The change in pressure wave resulting from the movement of a particle in a sound wave can be derived using the principles of wave propagation and fluid dynamics. Here's a simplified explanation of the process:
Understanding Sound Waves: Sound waves are mechanical waves that propagate through a medium, typically air, as a series of compressions (regions of high pressure) and rarefactions (regions of low pressure). These waves are generated by the vibration of a source, such as a loudspeaker or a musical instrument.
Particle Displacement: As a sound wave travels through a medium, the individual particles of the medium (air molecules, in the case of air) oscillate back and forth around their equilibrium positions. This oscillation is in the same direction as the wave propagation. The displacement of a particle from its equilibrium position at a given time determines its instantaneous pressure.
Relationship between Particle Displacement and Pressure: According to the principles of fluid dynamics, the pressure exerted by a fluid (in this case, air) is directly proportional to the force exerted by the fluid on a surface. The force, in turn, is related to the displacement of the particles from their equilibrium positions.
Wave Equation: The relationship between pressure and particle displacement in a sound wave is described by the wave equation, which is a partial differential equation that governs the behavior of the wave. The wave equation takes into account factors such as wave speed, wavelength, and frequency.
Derivation: The precise derivation of the change in pressure wave from the movement of a particle in a sound wave involves solving the wave equation, which is beyond the scope of a simple explanation. It requires advanced mathematical techniques and knowledge of differential equations.
In summary, the change in pressure wave resulting from the movement of a particle in a sound wave is a complex process involving the principles of wave propagation and fluid dynamics. While a simplified explanation helps provide an intuitive understanding, the precise derivation requires more advanced mathematical analysis.