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Movie Title Year Distributor Notes Rev Formats Boots and Ant Flip Fuck 2007 grooby.com Facial Top Bottom O Boots in Thigh High Boots 2007 grooby.com MastOnly O Boots Strokes It 2007 grooby.com MastOnly O Boots Tickles You Pink 2007 grooby.com MastOnly O Boots Working Hard in Office 2007 grooby.com MastOnly Acoustic waves are a type of energy propagation through a medium by means of adiabatic compression and decompression. Important quantities for describing acoustic waves are acoustic pressure, particle velocity, particle displacement and acoustic intensity. Acoustic waves travel with a characteristic acoustic velocity that depends on the medium they're passing through. Some examples of acoustic waves are audible sound from a speaker (waves traveling through air at the speed of sound), ground movement from an earthquake (waves traveling through the earth), or ultrasound used for medical imaging (waves traveling through the body).
Contents 1 Wave properties 1.1 Acoustic wave equation 1.2 Phase 1.3 Propagation speed 1.4 Interference 1.5 Standing wave 1.6 Reflection 1.7 Absorption 2 See also Wave properties Acoustic waves are elastic waves that exhibit phenomena like diffraction, reflection and interference. Note that sound waves in air, however, don't have any polarization since they oscillate along the same direction as they move. Acoustic wave equation Main article: Acoustic wave equation The acoustic wave equation describes the propagation of sound waves. The acoustic wave equation for sound pressure in one dimension is given by

{\displaystyle {\partial ^{2}p \over \partial x^{2}}-{1 \over c^{2}}{\partial ^{2}p \over \partial t^{2}}=0} { \partial^2 p \over \partial x ^2 } - {1 \over c^2} { \partial^2 p \over \partial t ^2 } = 0 where {\displaystyle p}p is sound pressure in Pa {\displaystyle x}x is position in the direction of propagation of the wave, in m {\displaystyle c}c is speed of sound in m/s {\displaystyle t}t is time in s The wave equation for particle velocity has the same shape and is given by {\displaystyle {\partial ^{2}u \over \partial x^{2}}-{1 \over c^{2}}{\partial ^{2}u \over \partial t^{2}}=0} { \partial^2 u \over \partial x ^2 } - {1 \over c^2} { \partial^2 u \over \partial t ^2 } = 0 where {\displaystyle u}u is particle velocity in m/s For lossy media, more intricate models need to be applied in order to take into account frequency-dependent attenuation and phase speed. Such models include acoustic wave equations that incorporate fractional derivative terms, see also the acoustic attenuation article. D'Alembert gave the general solution for the lossless wave equation. For sound pressure, a solution would be {\displaystyle p=R\cos(\omega t-kx)+(1-R)\cos(\omega t+kx)} p = R \cos(\omega t - kx) + (1-R) \cos(\omega t+kx) where {\displaystyle \omega }\omega is angular frequency in rad/s {\displaystyle t}t is time in s {\displaystyle k}k is wave number in rad·m-1 {\displaystyle R}R is a coefficient without unit For {\displaystyle R=1}R=1 the wave becomes a travelling wave moving rightwards, for {\displaystyle R=0}R=0 the wave becomes a travelling wave moving leftwards. A standing wave can be obtained by {\displaystyle R=0.5}R=0.5. Phase Main article: Phase (waves) In a travelling wave pressure and particle velocity are in phase, which means the phase angle between the two quantities is zero. This can be easily proven using the ideal gas law {\displaystyle \ pV=nRT}\ pV = nRT where {\displaystyle p}p is pressure in Pa {\displaystyle V}V is volume in m3 {\displaystyle n}n is amount in mol {\displaystyle R}R is the universal gas constant with value {\displaystyle 8.314\,472(15)~{\frac {\mathrm {J} }{\mathrm {mol~K} }}}8.314\,472(15)~\frac{\mathrm{J}}{\mathrm{mol~K}} Consider a volume {\displaystyle V}V. As an acoustic wave propagates through the volume, adiabatic compression and decompression occurs. For adiabatic change the following relation between volume {\displaystyle V}V of a parcel of fluid and pressure {\displaystyle p}p holds {\displaystyle {\partial V \over V_{m}}={-1 \over \ \gamma }{\partial p \over p_{m}}} { \partial V \over V_m } = { -1 \over \ \gamma } {\partial p \over p_m } where {\displaystyle \gamma }\gamma is the adiabatic index without unit As a sound wave propagates through a volume, displacement of particles {\displaystyle x}x occurs along the wave propagation direction. {\displaystyle {\partial x \over V_{m}}A={\partial V \over V_{m}}={-1 \over \ \gamma }{\partial p \over p_{m}}}{\displaystyle {\partial x \over V_{m}}A={\partial V \over V_{m}}={-1 \over \ \gamma }{\partial p \over p_{m}}} where {\displaystyle A}A is cross-sectional area in m2 From this equation it can be seen that when pressure is at its maximum, displacement reaches zero. As mentioned before, the oscillating pressure for a rightward travelling wave can be given by {\displaystyle p=p_{0}cos(\omega t-kx)} p = p_0 cos(\omega t - kx) Since displacement is maximum when pressure is zero there is a 90 degrees phase difference, so displacement is given by {\displaystyle x=x_{0}sin(\omega t-kx)} x = x_0 sin(\omega t - kx) Particle velocity is the first derivative of particle displacement. Differentiation of a sine gives a cosine again {\displaystyle u=u_{0}cos(\omega t-kx)} u = u_0 cos(\omega t - kx) During adiabatic change, temperature changes with pressure as well following {\displaystyle {\partial T \over T_{m}}={\gamma -1 \over \ \gamma }{\partial p \over p_{m}}} { \partial T \over T_m } = { \gamma - 1 \over \ \gamma } {\partial p \over p_m } This fact is exploited within the field of thermoacoustics. Propagation speed Main article: Speed of sound The propagation speed, or acoustic velocity, of acoustic waves is a function of the medium of propagation. In general, the acoustic velocity c is given by the Newton-Laplace equation: {\displaystyle c={\sqrt {\frac {C}{\rho }}}\,} c = \sqrt{\frac{C}{\rho}}\, where C is a coefficient of stiffness, the bulk modulus (or the modulus of bulk elasticity for gas mediums), {\displaystyle \rho }\rho is the density in kg/m3 Thus the acoustic velocity increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of the material, and decreases with the density. For general equations of state, if classical mechanics is used, the acoustic velocity {\displaystyle c}c is given by {\displaystyle c^{2}={\frac {\partial p}{\partial \rho }}} c^2=\frac{\partial p}{\partial\rho} where differentiation is taken with respect to adiabatic change. where {\displaystyle p}p is the pressure and {\displaystyle \rho }\rho is the density Interference Interference is the addition of two or more waves that results in a new wave pattern. Interference of sound waves can be observed when two loudspeakers transmit the same signal. At certain locations constructive interference occurs, doubling the local sound pressure. And at other locations destructive interference occurs, causing a local sound pressure of zero pascals. Standing wave Main article: Standing wave A standing wave is a special kind of wave that can occur in a resonator. In a resonator superposition of the incident and reflective wave occurs, causing a standing wave. Pressure and particle velocity are 90 degrees out of phase in a standing wave. Consider a tube with two closed ends acting as a resonator. The resonator has normal modes at frequencies given by {\displaystyle f={\frac {Nc}{2d}}\qquad \qquad N\in \{1,2,3,\dots \}}f = \frac{Nc}{2d}\qquad\qquad N \in \{1,2,3,\dots\} where {\displaystyle c}c is the speed of sound in m/s {\displaystyle d}d is the length of the tube in m At the ends particle velocity becomes zero since there can be no particle displacement. Pressure however doubles at the ends because of interference of the incident wave with the reflective wave. As pressure is maximum at the ends while velocity is zero, there is a 90 degrees phase difference between them. Reflection An acoustic travelling wave can be reflected by a solid surface. If a travelling wave is reflected, the reflected wave can interfere with the incident wave causing a standing wave in the near field. As a consequence, the local pressure in the near field is doubled, and the particle velocity becomes zero. Attenuation causes the reflected wave to decrease in power as distance from the reflective material increases. As the power of the reflective wave decreases compared to the power of the incident wave, interference also decreases. And as interference decreases, so does the phase difference between sound pressure and particle velocity. At a large enough distance from the reflective material, there is no interference left anymore. At this distance one can speak of the far field. The amount of reflection is given by the reflection coefficient which is the ratio of the reflected intensity over the incident intensity {\displaystyle R={\frac {I_{\mathrm {reflected} }}{I_{\mathrm {incident} }}}}R = \frac{ I_{\mathrm{reflected}} }{ I_{\mathrm{incident}} } Absorption Acoustic waves can be absorbed. The amount of absorption is given by the absorption coefficient which is given by {\displaystyle \alpha =1-R^{2}}\alpha = 1 - R^2 where {\displaystyle \alpha }\alpha is the absorption coefficient without a unit {\displaystyle R}R is the reflection coefficient without a unit Often

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