetry properties of the second harmonic of Lamb wave propagation,^[73] and the symmetry properties for the second-harmonic field of primary Lamb waves were obtained when the primary Lamb wave mode was either symmetric or antisymmetric. The analytical solution of second harmonics of primary Lamb waves shows that only the symmetric second-harmonic modes at double frequency exist, even if the antisymmetric Lamb wave modes are selected as the primary guided waves. It can be concluded that neither symmetric nor antisymmetric primary Lamb wave mode can generate an antisymmetric Lamb wave mode at double frequency.

In order to ensure generation of cumulative second harmonic of guided waves, it is well accepted that the conditions of phase velocity matching and non-zero power flux from primary mode to secondary mode should be satisfied.^[29–32] In addition, some researchers pointed out that the requirement of group velocity matching between the primary guided wave mode and secondary ones is also needed for generation of a cumulative second harmonic.^[76–78] However, this idea is still in dispute.^[79,80]

The theoretical analysis of SHG of guided waves propagation at the condition of non-matched group velocity of primary mode and secondary ones was firstly reported by Deng et al.^[79,80] Figure 12 shows the mode pair that satisfies the phase matching condition, while the group velocity of primary wave mode ( ) is higher than that of secondary one ( ). The process of generation of the second harmonic under the condition of phase velocity matching ( ) and group velocity mismatching ( ) is schematically illustrated in Fig. 13, where and are the phase velocity of the primary Lamb wave and DFLW, respectively. The tone-burst signal of DFLW will be generated when the tone burst of the primary Lamb wave is excited at the position Tx, assuming that the time-of-flight of the DFLW tone burst from z_p to Rx is , while the time-of-flight for the primary wave tone burst from z₀ to z_p is . Figure 13 indicates that at the position Rx, the time range of the envelope of the DFLW tone burst is from to , where the corresponding envelope function is denoted by A_2f(t). The cumulative second- harmonic field mainly dependent on the DFLW mode due to can be generated. Based on the analysis shown in Fig. 13, the physical process of SHG can be clearly illustrated in the time range . From the time range of to , the overlapping degree of the DFLW tone burst will increase linearly with time. Thus, the second-harmonic amplitude increases linearly with time in this time range. From the time range of to , the overlapping degree of the DFLW tone burst generated is kept unchanged, and thus the amplitude is kept constant in this specified time range. From the time range of to , the overlapping degree of the DFLW tone burst decreases linearly with time. Actually, the second harmonic generated in the time range from to is equivalent to the second harmonic generated by a continuous Lamb wave passing through the path z. The result of the analysis shown in Fig. 13 has been confirmed by the experiment.^[79]

Fig. 12.

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	Fig. 12. (color online) Group velocity dispersion curves for primary Lamb waves and DFLWs.[79]

Recently, Xiang et al. further studied the generation of cumulative second harmonic by guided wave propagation with group velocity mismatching by numerical and experimental methods.^[81] As illustrated in their results, it is found that the cumulative effect of SHG with propagation distance can be obtained for the phase matched mode pair under the condition of group velocity mismatching. Thus, it can be concluded that the integrated amplitude of the second-harmonic signal (i.e., in Fig. 13), which can be used to quantify the SHG efficiency, grows linearly with propagation distance even when the group velocity matching condition is not satisfied.

Fig. 13.

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	Fig. 13. (color online) Process of generation of the second harmonic under the condition of and .[79]