## 2 Principles  
The acoustic nonlinearity parameter of a surface acoustic wave (\(\beta\)) can be derived in terms of the out-of-plane displacement amplitude for the fundamental and second-order harmonic frequency components of the surface acoustic wave as follows [1, 6, 18, 19],  
\[\beta=\frac{8}{k_{L}^{2}x}\frac{A_{2}}{A_{1}^{2}}\frac{k_{T}^{2}\sqrt{k_{S}^{2}- k_{L}^{2}}}{k_{S}(2k_{S}^{2}-k_{T}^{2})} \tag{1}\]  
Here, \(A_{1}\) and \(A_{2}\) are the out-of-plane displacement amplitudes of the fundamental and second-order harmonic components of surface acoustic waves, respectively; \(x\) is the wave propagation distance; and \(k_{L}\), \(k_{T}\), and \(k_{S}\) are the wavenumbers of longitudinal, transverse, and surface acoustic waves, respectively. If the term consisting only of wavenumbers in Equation (1) is defined as a parameter \(F\) as in Equation (2),  
\[F=\frac{k_{T}^{2}\sqrt{k_{S}^{2}-k_{L}^{2}}}{k_{S}(2k_{S}^{2}-k_{T}^{2})} \tag{2}\]  
Equation (1) can then be expressed as Equation (3).  
\[\beta=\frac{8}{k_{L}^{2}x}\frac{A_{2}}{A_{1}^{2}}F \tag{3}\]  
In contrast, the relative acoustic nonlinearity parameter of surface acoustic wave \(\beta^{{}^{\prime}}_{SAW}\) is defined by the voltage amplitudes as follows.  
\[\beta^{\prime}_{SAW}=\frac{A^{\prime}_{2}}{A^{\prime 2}_{1}} \tag{4}\]  
where \(A_{1}\)' and \(A_{2}\)' are the detected signal amplitudes of the fundamental and second-order harmonic components of the surface acoustic wave, respectively.  
In a previous study, it has been proved that when a laser interferometer is used as a receiver, the detected signal amplitude is proportional to the displacement amplitude regardless of the difference in materials. Thus, the VDC is not dependent on the material, and the displacement amplitudes of the fundamental and second-  
The second term of Equation (6) was obtained by substituting Equation (5) into Equation (3) for the test and reference materials. This can be simplified to the third term when the propagation distance \(x\) is fixed. The VDCs of the two materials are canceled out. The fourth term is a rearrangement of the third term using Equation (4), where \(k^{\prime}\) is a factor representing the ratio of \(F\) values in the two materials and is defined in Equation (7).  
\[k^{\prime}=\frac{F_{t}\cdot k_{r,L}^{2}}{F_{r}\cdot k_{t,L}^{2}} \tag{7}\]  
Finally, the absolute acoustic nonlinearity parameter of the test material can be determined from the relationship in Equation (6).  
\[\beta_{t}=k^{\prime}\frac{\beta_{SAW,t}^{\prime}}{\beta_{SAW,r}^{\prime}}\beta _{r} \tag{8}\]  
Equation (8) indicates that the absolute acoustic nonlinearity parameter of the test material can be estimated from the ratio of the relative acoustic nonlinearity parameters of the test and reference materials. The absolute acoustic nonlinearity parameter of the reference material as well as the wavenumber-dependent factor \(k^{\prime}\) (referred to as the wavenumber compensation factor in this paper) should be known in advance. If the test material and the reference material are similar, \(k^{\prime}\) is almost one and thus it can be ignored. However, if they are different, the wavenumbers in the two materials are different and they should be taken into account.