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Wave equation in fluids ELEC-E5610 Acoustics and the Physics of Sound, Lecture 5 Henna Tahvanainen Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering November 10, 2015 1 Fluids Wave equation in fluids Henna Tahvanainen Aalto SPA 2/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Fluids Fluids Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity Fluids consist of “particles that easily move and change their relative position without a separation of the mass and that easily yield to pressure” (Merriam-Webster) in practice, gases and liquids gases fill all the volume they’re enclosed in negligible particle interaction unless they collide the relative deformations of volume and pressure about the same magnitude liquids certain volume in a certain temperature strong interaction between particles the relative deformation of volume smaller than deformation of pressure Wave equation in fluids Henna Tahvanainen Aalto SPA 3/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Ideal Fluid Fluids Ideal Fluid Ideal Gas The ideal fluid is a simplification of real fluids. In particular, it possesses some interesting properties: Continuums Wave Equation Sound Velocity it is lossless zero viscosity zero shear stress no heat conduction Bose-Einstein condensates (a.k.a superfluids, e. g. Helium below 2.2K ◦ ) fully act as ideal fluids (http://www.youtube.com/watch?v=2Z6UJbwxBZI). Wave equation in fluids Henna Tahvanainen Aalto SPA 4/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Ideal Gas Fluids Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity The ideal gas is an ideal fluid that obeys the ideal gas law: PV = constant = nR , T where P is pressure, V is volume, and T is temperature,n is number of molecules, R is universal gas constant. Although the ideal gas in an idealization, it is a good approximation to the behavior of many gases under many conditions. Remember that pressure in gases arises from random molecular motion, it is scalar Wave equation in fluids Henna Tahvanainen Aalto SPA 5/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Ideal gas Written in another way: Fluids P = R ρT , Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity We see that for constant density, pressure increases with temperature (the molecules have a higher average kinetic energy -> a greater push on their surroundings) for constant temperature, pressure increases with density (the more molecules per unit volume, the greater the push exerted by collisions) for constant pressure, temperature and density are inversely related (fewer molecules in a given volume, need to travel at a greater average speed to exert the same pressure) Wave equation in fluids Henna Tahvanainen Aalto SPA 6/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Continuums Fluids Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity according to our current understanding, all matter consists of particles in theory, sound propagation laws could be derived from the “averaged” conservation laws of mass, momentum, and energy for these particles however, a more convenient model can be obtained by considering the medium as a continuum ⇒ forget the molecular structure of the fluid. Valid assumption, if Λ 1, L where Kn is Knudsen’s number, Λ is the average free space between molecular collisions, and L is the length in the dimension of observation. Kn = Wave equation in fluids Henna Tahvanainen Aalto SPA 7/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Continuums II Fluids Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity in the standard atmosphere Λ = 6 × 10−8 m for sound fields L = λ can the air be considered continuous for audio frequencies? upper limit of audio frequencies f = 20 kHz ⇒ λ = 2×c104 ≈ 0.0172 m 6 × 10−8 ≈ 3.5 × 10−6 1 L 0.0172 ⇒ yes, air can be considered continuous for audio frequencies! Kn = Λ = Wave equation in fluids Henna Tahvanainen Aalto SPA 8/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Disturbance Fluids Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity Sound is a disturbance of air density and pressure it is so rapid that no flow occurs no heat convection between compression and rarefaction occurs in a linear case we assume that the disturbance is the same regardless of the static air pressure So, let’s add a small disturbance to the static pressure P0 and density ρ0 P = P0 + Pe (1) ρ = ρ0 + ρe (2) so that P0 Pe and ρ0 ρe Wave equation in fluids Henna Tahvanainen Aalto SPA 9/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Features Fluids Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity The physics of the sound waves involves three features I The gas moves and changes density II The change in density corresponds to a change in pressure (adiabatic process) III Pressure inequalities (spatial pressure gradients) generate gas motion Let’s solve for one-dimensional case as group work: 1. Divide into groups of three (should be six groups) 2. Each group takes one feature (two groups per feature) 3. There is prematerial for each feature. 4. Work out the math and physics of the feature so that you can explain it to another group. 5. Let’s come back here in 30 min. Wave equation in fluids Henna Tahvanainen Aalto SPA 10/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Explaining features Fluids Ideal Fluid Ideal Gas 1. Regroup according to 1-2-3, 1-2-3 2. First Group 1 explains their feature, then Group 2.. Continuums Wave Equation Sound Velocity Wave equation in fluids Henna Tahvanainen Aalto SPA 11/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Combining the features III Spatial gradients generate gas motion Fluids Ideal Fluid Ideal Gas ρ ∂2x ∂ Pe =− 2 ∂t ∂x Continuums Wave Equation Sound Velocity combined with II Change in density is change in pressure Pe = κρe ∂2x ∂ρe ρ 2 = −κ ∂t ∂x and finally I Gas moves and changes density ρe = −ρ0 ∂∂Xx ∂2X ∂2X = κ ∂t 2 ∂x 2 where κ = 1/c 2 = ρ/γ P for an adiabatic process Wave equation in fluids Henna Tahvanainen Aalto SPA 12/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Combining features Fluids Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity So, our one-dimensional wave equation for a fluid looks very familiar: ∂2X ∂2X = κ ∂t 2 ∂x 2 We have the inertial force from the fluid mass, and restoring force from the pressure Note that we now derived the equation for displacement, the same applies for pressure https://vimeo.com/59422886 -> Check for alternative in Fahy’s book Chapter 3. Wave equation in fluids Henna Tahvanainen Aalto SPA 13/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Adiabatic process - nonlinearity Fluids Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity Note that in the adiabatic process the pressure-density relation is not linear P = αργ (we can also draw many isothermal curves, as γ depends on temperature) Wave equation in fluids Henna Tahvanainen Aalto SPA 14/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Sound Velocity Fluids Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity c= q K ρ, how to get K in practice? depends on the thermodynamic behaviour of the medium isothermal: heat transfer between nodes and antinodes, K = P0 , P0 is static air pressure adiabatic: no heat transfer, K = γ P0 , γ is the adiabatic index (gas property) normally, air behaves as adiabatic (contrary to what Newton thought) near solids and in porous materials air behaves as isothermal ⇒ porosity slows down sound Wave equation in fluids Henna Tahvanainen Aalto SPA 15/21 November 10, 2015 ELEC-E5610 Lecture 5 1 Sound Velocity in Liquids Fluids Ideal Fluid Ideal Gas Continuums Wave Equation Sound Velocity only small difference between adiabatic and isothermal compression special case: liquids with bubbles gas determines compressibility (high ⇒ small bulk modulus) the liquid determines the density (high for liquids) the effect of dissolved gases is low... ...except for very high sound pressures ⇒ acoustic cavitation! the dissolved gases form a bubble at the antinode sonoluminescence, bubble fusion (http://www.youtube.com/watch?v=LWO93G-zLZ0) Wave equation in fluids Henna Tahvanainen Aalto SPA 16/21 November 10, 2015 ELEC-E5610 Lecture 5 2 Loss Mechanisms Wave equation in fluids Henna Tahvanainen Aalto SPA 17/21 November 10, 2015 ELEC-E5610 Lecture 5 2 Loss Types Loss Types Viscosity Heat Conduction Molecular Energy Transfer In real fluids, various phenomena attenuate sound waves: geometric attenuation losses due to boundaries losses within the medium viscosity heat conduction relaxation Wave equation in fluids Henna Tahvanainen Aalto SPA 18/21 November 10, 2015 ELEC-E5610 Lecture 5 2 Viscosity Loss Types Viscosity Heat Conduction Molecular Energy Transfer Viscosity causes a shear stress (force) σxy in the direction of the movement. This stress force tries to cancel the velocity difference of adjacent fluid “layers” in the direction orthogonal to the movement (y -direction in the figure) ∂ ux σxy = µ , where ∂y µ is the shear viscosity (property of the fluid) ux is the velocity of the fluid Wave equation in fluids Henna Tahvanainen Aalto SPA 19/21 November 10, 2015 ELEC-E5610 Lecture 5 2 Heat Conduction Loss Types Viscosity Heat Conduction Molecular Energy Transfer heat flows from compressed areas to uncompressed ones reduces pressure difference between wave maxima and minima ⇒ wave attenuates compressibility no longer adiabatic typical case near solids Wave equation in fluids Henna Tahvanainen Aalto SPA 20/21 November 10, 2015 ELEC-E5610 Lecture 5 2 Molecular Energy Transfer Loss Types Viscosity Heat Conduction Molecular Energy Transfer When a fluid is compressed, its pressure, density, and temperature increase some of the energy of the compression is transferred into rotational and vibrational movement of the molecules ⇒ pressure decreases! this phenomenon is called relaxation irreversible process it takes a certain time until the energy transfer is finished and an equilibrium is found this time is called relaxation time if wave period is greater than relaxation time, losses occur (why?) Wave equation in fluids Henna Tahvanainen Aalto SPA 21/21 November 10, 2015 ELEC-E5610 Lecture 5