## 1 Introduction  
The membranes of variable shape are used both indoors and outdoors where membranes have become part of roofing and shielding elements as well as of facades. A wide range of solutions for structural skins exists, whether for single, second skin or double skin facades, atria coating or exterior environment covering [1]. Moreover, lightweight solutions such as membrane-based constructions are considered as a sustainable solution in architecture [2]. From an acoustic point of view, every new construction solution brings specific pitfalls associated with it. For example, the impact of the frequently used double transparent facades (DTF) on administrative buildings for sound insulation improves the sound reduction index, but has possible negative effects in the low frequency range caused by standing wave resonances occurring in the air cavity inside the DTF. DTFs can also cause problems related to speech privacy in the interior caused by flanking transmission paths between attached offices, which can be problematic mainly in cases of low background noise level in the DTF cavity [3, 4]. Nowadays, when the influence of facades on environmental noise is increasingly being discussed, adaptive facade (or smart facade-facades that adapt to the climate and environment through their skin) solutions with implemented membrane structures are a topic of growing interest. The properties of membrane structures are very dependent on their mass, dimensions and mechanical stresses. The present work was motivated by the idea of [PERSON] et al. [5], where authors investigated in the determination of the sound energy absorbed by the plant leaves based on the measured surface velocity and sound pressure. The goal was to verify it that would be possible to apply the proposed theory in an experimental way. The theory of circular and annular membranes was described by [PERSON] [6]. For more complex shapes, several approximate methods have been proposed by [PERSON] [7]. Later, the impact of the circular core fixed to the membrane was investigated [8]. A number of previous works were published with a focus on the variations of the membrane boundary conditions. For example, [PERSON] investigated the impact of a pinned large axisymmetric mass at the centre of the membrane on its natural frequency [9; 10]. These kinds of membranes were also called membrane-type metamaterial (MAMs) with so-called negative dynamic mass [11; 12; 13; 14]. By stacking together membrane panels with attached mass (circular shaped weights) in the centre of membrane surfaces, the sound transmission loss of the membrane composition can be increased in a broad band frequency range from 50 to 1000 [15]. The effect of the mass of the membrane and the membrane tension on the transmission loss was investigated by a number of researchers [15; 16]. Normally, the bending stiffness of membranes can safely be ignored. The restoring force arises entirely from the applied tension, not from bending stiffness as is the case for plates. Relatively popular MAMs or locally resonant sonic materials (LRSMs) contain arrays of elastic resonators composed of a heavy core surrounded by a soft coating layer. In principle, by loading the membrane by a relatively heavy object the so-called dipole resonance occurs. The dipole resonance enhances the acoustic performance [17; 18]. These resonant materials are able to control low-frequency sound reflection and transmission very effectively. The negative dynamic mass causes the subwavelength attenuation of sound in the audible frequency band and breaks the mass density law. In the frequency range from 100 to 1000 Hz, a significant increase of the normal incidence sound transmission loss (_nTL_) can be achieved [19]. Spatially averaged force and acceleration are opposite in phase, which leads to near-total reflection (anti-resonance) at the frequency between two eigenmodes. In this region, the in-plane average of normal displacement is close to zero [19].