Because of the increasing concern with vehicle weight, there is an interest in lightweight materials that can serve several functions at once. Here we consider the vibration damping performance provided by an “acoustical” material (i.e., a fibrous layer that would normally be used for airborne noise control). It has been previously established that the vibration of panel structures creates a non-propagating nearfield in the region close to the panel. In that region, there is an oscillatory, incompressible fluid flow parallel to the panel whose strength decays exponentially with distance from the panel. When a fibrous medium is placed close to the panel in the region where the oscillatory nearfield is significant, energy is dissipated by the viscous interaction of the flow and the fibers, and hence the panel vibration is damped. The degree of panel damping is then proportional to the energy removed from the nearfield by the viscous interaction with the fibrous medium. In his paper, experiments are described that demonstrate this effect. Fibrous layers were placed next to a lightly damped panel driven by a shaker, and the vibration of the panel was quantified by using a scanning laser vibrometer. These experiments showed that it is possible to achieve a strong damping effect by using fibrous layers. In addition a new theory that can be used to predict the depth of treatment needed to achieve a damping effect is presented. The theory is based on analyzing the wave number transforms of the panel motion in terms of radiating and non-radiating components, and by using that approach to identify the spatial extent of the oscillatory nearfield, and hence the depth of the fibrous layer required to provide effective structural damping.