Побудова математичних моделей збудження хрестоподібних поверхневих хвиль у воді в «співаючих бокалах» при випромінюванні бокалами звукових хвиль

Abstract

The appearance and structure of cross-shaped waves on the free surface of the liquid contained in the ‘singing glass’ is explained. It was found that the main mode of vibration of the glass is a forced wave, with four nodes evenly spaced along the cylindrical glass wall. When such a singing glass is partially filled with a liquid (which changes the frequency of sound), one can observe another amazing feature: a complex pattern of waves on the free surface of the liquid. With sufficient illumination, one can observe waves directly caused by the vibration of the wall in the four-node mode moving in the circumferential direction at the same speed as the moving finger. However, this picture is dominated by so-called ‘cross waves’ or ‘edge waves’, the crests of which are perpendicular to the vibrating glass wall. Although cross-shaped waves were described by Faraday (1831), until now there was no theoretical justification for the possibility of their excitation in a ‘singing glass’. This paper fills this gap: the appearance of cross waves in a ‘singing glass’ has been mathematically described as resonant oscillations. Non-linear parametric equations describing them were obtained for cross-shaped waves. Two mathematical models were built. The first theoretical model describes the appearance of cross-shaped waves, where only one eigenmode is present on the free surface of the liquid, and the second model applies to the situation where at least two or three eigenmodes are present. The oscillations of the free surface in approximations with three eigenmodes reflect the main features of the wave patterns observed experimentally in the ‘singing glass’.