Intensity and Pressure of Sound Waves

Sound intensity is the time average of power per unit area, calculated as the product of pressure and particle velocity. The particle velocity is derived from the time derivative of particle displacement. By substituting pressure and velocity expressions into the power equation and averaging over time, intensity can be expressed in terms of pressure amplitude, medium density, and wave velocity.

A sinusoidal term oscillates equally between positive and negative values over each cycle, so its time average is zero. However, the square of a sinusoidal term is always positive, yielding a time average of one-half. This distinction is critical when deriving intensity expressions, as squaring eliminates the oscillating component and captures the sustained energy transfer.

Sound intensity expressed via displacement amplitude depends directly on frequency. High-frequency waves produce the same intensity as low-frequency waves even with smaller amplitude vibrations because intensity increases with the square of frequency. This frequency dependence explains why high-pitched sounds can be perceived as intense despite small particle displacements.

Sound intensity expressed through pressure amplitude is independent of wave frequency, making it easier to relate intensity directly to medium properties like density and wave velocity. This frequency-independent relationship simplifies discussions of how intensity varies with the medium's physical characteristics, unlike displacement-based expressions that require frequency information.

The bulk modulus relates to wave velocity, which is essential for expressing intensity in terms of pressure amplitude and medium properties. By writing the wave number in terms of wave velocity and connecting it to the bulk modulus, the intensity formula becomes independent of frequency and directly dependent on the medium's resistance to compression.

Pressure amplitude and displacement amplitude are connected through the wave properties of the medium. The relationship between these amplitudes allows intensity to be expressed in either form. Using this connection, intensity can be derived from displacement amplitude or pressure amplitude, depending on which form is more convenient for the application.

Averaging power over time eliminates oscillations and captures the sustained energy delivery of the sound wave. Since instantaneous power varies sinusoidally, the time average provides a constant, meaningful measure of energy transfer. This averaging process is essential for defining intensity as a practical, measurable quantity independent of instantaneous fluctuations.