17.2
Sound as Pressure Waves
Sound waves, which are longitudinal waves, can be modeled as the displacement amplitude varying as a function of the spatial and temporal coordinates.…
Consider sound traveling through a medium. The longitudinal disturbances create a pressure difference between its successive columns, which then undergo oscillations.
Consider an undisturbed cylinder of the medium, with cross-sectional area A along the x-axis. Its longitudinal displacement, given by y, is a wave function.
As the wave travels, its ends at x-1 and x-2 are displaced by y-1 and y-2, respectively. If the latter is greater, the cylinder expands, and the pressure falls from the surrounding pressure.
Its initial volume is known, and its change in volume is derived. The fractional change in volume is then obtained.
Recall the definition of bulk modulus, from which the gauge pressure is obtained. On simplification, the gauge pressure is observed to be a wave.
The gauge pressure is maximum at points of zero displacement and minimum at points where the displacement is maximum.
Its amplitude is proportional to the displacement amplitude, the bulk modulus of the medium, and the wave number. So, it is inversely proportional to the wavelength.
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Q1: How do sound waves create pressure differences in a medium?
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.
Q2: What is the relationship between displacement and gauge pressure in sound waves?
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.
Q3: Why is gauge pressure zero at maximum displacement points?
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.
Q4: How does wavelength affect pressure amplitude in sound waves?
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.
Q5: What role does bulk modulus play in sound wave pressure?
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.
Q6: How do compression and rarefaction zones differ in pressure?
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.
Q7: What determines the amplitude of pressure fluctuations in 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.