This paper presents a new approach to solving acoustic scattering problems: the Unified Transform method. This spectral, boundary-based collocation method can be readily applied to acoustic scattering by disjoint two-dimensional structures, and, for the purposes of this paper, is illustrated in the case of multiple flat plates, which also addresses the additional difficulty of mathematical singularities in the scattered field due to diffraction at sharp edges. Fluid-structure interaction may also be incorporated into the method, such as plate elasticity, which when applied to aerofoil trailing edges, is known to reduce aerodynamic noise. While a range of examples are illustrated to show the versatility of the method, attention is in particular given to the scattering of quadrupole sources by rigid plates with finite elastic extensions. It is seen that whilst a fully elastic plate is most beneficial acoustically, plates with only small extensions can considerably reduce the far-field sound power versus a fully rigid plate.