Tuesday, August 16, 2011
Sound Waves Sound is made of longitudinal mechanical waves. Mechanically, the molecules of some medium are induced to oscillate with a certain frequency and amplitude, along the “longitude” or axis of propagation (as opposed to transverse waves, where the oscillations are perpendicular to the axis of propagation). There are two ways to describe the motion of sound waves - through the displacement (position) of the molecules of the medium or the pressure caused by the moving molecules. The relationship between them is that where the pressure amplitude is the highest (a wave crest or trough) the displacement amplitude is zero, and vice versa. Points of zero amplitude oscillation, where a molecule has returned to the equilibrium position or the pressure is at the equilibrium pressure, are called nodes, while points of maximum amplitude (both crests and troughs) are called antinodes. A pressure antinode always coincides with a displacement node. You can see this by putting the equations for the pressure and displacement amplitudes next to each other. Let x and t be the place and time that you are measuring the oscillation amplitudes, ω and k be the frequencies in radians/sec and radians/meter, and p0 and x0 be the equilibrium pressure and position: Ap(x,t) = P*cos(ωt + kx) + p0As(x,t) = S*cos(ωt + kx ± π/2) + x0 The constants ω and k are related to the frequency f and wavelength λ of a sound wave by f (waves/sec or Hz) = ω (radians/sec) / 2π (radians/wave) = ω/2πλ (meter/wave) = 2π (radians/wave) / k (radians/meter) = 2π/k Following the wave model, sound has a constant speed in a certain medium and its wavelength is proportional to its frequency: λf = vs where vs is the speed of sound.

Sound Waves

Sound is made of longitudinal mechanical waves. Mechanically, the molecules of some medium are induced to oscillate with a certain frequency and amplitude, along the “longitude” or axis of propagation (as opposed to transverse waves, where the oscillations are perpendicular to the axis of propagation). There are two ways to describe the motion of sound waves - through the displacement (position) of the molecules of the medium or the pressure caused by the moving molecules. The relationship between them is that where the pressure amplitude is the highest (a wave crest or trough) the displacement amplitude is zero, and vice versa. Points of zero amplitude oscillation, where a molecule has returned to the equilibrium position or the pressure is at the equilibrium pressure, are called nodes, while points of maximum amplitude (both crests and troughs) are called antinodes. A pressure antinode always coincides with a displacement node.

You can see this by putting the equations for the pressure and displacement amplitudes next to each other. Let x and t be the place and time that you are measuring the oscillation amplitudes, ω and k be the frequencies in radians/sec and radians/meter, and p0 and x0 be the equilibrium pressure and position:

Ap(x,t) = P*cos(ωt + kx) + p0
As(x,t) = S*cos(ωt + kx ± π/2) + x0

The constants ω and k are related to the frequency f and wavelength λ of a sound wave by

f (waves/sec or Hz) = ω (radians/sec) / 2π (radians/wave) = ω/2π
λ (meter/wave) = 2π (radians/wave) / k (radians/meter) = 2π/k

Following the wave model, sound has a constant speed in a certain medium and its wavelength is proportional to its frequency:

λf = vs

where vs is the speed of sound.

Posted at 8:06 PM 42 notes Tags: antinodes dimensional analysis longitudinal wave mechanical wave nodes physics position wave pressure wave propagation of sound sound sound waves standing waves waves science

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