## 3 Propagation of Particle Rotation Waves
### Wave Equations  
By substituting the macroscale constitutive relations associated with particle rotation, Equations (23)-(25), into the governing equations, Equations (10c) and (10d), and ignoring the body forces and moments, we can obtain the following 12 wave equations:\[[(5+2a)(b_{1}+p_{1})+(16+10a)(b_{2}+p_{2})]\omega_{1,11}-[(5+2a)p_{1}+(16+10a)p_{2} ]B_{11,1}-(7+a)p_{2}(B_{22,1}+B_{33,1})=\frac{\rho\beta}{l^{2}N_{p}}\bar{\omega}_ {1}, \tag{26}\]  
\[[(5+2a)(b_{1}+p_{1})+(2-a)(b_{2}+p_{2})]\omega_{2,11}-[(5+2a)p_{1}+(2-a)p_{2}]B_ {21,1}-(7+a)p_{2}B_{12,1}=\frac{\rho\beta}{N_{p}l^{2}}\bar{\omega}_{2}, \tag{27}\]  
\[[(5+2a)(b_{1}+p_{1})+(2-a)(b_{2}+p_{2})]\omega_{3,11}-[(5+2a)p_{1}+(2-a)p_{2}]B _{31,1}-(7+a)p_{2}B_{13,1}=\frac{\rho\beta}{N_{p}l^{2}}\bar{\omega}_{3}, \tag{28}\]  
\[\frac{l^{2}}{4}[(21w_{1}+114w_{2}+12w_{1}a+78w_{2}a)B_{11,11}+(27w_{2}+9w_{2}a )(B_{22,11}+B_{33,11})]\]  
\[+[(5p_{1}+16p_{2}+4p_{1}a+8p_{2}a)\omega_{1,1}-(5p_{1}+16p_{2}+4p_{1}a+8p_{2}a )B_{11}-(7p_{2}+p_{2}a)(B_{22}+B_{33})]=\frac{\rho T}{l^{2}N_{p}}\bar{B}_{11}\ \, \tag{29}\]  
\[\frac{l^{2}}{4}[(7w_{1}+20w_{2}-w_{2}a+w_{1}a)B_{22,11}+(27w_{2}+9w_{2}a)B_{11,11}+9w_{2}B_{33,11}]+(7p_{2}+p_{2}a)(\omega_{1,1}-B_{11})-\ \, \tag{30}\]  
\[(5p_{1}+16p_{2}-2p_{1}a-4p_{2}a)B_{22}-(7p_{2}-2p_{2}a)B_{33}=\frac{\rho T}{l ^{2}N_{p}}\bar{B}_{22}\]  
\[\frac{l^{2}}{4}[(7w_{1}+20w_{2}-w_{2}a+w_{1}a)B_{33,11}+(27w_{2}+9w_{2}a)B_{11,11}+9w_{2}B_{22,11}]+(7p_{2}+p_{2}a)(\omega_{1,1}-B_{11})-\ \, \tag{31}\]  
\[(5p_{1}+16p_{2}-2p_{1}a-4p_{2}a)B_{33}-(7p_{2}-2p_{2}a)B_{22}=\frac{\rho T}{l ^{2}N_{p}}\bar{B}_{33}\]  
\[\frac{l^{2}}{4}[7w_{1}+20w_{2}+8w_{2}a+w_{1}a]B_{12,11}+\frac{l^{2}}{4}(27+9u _{2})w_{2}B_{21,11}+(7+a)p_{2}(\omega_{2,1}-B_{21})-(5p_{1}+2p_{2}+p_{1}a)B_{ 12}=\frac{\rho T}{l^{2}N_{p}}\bar{B}_{12}, \tag{32}\]  
\[\frac{l^{2}}{4}[\left[ \begin{array}{c}3(7w_{1}+2w_{2}-w_{2}a+4w_{1}a)B_{21,11}\\ +3(9w_{2}+3w_{2}a)B_{12,11}\end{array} \right]+[(5p_{1}+2p_{2}+p_{1}a)(\omega_{ 2,1}-B_{21})-(7p_{2}+p_{2}a)B_{12}]=\frac{\rho\gamma}{l^{2}N_{p}}\bar{B}_{21}, \tag{33}\]  
\[\frac{l^{2}}{4}(7w_{1}+20w_{2}+8w_{2}a+w_{1}a)B_{13,11}+\frac{l^{2}}{4}(27+9u _{2})w_{2}B_{31,11}+(7+a)p_{2}(\omega_{3,1}-B_{31})-(5p_{1}+2p_{2}+p_{1}a)B_{ 13}=\frac{\rho\gamma}{l^{2}N_{p}}\bar{B}_{13}, \tag{34}\]  
\[\frac{l^{2}}{4}[\left[ \begin{array}{c}3(7w_{1}+2w_{2}-w_{2}a+4w_{1}a)B_{31,11}\\ +3(9w_{2}+3w_{2}a)B_{13,11}\end{array} \right]+[(5p_{1}+2p_{2}+p_{1}a)(\omega_{ 3,1}-B_{31})-(7p_{2}+p_{2}a)B_{13}]=\frac{\rho\gamma}{l^{2}N_{p}}\bar{B}_{31}, \tag{35}\]  
\[\frac{l^{2}}{4}[(7w_{1}+2w_{2}-w_{2}a+w_{1}a)B_{23,11}+9w_{2}B_{32,11}]-[(5p_{1 }+2p_{2}-2p_{1}a)B_{23}+(7-2a)p_{2}B_{32}]=\frac{\rho\gamma}{l^{2}N_{p}}\bar{B }_{23}, \tag{36}\]  
\[\frac{l^{2}}{4}[(7w_{1}+2w_{2}-w_{2}a+w_{1}a)B_{32,11}+9w_{2}B_{23,11}]-[(5p_{1 }+2p_{2}-2p_{1}a)B_{32}+(7-2a)p_{2}B_{23}]=\frac{\rho\gamma}{l^{2}N_{p}}\bar{B }_{32}. \tag{37}\]  
To simplify the derivation without the loss of generality, here we consider the case where the rotation waves propagate in one direction only (plane waves). Hence, the general solutions of above wave equations can be assumed in a harmonic form as follows:  
\[\mathbf{\omega}=\mathbf{\Omega}\text{i}e^{\text{i}(kx_{1}-\omega t)},\mathbf{\mathsf{B}}= \mathbf{\Theta}\text{e}^{\text{i}(kx_{1}-\omega t)}, \tag{38}\]  
where \(\mathbf{\Omega}\) and \(\mathbf{\Theta}\) are the amplitudes of macroscale rotation and the gradient of rotation, respectively, \(\omega\) is the angular frequency, and \(k\) is the wave number.