# The Effects of Stress on Second Harmonics in Plate-Like Structures
## 2 Theoretical Basis
### Amplitude Coefficient of Second Harmonic Mode  
Presume the waveguide cross-section is the y-z plane, and the wave propagation direction is x direction. The expression of \({u_{i}}^{(2)}\)[20; 37] is as follows:  
\[{u_{i}}^{(2)}=\sum\limits_{n=1}^{N}{A_{n}(x)u_{n}e^{-2\omega t}} \tag{13}\]  
where \(N\) is the number of modes with an angular frequency of \(2\omega\), \(n\) represents the nth mode at the angular frequency \(2\omega\), \(u_{n}\) is the mode shape of the nth mode, and \(A_{n}(x)\) is the amplitude function of the nth mode. The expression of \(A_{n}(x)\) is as follows:  
\[{A_{n}}(x)=\left\{ \begin{array}{ll}\frac{{f_{n}}^{surf}+{f_{n}}^{vol}}{2{P_{ mn}}({u_{n}}^{*}-2k)}\sin\!\left({\frac{1}{2}({k_{n}}^{*}-2k)x}\right){u_{n}}(x)e^{i \left((1/2)(2k+{k_{n}}^{*})x-2\omega t\right)},{k_{n}}^{*}\
eq 2k_{1}\\ \frac{{f_{n}}^{surf}+{f_{n}}^{vol}}{4{P_{mn}}}x{u_{n}}(x)e^{2i(kx-\omega t)}, {k_{n}}^{*}=2k_{1}\end{array} \right. \tag{14}\]  
where  
\[{P_{mn}}=-\frac{1}{2}\int\limits_{\Omega}{({v_{n}}^{T*}\cdot{T_{ij(n)}}^{L}) \cdot{n_{x}}d\Omega} \tag{15}\]  
\[{f_{n}}^{surf}=\int\limits_{S}{({v_{n}}^{T}\cdot{S_{ij(1)}}^{NL})\cdot\overrightarrow {n}dS} \tag{16}\]  
\[{f_{n}}^{vol}=\int\limits_{\Omega}{{v_{n}}^{T}\cdot{f_{i(1)}}d\Omega} \tag{17}\]  
where \({f_{n}}^{surf}\) and \({f_{n}}^{vol}\) represent the power flux from the primary excitation mode to the nth secondary mode through the surface and the volume of the waveguide, respectively. \({P_{mn}}\) represents the average power flow along the wave propagation direction. \({k_{1}}\) is the wavenumber of the primary excitation mode, \({k_{n}}^{*}\) is the complex conjugate of wavenumber of the nth secondary mode, \({v_{n}}\) is the particle velocity of the nth secondary mode, \({T_{ij(n)}}^{L}={T_{ij}}^{L}({u_{n}})\), \({S_{ij(1)}}^{NL}={S_{ij}}^{NL}({u_{n}}^{(1)})\), and \({f_{i(1)}}={f_{i}}({u_{n}}^{(1)})\). \({u_{n}}^{(1)}\) is the mode shape of the primary excitation mode corresponding to the nth secondary mode. \({n_{x}}\) is the unit vector in the wave propagation direction. \(\overrightarrow{n}\) is the unit vector normal to the surface of the waveguide, \(S\), and \(\Omega\) is the waveguide cross-section area. The amplitude coefficient of the second harmonic mode, \(\overline{A}_{n}(x)\), can be defined as follows:  
\[\overline{A}_{n}(x)=\left\{ \begin{array}{ll}\frac{{f_{n}}^{surf}+{f_{n}}^{vol }}{2{P_{mn}}({k_{n}}^{*}-2k)}\sin\!\left({\frac{1}{2}({k_{n}}^{*}-2k)x} \right)\!\left|{k_{n}}^{*}\
eq 2k_{1}\\ \left|{\frac{{f_{n}}^{surf}+{f_{n}}^{vol}}{4{P_{mn}}}x}\right|{k_{n}}^{*}=2k_{1 }\end{array} \right. \tag{18}\]  
where \(\overline{A}_{n}(x)\) determines how large the amplitude of the second harmonic mode is [37]. Equation (18) shows that, for the mode with an angular frequency of \(2\omega\), the amplitude will increase with the rise of the propagation distance only when the following two conditions are satisfied:  
* Phase matching: \({k_{n}}^{*}=2k_{1}\);
* Non-zero power flux: \({f_{n}}^{surf}+{f_{n}}^{vol}\
eq 0\). At present, in the researches and applications of nonlinear ultrasonic guided waves, group velocity matching does not represent a necessary requirement for cumulative second-harmonic generation. For this reason, only phase matching and power transfer are considered in detail in this article. The amplitude coefficient of second harmonic mode, \(\overline{A}_{n}(x)\), is directly related to the mode shapes of fundamental mode and second harmonic mode. The relevant formulas are as follows:  
\[v_{n}=-i2\omega u_{n} \tag{19}\]  
\[{T_{ij(n)}}^{L}={T_{ij}}^{L}(u_{n})=\lambda(E_{1}\cdot a)I_{3\times 3}+\mu(b+b^{ T}) \tag{20}\]  
\[ \begin{array}{l}{S_{ij(1)}}^{NL}={S_{ij}}^{NL}({u_{n}}^{(1)})=(\frac{3}{2}(c^ {T}\cdot c)+C(E_{1}\cdot c)^{2})I_{3\times 3}+B(E_{1}\cdot c)d^{T}+\frac{A}{4}(d^{T} \cdot d^{T})\\ \quad+\frac{B}{2}(c^{T}\cdot c+c^{T}\cdot E_{2}\cdot c)I_{3\times 3}+(\lambda+B)(E_{1} \cdot c)d+(\mu+\frac{A}{4})(d\cdot d^{T}+d^{T}\cdot d+d\cdot d)\end{array}  \tag{21}\]  
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