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Transcript
```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
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
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
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
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
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
```