Sound, physics standing waves

This model is a physically motivated improvement over the Einstein crystal model. Debye sought a realistic way to assign different frequencies to the vibrational normal modes. He assumed that the normal modes could be represented by standing waves that vanish at the surfaces of the crystal. The quanta of energy of these waves are called phonons since the waves are essentially sound waves. Consider a cubic crystal with side L. The amplitude of a standing wave that vanishes at the boundaries of the cube is represented by... 

The familiar test for antinodes due to standing sound waves in a closed horizontal cylinder is well known to students of elementary physics. Fine dust initially scattered through the cylinder gathers at... 

In a mathematical sense, a periodic wave is any function f(x) whose value varies in a repetitive and perfectly predictable manner over discrete intervals of some variable x. A physical way of describing waves is that they are some property of the medium in which they exist that changes in a regular and periodic manner as a function of the distance from some point, or as a function of time if one stands at a fixed point in space and measures the unique property. For sound, the property may be pressure for waves in the water, it may be height above or below the surface for light or X rays, the properties are electromagnetic. 

Coherent reflections at the top and bottom boundaries of the plate give way for a set of standing acoustic waves between the two main surfaces of the plate. Due to the piezoelectric nature of quartz two sets of resonance frequencies exist for each mode, depending on the electrical boundary conditions. The first set corresponds to a plate with open-circuit boundary conditions. From the physical point of view charges will be collected on the electrodes building up a potential difference and hence an electrical field from the electrical point of view the electrodes are unconnected. This resonance is termed anti-resonance in the piezoelectric literature and parallel resonance in electronics literature. The second set of resonance frequencies corresponds to a plate with short-circuit boundary conditions. The electrodes are connected and a potential difference cannot be built up. The respective names are resonance in piezoelectric and series resonance in electronics Hterature. The differences arise from piezoelectric stiffening accompanied by differences in the sound velocity. The anti-resonance (parallel) frequencies of each of the three acoustic modes are completely decoupled giving ... 

Borisov Yu Ya, Gynkina NM. On acoustic drying in a standing sound wave. Soviet Physics-Acoustics 8(1) 129-131, 1962. 

---