Solution of the Dynamic Interaction Problem between a Framed Structure and an Acoustic Cavity Using Imposed Deformation Functions at the Interface

Table 1

Resume of possible solutions according to $\beta$.

	$\alpha$	$G(y)$	$F(x)$
$\beta =0$	$\frac{\omega }{c}$	$C_1+C_{2}y$	$C_3·e^{i\alpha x}+C_4·e^{-i\alpha x}$
			or${}^{*}$
			$A·\cos \hspace{0.17em}(\alpha ·x)+B·\sin (\alpha ·x)$
$\beta =-{\kappa }^2$	$\sqrt{{\kappa }^2+\frac{{\omega }^2}{c^2}}$	$C_1{e}^{\kappa ·y}+C_2{e}^{-\kappa ·y}$	same for $\beta =0$
$\beta ={\kappa }^2$	$\frac{{\omega }^2}{c^2}-{\kappa }^2$	$C_1{e}^{i\kappa ·y}+C_2{e}^{-i\kappa ·y}$	$C_3·e^{i\sqrt{\alpha }x}+C_4·e^{-i\sqrt{\alpha }x}$
		or ${}^{*}$	or ${}^{*}$
		$A·\cos \hspace{0.17em}(\kappa ·y)+B·\sin (\kappa ·y)$	$C·\cos \hspace{0.17em}(\sqrt{\alpha }·x)+D·\sin (\sqrt{\alpha }·x)$

${}^{*}$Valid only for nonnull constants forming a complex conjugated pair.