# Acoustic Method of Quality Control of Two-Component Composite Materials
## 3 Determination of the Component Concentration of a Monodisperse Composite
### Long Wave Approach  
For the acoustic sounding technique, it is advisable to consider the asymptotic behavior of the waveguide frequencies and phase velocities of the creeping mode, provided that the wavelength is significantly greater than the spatial period of the bubble chain. The wavelength \(L_{w}\) of the creeping waveguide mode corresponding to the waveguide frequency \(\lambda_{1}(\xi,k,\tau)\), has the form \(L_{w}=2\pi/\xi\).  
At large values \(L_{w}\), the wave number of the waveguide mode is close to zero. For small \(\xi\) (6) takes the form:  
\[\lambda_{1}(\xi,k,\tau)=\xi\sqrt{\tau}/\sqrt{(k+\tau-k\tau)(k\tau-k\kappa+k^{2 })}. \tag{10}\]  
Dimensionless phase velocity \(C_{ph}^{(1)}(\xi,k,\tau)\) in the propagation of a long wave in the first bandwidth is defined as \(C_{ph}^{(1)}(\xi,k,\tau)=\lambda_{1}(\xi,k,\tau)/\xi\), and for small values of \(\xi\), the dimensionless wavenumber and small values of \(\tau\), have the form:  
\[C_{ph}^{(1)}(\xi,k,\tau)=\sqrt{\tau}/\left\lfloor k\sqrt{k(1-k)}\right\rfloor =(c_{2}/c_{1})\sqrt{\tau/k(1-k)} \tag{11}\]  
The phase velocity of the creeping mode depends only on the concentration, the ratio of the speeds of sound, and the ratio of the densities of the two media forming the chain. This allows using the known phase velocity of long waves and the known properties of the components of the medium to determine the concentration of the components using the relation [10]:  
\[k(1-k)=\tau\Big{[}C_{ph}^{(1)}(\xi,k,\tau)/(c_{2}/c_{1})\Big{]}^{-2} \tag{12}\]  
Determination of the Concentration of Components from the Known Acoustic Phase Velocity of Long Waves in Composite Materials  
Let one period of the chain contain two inclusions of the medium M1 (water) with sizes \(k_{1}\) and \(k_{3}\) (\(k_{1}+k_{3}=k\)), distances between them--\(k_{2}\) and \(k_{4}\) (Figure 2) [6].  
Elementary wave packets are described in one spatial period by relations (1), (2), and phase shift conditions (4) or equivalent (7), which are equivalent to a system of eight equations for eight unknowns.  
The smallest waveguide frequency of oscillations of a polydisperse chain is found by direct calculation; for small values of frequency, this frequency has the form:  
\[\lambda_{1}(\xi,k,\tau)=\sqrt{2\tau[1-cos(\xi)]/\left[(k+\tau-k\tau)(k\tau-k^{ 2}+k^{2})\right]} \tag{13}\]  
The expression for the phase velocity is obtained by direct calculation using dispersion relations for low frequencies and for long waves in a polydisperse phonon crystal with two and three different dimensions of the reinforcement and, accordingly, the matrix:  
\[U_{phase}=\sqrt{\frac{\tau}{(k\tau-k-\tau)(\kappa^{2}k-k\tau-\kappa^{2})}} \tag{14}\]  
This means that the concentration of component 1 in the composite material is fully determined by the quadratic equation for its calculation:  
\[\Big{(}\kappa^{2}\tau-\kappa^{2}-\tau^{2}+\tau\Big{)}k^{2}+\Big{(}-2\kappa^{2 }\tau+\kappa^{2}+\tau^{2}\Big{)}k+\kappa^{2}\tau-\frac{\tau}{U_{phase}^{2}}=0. \tag{15}\]  
Since the phase velocity \(U_{phase}\) is known, this relationship allows one to determine the concentration of the components from the known phase velocity of long waves in the composite material.  
It should be noted that the influence of the polydispersity of the distribution of the components of the matrix and reinforcing material in this case is insignificant.