Sound as Pressure Waves

Sound waves are longitudinal disturbances that displace successive columns of a medium by different amounts. When one column displaces more than an adjacent column, the medium expands or compresses, creating a pressure difference from the surrounding pressure. This pressure variation propagates through the medium as the wave travels, forming the basis of sound wave behavior.

Gauge pressure in a sound wave is directly related to particle displacement through the medium's bulk modulus. Gauge pressure is maximum at points where particle displacement is zero (compression and rarefaction zones) and minimum where displacement is maximum. The pressure amplitude depends on displacement amplitude, bulk modulus, and wave number, making shorter wavelengths produce greater pressure amplitudes.

At maximum displacement points, particles in the medium have moved farthest from their equilibrium positions but are not compressing or expanding relative to surrounding columns. This creates a neutral pressure state where gauge pressure equals zero. Compression occurs at zero displacement, producing maximum positive pressure, while rarefaction produces maximum negative pressure.

Pressure amplitude is inversely proportional to wavelength. Shorter wavelengths produce greater pressure amplitudes, while longer wavelengths produce smaller pressure amplitudes. This relationship arises because pressure amplitude depends on the wave number, which is inversely related to wavelength, making high-frequency sound waves generate larger pressure fluctuations.

Bulk modulus quantifies a medium's resistance to compression and directly determines gauge pressure from particle displacement. The relationship between instantaneous displacement and gauge pressure is derived through bulk modulus, which links the material's mechanical properties to pressure fluctuations. A higher bulk modulus produces greater pressure changes for the same displacement amplitude.

Compression zones occur where medium particles aggregate closely together, producing the most positive pressure. Rarefaction zones occur where particles are farthest apart, producing the most negative pressure. Between these zones, at maximum particle displacement, pressure returns to zero, creating the oscillating pressure pattern characteristic of sound waves.

Pressure amplitude is proportional to three factors: displacement amplitude, the bulk modulus of the medium, and the wave number. Since wave number is inversely proportional to wavelength, shorter wavelengths generate larger pressure amplitudes. These relationships show that stiffer materials and higher-frequency waves produce greater pressure variations.