17.9
Intensity and Pressure of Sound Waves
The intensity of sound waves can be related to displacement and pressure amplitudes by using their wave expressions and the definition of intensity. T…
Consider a sound wave propagating in the x-direction and the displacement of the particles denoted by the wave function y. Recall the expression for a pressure wave.
The particle velocity is given by the time derivative of the displacement, which is obtained.
Recall the expression for power as a product of force and velocity. The power per unit area is given by the product of the force per unit area—which is the pressure—and the velocity. The intensity is the time average of this quantity.
Substituting the particle velocity and pressure expressions and taking the average over time, the intensity is derived.
It is further simplified by writing the wave number in terms of the wave velocity, which is related to the bulk modulus.
Recall the relationship between the pressure and displacement amplitudes. Using these three equations, the intensity can be expressed in terms of the pressure amplitude, the density of the medium, and the wave velocity.
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Q1: How is sound intensity related to particle velocity and pressure?
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.
Q2: Why does the time average of a sinusoidal term equal zero while its square equals one-half?
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.
Q3: How does frequency affect sound intensity expressed in terms of displacement amplitude?
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.
Q4: Why is pressure amplitude more useful than displacement amplitude for describing intensity?
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.
Q5: What role does the bulk modulus play in deriving the intensity expression?
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.
Q6: How are pressure amplitude and displacement amplitude related in sound wave equations?
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.
Q7: What is the mathematical significance of averaging power over time when calculating intensity?
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.