Propagation of flexural waves along periodic chain plate systems (Q2760578)

An infinite plate constrained by two mutually perpendicular systems of axial hinges is considered. Distances between hinges are \(a\) and \(b\). Harmonic waves propagate in the positive \(x\) direction and are \(2b\)-periodic odd functions in the \(y\) direction. The problem is reduced to finding a solution of the equation of stationary plate vibrations \(\Delta\Delta w - k^4 w = 0\), \(k^4 = \omega^2\rho h/D\) for the plate \(0\leq x\leq a\), \(0\leq y\leq b\) under proper boundary and periodicity conditions. The solution for the transverse displacement \(w\) is sought in the form \(w(x,y) = \sum_{n=1}^\infty X_n(x) Y_n(y)\) with \(Y_n(x) = \sin n\pi y/b\), \(X_n(x) = A_n\cosh\gamma_n x+ B_n\sinh\gamma_n x + C_n\cosh\mu_n x + D_n\sinh\mu_n x\), where \(\gamma_n =[(n\pi/b)^2-k^2]^{1/2}\), \(\mu_n =[(n\pi/b)^2+k^2]^{1/2}\). For different forms of plate support analytic solutions are found and dependence of the multiplicator of the problem on the wave frequency \(\omega\) is determined.