Characteristics of Progressive Waves

Six properties together characterise any progressive (travelling) wave. First, every particle in the medium vibrates about its own mean (equilibrium) position, and every particle shares the same amplitude of vibration. Second, the phase of each individual particle's oscillation ranges continuously through the full cycle from $0$ to $2\pi$ radians as the wave passes. Third, no particle of the medium ever remains permanently at rest during wave propagation; each particle comes to instantaneous rest only twice per full oscillation cycle, at the two extreme points of its displacement (the momentary turning points of its simple-harmonic-like motion). Fourth, transverse progressive waves are visually characterised by their crests and troughs, whereas longitudinal progressive waves are instead characterised by their compressions and rarefactions. Fifth, as any given particle passes through its own mean position during oscillation, it always does so moving at its own maximum possible speed (exactly as for simple harmonic motion generally). Sixth, any two particles of the medium that are separated by a distance equal to a whole number of wavelengths, $n\la …