ENGINEERING ACOUSTICS EE 363NINDEX (p,q,r) modes pdf

From the ideal gas equation: ρ P adiabatic bulk modulus, approximately equal to the isothermal bulk modulus, 2.18×109 for water [Pa] c = the phase speed speed of sound [m/s] γ = ratio of

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Tom Penick tom@tomzap.com www.teicontrols.com/notes EngineeringAcoustics.pdf 12/20/00 Page 1 of 36

ENGINEERING ACOUSTICS EE 363N

INDEX

(p,q,r) modes 28

2θHP half-power beamwidth .16

A absorption 27

a absorption coefficient 21

absorption 27

average 27

measuring 27

absorption coefficient 21, 28 measuring 21

acoustic analogies 8

acoustic impedance 3, 10 acoustic intensity 10

acoustic power 10

spherical waves 11

acoustic pressure 5, 9 effective 5

adiabatic 7, 36 adiabatic bulk modulus 6

ambient density 2, 6 amp 3

amplitude 4

analogies 8

anechoic room 36

arbitrary direction plane wave 9

architectural absorption coefficient 28

area sphere 36

average absorption 27

average energy density 26

axial pressure 19

B bulk modulus 6

band frequency 12

bandwidth 12

bass reflex 19

Bessel J function 18, 34 binomial expansion 34

binomial theorem 34

bulk modulus 6

C compliance 8

c speed of sound 3

calculus 34

capacitance 8

center frequency 12

characteristic impedance 10

circular source 15

cocktail party effect 30

coincidence effect 22

complex conjugate 33

complex numbers 33

compliance 8

condensation 2, 6, 7 conjugate complex 33

contiguous bands 12

coulomb 3

C p dispersion 22

Cramer's rule 23

critical gradient 32

cross product 35

curl 36

D(r) directivity function 16

D(θ) directivity function 14,

15, 16 dB decibels 2, 12, 13 dBA 13

decibel 2, 12, 13 del 35

density 6

equilibrium 6

dependent variable 36

diffuse field 28

diffuse field mass law 22

dipole 14

direct field 29, 30 directivity function 14, 15, 16 dispersion 22

displacement particle 10

divergence 35

dot product 35

double walls 23

E energy density 26

E(t) room energy density 26

effective acoustic pressure 5

electrical analogies 8

electrical impedance 18

electrostatic transducer 19

energy density 26

direct field 29

reverberant field 30

enthalpy 36

entropy 36

equation of state 6, 7 equation overview 6

equilibrium density 6

Euler's equation 34

even function 5

expansion chamber 24, 25 Eyring-Norris 28

far field 16

farad 3

f c center frequency 12

f l lower frequency 12

flexural wavelength 22

flow effects 25

focal plane 16

focused source 16

Fourier series 5

Fourier's law for heat conduction 11

frequency center 12

frequency band 12

frequency band intensity level 13

f u upper frequency 12

gas constant 7

general math 33

glossary 36

grad operator 35

gradient thermoacoustic 32

gradient ratio 32

graphing terminology 36

H enthalpy 36

h specific enthalpy 36

half-power beamwidth 16

harmonic wave 36

heat flux 11

Helmholtz resonator 25

henry 3

Hooke's Law 4

horsepower 3

humidity 28

hyperbolic functions 34

I acoustic intensity10, 11, 12 I f spectral frequency density 13

IL intensity level 12

impedance 3, 10 air 10 due to air 18

mechanical 17

plane wave 10

radiation 18

spherical wave 11

incident power 27

independent variable 36

inductance 8

inertance 8

instantaneous intensity 10

instantaneous pressure 5

intensity 10, 11 intensity (dB) 12, 13 intensity spectrum level 13

intervals musical 12

Iref reference intensity 12

isentropic 36

ISL intensity spectrum level 13

isothermal 36

isotropic 28

joule 3

k wave number 2

k wave vector 9

kelvin 3

L inertance 8

Laplacian 35

line source 14

linearizing an equation 34

L M mean free path 28

m architectural absorption coefficient 28

magnitude 33

mass radiation 18

mass conservation 6, 7 material properties 20

mean free path 28

mechanical impedance 17

mechanical radiation impedance 18

modal density 28

modes 28

modulus of elasticity 9

momentum conservation 6, 8 monopole 13

moving coil speaker 17

m r radiation mass 18

mufflers 24, 25 musical intervals 12

N fractional octave 12

n number of reflections 28

N(f) modal density 28

nabla operator 35

natural angular frequency 4

natural frequency 4

newton 3

Newton's Law 4

noise 36

noise reduction 30

NR noise reduction 30

number of reflections 28

octave bands 12

odd function 5

p acoustic pressure 5, 6 Pa 3

particle displacement 10, 22 partition 21

pascal 3

paxial axial pressure 19

P e effective acoustic pressure 5

perfect adiabatic gas 7

phase 33

phase angle 4

phase speed 9

phasor notation 33

piezoelectric transducer 19

pink noise 36

plane wave impedance 10

velocity 9

plane waves 9

polar form 4

power 10, 11 SPL 29

power absorbed 27

Pref reference pressure 13

pressure 6, 9 progressive plane wave 9

progressive spherical wave 11 propagation 9

propagation constant 2

Q quality factor 29

quality factor 29

r gas constant 7

R room constant 29

radiation impedance 18

radiation mass 18

radiation reactance 18

rayleigh number 16

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rayls 3

r d reverberation radius 29

reflection 20

reflection coefficient 20

resonance modal 28

reverberant field 30

reverberation radius 29

reverberation room 36

reverberation time 28

rms 5, 34 room acoustics 26

room constant 29

room energy density 26

room modes 28

root mean square 34

s condensation 2, 6 Sabin formula 28

sabins 27

series 34

sidebranch resonator 26

simple harmonic motion 4

sound 3

sound decay 26

sound growth 26

sound power level 29

sound pressure level (dB) 13

source 13, 14 space derivative 35

space-time 33

speaker 17

specific acoustic impedance .10

specific enthalpy 36

specific gas constant 7

spectral frequency density 13 speed amplitude 4

speed of sound 3

sphere 36

spherical wave 11

impedance 11

velocity 11

spherical wave impedance.11 SPL sound power level 29

SPL sound pressure level 13

spring constant 4

standing waves 10

Struve function 18

surface density 21

T60 reverberation time 28

TDS 36

temperature 3

temperature effects 25

tesla 3

thermoacoustic cycle 31

thermoacoustic engine 31

thermoacoustic gradient 32

thin rod 9

time constant 26

time delay spectrometry 36

time-average 33

time-averaged power 33

TL transmission loss 21, 22 trace wavelength 22

transducer electrostatic 19

piezoelectric 19

transmission 20

transmission at oblique incidence 22

transmission coefficient 20

transmission loss 21

composite walls 22

diffuse field 22

expansion chamber 25

thin partition 21

trigonometric identities 34

u velocity 6, 9, 11 U volume velocity 8

vector differential equation35 velocity 6

plane wave 9

spherical wave 11

volt 3

volume sphere 36

volume velocity 8

w bandwidth 12

Wabs power absorbed 27

watt 3

wave progressive 11

spherical 11

wave equation 6

wave number 2

wave vector 9

wavelength 2

temperature effects 25

weber 3

weighted sound levels 13

white noise 36

Wincident incident power 27

Young's modulus 9

z acoustic impedance 10

z impedance 10, 11 z0 rayleigh number 16

Z A elec impedance due to air 18

Z M elec impedance due to mech forces 18

Z m mechanical impedance 17 Z r radiation impedance 18

Γ gradient ratio 32

Π acoustic power 10, 11 γ ratio of specific heats 6

λ wavelength 2

λp flexural wavelength 22

λtr trace wavelength 22

ρ0 equilibrium density 6

ρs surface density 21

τ time constant 26

ξ particle displacement 10,

22 ∇ del 35

∇× · curl 36

∇2 · Laplacian 35

∇ · divergence 35

DECIBELS [dB]

A log based unit of energy that makes it easier to

describe exponential losses, etc The decibel means

10 bels, a unit named after Bell Laboratories

energy

10 log reference energy

One decibel is approximately the minimum discernable

amplitude difference that can be detected by the human ear

over the full range of amplitude

λλ WAVELENGTH [m]

Wavelength is the distance that a

wave advances during one cycle

At high temperatures, the speed of

sound increases so λ changes T k is

temperature in Kelvin

2

c

π

343 293

k

T f

λ =

k WAVE NUMBER [rad/m]

The wave number of propagation constant

is a component of a wave function representing the wave density or wave spacing relative to distance Sometimes represented by the letter β See also WAVE VECTOR p9

2

k

c

π ω

λ

s CONDENSATION [no units]

The ratio of the change in density to the ambient density, i.e the degree to which the medium has condensed (or expanded) due to sound waves For

example, s = 0 means no condensation or expansion

of the medium s = -½ means the density is at one half the ambient value s = +1 means the density is at

twice the ambient value Of course these examples are unrealistic for most sounds; the condensation will typically be close to zero

0 0

s = ρ − ρ ρ

ρ = instantaneous density [kg/m3]

ρ = equilibrium (ambient) density [kg/m3]

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In liquids:

0

T

c = γ ρ

B where B = γ BT

γ = ratio of specific heats (1.4 for a diatomic gas) [no units]

P0 = ambient (atmospheric) pressure (

0

p= ) At sea Plevel,

0≈101 kPa

ρ0 = equilibrium (ambient) density [kg/m3]

r = specific gas constant [J/(kg· K)]

P adiabatic bulk modulus [Pa]

BT = isothermal bulk modulus, easier to measure than the adiabatic bulk modulus [Pa]

Two values are given for the speed of sound in solids, Bar and Bulk The Bar value provides for the ability of sound to distort the dimensions of solids having a small-cross-sectional area Sound moves more slowly in Bar material

The Bulk value is used below where applicable

Speed of Sound in Selected Materials [m/s]

Air @ 20°C 343 Copper 5000 Steel 6100 Aluminum 6300 Glass (pyrex) 5600 Water, fresh 20°C 1481

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SIMPLE HARMONIC MOTION

Restoring force on a spring

π is the natural frequency in Hz

The general solution takes the form

By differentiation, it can be found that the speed of the mass

is u = − U sin ( ω + φ0t ), where U = ω0A is the speed amplitude The acceleration is a = −ω0U cos ( ω + φ0t ) Using the initial conditions, the equation can be written

x0 = the initial position [m]

u0 = the initial speed [m/s]

0

s m

ω = is the natural angular frequency in rad/s

It is seen that displacement lags 90° behind the speed and that the acceleration is 180° out of phase with the

Speed

u

a

Acceleration

Initial phase angle φφ=0°

The speed of a simple oscillator leads the displacement by 90° Acceleration and displacement are 180° out of phase with each other

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FOURIER SERIES

The Fourier Series is a method of describing a

complex periodic function in terms of the frequencies

and amplitudes of its fundamental and harmonic

frequencies

where T = the period

ω = = the fundamental frequency

A0 = the DC component and will be zero provided the

function is symmetric about the t-axis This is almost

always the case in acoustics

odd function, i.e f(t)=-f(-t),

the right-hand plane is a mirror image of the left-hand plane provided one of them is first flipped about the horizontal axis, e.g

even function, i.e f(t)=f(-t),

the right-hand plane is a mirror image of the left-hand plane, e.g cosine function

where t0= an arbitrary time

p ACOUSTIC PRESSURE [Pa]

Sound waves produce proportional changes in pressure, density, and temperature Sound is usually measured as a change in pressure See Plane Waves p9

P = instantaneous pressure [Pa]

0

p= ) At sea Plevel,

0≈101 kPa

P = peak acoustic pressure [Pa]

x = position along the x-axis [m]

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From the ideal gas equation:

ρ P adiabatic bulk modulus, approximately equal

to the isothermal bulk modulus, 2.18×109 for water [Pa]

c = the phase speed (speed of sound) [m/s]

0

p= ) At sea Plevel,

ρ0 Equilibrium Density of Selected Materials [kg/m3]

Air @ 20°C 1.21 Copper 8900 Steel 7700

Aluminum 2700 Glass (pyrex) 2300 Water, fresh 20°C 998

B ADIABATIC BULK MODULUS [Pa]

B is a stiffness parameter A larger B means the

material is not as compressible and sound travels

faster within the material

c = the phase speed (speed of sound, 343 m/s in air) [m/s]

P = instantaneous (total) pressure [Pa or N/m2]

0

p= ) At sea Plevel,

0≈101 kPa

B

B Bulk Modulus of Selected Materials [Pa]

Aluminum 75×109 Iron (cast) 86×109 Rubber (hard) 5×109

From the above 3 equations and 3 unknowns (p, s, u)

we can derive the Wave Equation

2 2

EQUATION OF STATE - GAS

An equation of state relates the physical properties describing the thermodynamic behavior of the fluid In acoustics, the temperature property can be ignored

In a perfect adiabatic gas, the thermal conductivity of

the gas and temperature gradients due to sound waves are so small that no appreciable thermal energy transfer occurs between adjacent elements of the gas

Perfect adiabatic gas:

P = instantaneous (total) pressure [Pa]

P0 = ambient (atmospheric) pressure (p= ) At sea P0level, P0≈101 kPa [Pa]

p = P - P0 acoustic pressure [Pa]

s = 0 0

1

ρ − ρ

ρ = condensation [no units]

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EQUATION OF STATE – LIQUID

An equation of state relates the physical properties

describing the thermodynamic behavior of the fluid In

acoustics, the temperature property can be ignored

adiabatic bulk modulus, approximately equal

to the isothermal bulk modulus, 2.18×109 for water [Pa]

r SPECIFIC GAS CONSTANT [J/(kg· K)]

The specific gas constant r depends on the universal

gas constant R and the molecular weight M of the

particular gas For air r≈287 J/ kg·K( )

r M

= R

R = universal gas constant

M = molecular weight

MASS CONSERVATION – one dimension

For the one-dimensional problem, consider sound waves traveling through a tube Individual particles of

the medium move back and forth in the x-direction

tube area

x

A =

x + dx

(ρuA)xis called the mass flux [kg/s]

(ρuA)x dx+ is what's coming out the other side (a different

value due to compression) [kg/s]

The difference between the rate of mass entering the center

volume (A dx) and the rate at which it leaves the center

volume is the rate at which the mass is changing in the center volume

A = area of the tube [m2]

MASS CONSERVATION – three dimensions

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MOMENTUM CONSERVATION – one

dimension (5.4) For the one-dimensional problem, consider sound

waves traveling through a tube Individual particles of

the medium move back and forth in the x-direction

( )PA x dx+ is the force due to sound pressure at location

x + dx in the tube (taken to be in the positive or

Force in the tube can be written in this form, noting that this

is not a partial derivative:

∂ often discarded in acoustics

P = instantaneous (total) pressure [Pa or N/m2]

A = area of the tube [m2]

u = particle velocity (due to oscillation, not flow) [m/s]

∂

v

is quadratic after multiplication

U = volume velocity (not a vector) [m3/s]

Z A = acoustic impedance [Pa·s/m3

The springiness of the system; a higher value means

softer Analogous to electrical capacitance

0

V

C = γρ

V = volume [m3]

ρ0 = ambient density [kg/m3]

U VOLUME VELOCITY [m3/s]

Although termed a velocity, volume velocity is not a

vector Volume velocity in a (uniform flow) duct is the product of the cross-sectional area and the velocity

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PLANE WAVES PLANE WAVES (2.4, 5.7)

A disturbance a great distance from the source is

approximated as a plane wave Each acoustic

variable has constant amplitude and phase on any

plane perpendicular to the direction of propagation

The wave equation is the same as that for a

disturbance on a string under tension

p x t = 1424 Ae ω −3 1424 + Be ω +3

A = magnitude of the positive-traveling wave [Pa]

B = magnitude of the negative-traveling wave[Pa]

ω = frequency [rad/s]

t = time[s]

k = wave number or propagation constant[rad./m]

x = position along the x-axis[m]

PROGRESSIVE PLANE WAVE (2.8)

A progressive plane wave is a unidirectional plane

wave—no reverse-propagating component

( ) j( )

p x t = Ae ω −

ARBITRARY DIRECTION PLANE WAVE

The expression for an arbitrary direction plane wave

contains wave numbers for the x, y, and z

components

, t k x k y k z x y z

p x t = Ae ω − − − where

u VELOCITY, PLANE WAVE [m/s]

The acoustic pressure divided by the impedance, also from the momentum equation:

z = wave impedance [rayls or (Pa· s)/m]

c = dx dt is the phase speed (speed of sound) [m/s]

k = wave number or propagation constant [rad./m]

r = radial distance from the center of the sphere [m]

PROPAGATION (2.5) F( x-c ∆ t )

a disturbance

x x

c = dx dt is the phase speed (speed of sound) at which F is

translated in the +x direction. [m/s]

k v

WAVE VECTOR [rad/m or m-1]

The phase constant k is converted to a vector For

plane waves, the vector k v

is in the direction of propagation

THIN ROD PROPAGATION

A thin rod is defined as λ ? a

rod radius

a =

a

ϒ = Young's modulus, or modulus of elasticity, a characteristic property of the material[Pa]

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STANDING WAVES

Two waves with identical frequency and phase

characteristics traveling in opposite directions will

cause constructive and destructive interference:

Specific acoustic impedance or characteristic

impedance z is a property of the medium and of the

type of wave being propagated It is useful in

calculations involving transmission from one medium

to another In the case of a plane wave, z is real and

is independent of frequency For spherical waves the

opposite is true In general, z is complex

0

p

u

= = ρ (applies to progressive plane waves)

Acoustic impedance is analogous to electrical

impedance:

impedance

In a sense this is reactive,

in that this value represents

an impediment to propagation.

In a sense this is resistive,

i.e a loss since the wave

departs from the source.

z = r+ x j

ρ0c Characteristic Impedance, Selected Materials (bulk) [rayls]

Air @ 20°C 415 Copper 44.5×106 Steel 47×106

Aluminum 17×106 Glass (pyrex) 12.9×106 Water, fresh 20°C 1.48×106

Brass 40×106 Ice 2.95×106 Water, sea 13°C 1.54×106

Concrete 8×106 Steam @ 100°C 242 Wood, oak 2.9×106

∂ξ

=

∂

v v

0

p c

ξ = ωρ

propagation; power per unit area Note that I = 〈pu〉T

is a nonlinear equation (It’s the product of two functions of space and time.) so you can't simply use

jωt or take the real parts and multiply, see Average p33

T T

I(t) = instantaneous intensity [W/m2]

|p| = peak acoustic pressure [Pa]

P e = effective or rms acoustic pressure [Pa]

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FOURIER'S LAW FOR HEAT

CONDUCTION, HEAT FLUX

Sound waves produce proportional changes in

pressure, density, and temperature Since the

periodic change in temperature is spread over the

length of a wavelength, the change in temperature per

unit distance is very small

General solution for a symmetric spherical wave:

A = magnitude of the positive-traveling

wave[Pa]

B = magnitude of the

negative-traveling wave[Pa]

SPHERICAL WAVE BEHAVIOR

Spherical wave behavior changes markedly for very

small or very large radii Since this is also a function

of the wavelength, we base this on the kr product

where kr ∝ r/λ

For kr ? 1, i.e r ? λ (far from the source):

In this case, the spherical wave is much like a plane

wave with the impedance z ; ρ0c and with p and u in

phase

For kr = 1, i.e r = λ (close to the source):

In this case, the impedance is almost purely reactive

0

j

z ; ωρ r and p and u are 90° out of phase The

source is not radiating power; particles are just sloshing

back and forth near the source

IMPEDANCE [rayls or (Pa· s)/m]

Spherical wave impedance is frequency dependent:

( )

0

1 j /

c p

u VELOCITY, SPHERICAL WAVE [m/s]

0

j 1

z = wave impedance [rayls or (Pa· s)/m]

Π ACOUSTIC POWER, SPHERICAL

4 r I

Π = π for spherical dispersion

hemispherical surface

2 r I

Π = π for hemispherical dispersion

S = surface surrounding the sound source, or at least the surface area through which all of the sound passes [m2]

I = acoustic intensity [W/m2]

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FREQUENCY BANDS

fu fl FREQUENCY BANDS

The human ear perceives different frequencies at

different levels Frequencies around 3000 Hz appear

loudest with a rolloff for higher and lower frequencies

Therefore in the analysis of sound levels, it is

necessary to divide the frequency spectrum into

f u = the upper frequency in the band [Hz]

f l = the lowest frequency in the band [Hz]

N = the bandwidth in terms of the (inverse) fractional portion

of an octave, e.g N=2 describes a ½-octave band

f u = the upper frequency in the band [Hz]

f l = the lowest frequency in the band [Hz]

w BANDWIDTH [Hz]

The width of a frequency band

1k300

f f

+

=

Octave bands are the most common contiguous bands:

12

n c n c

f f

+

=

2

c l

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If SPECTRAL FREQUENCY DENSITY

[W/m2]

The distribution of acoustic intensity over the

frequency spectrum; the intensity at frequency f over a

bandwidth of ∆f The bandwidth ∆f is normally taken

to be 1 Hz and may be suppressed

ISL INTENSITY SPECTRUM LEVEL [dB]

The spectral frequency density expressed in decibels

This is what you see on a spectrum analyzer

Iref = the reference intensity 10-12 [Pa]

ILBAND FREQUENCY BAND INTENSITY

SPL SOUND PRESSURE LEVEL [dB]

Acoustic pressure in decibels Note that IL = SPL

when IL is referenced to 10-12 and SPL is referenced to

20×10-6 An increase of 6 dB is equivalent to doubling

the amplitude A spherical source against a planar

surface has a 3 dB advantage over a source in free

space, 6 dB if it's in a corner, 9 dB in a 3-wall corner

Sound Pressure Level:

P e = effective or rms acoustic pressure [Pa]

Pref = the reference pressure 20×10-6 in air, 1×10-6 in water

[Pa]

N = the number of sources

PSL PRESSURE SPECTRUM LEVEL

[dB]

Same as intensity spectrum level

PSL f = ISL f = SPL

SPL = sound pressure level [dB]

dBA WEIGHTED SOUND LEVELS (13.2) Since the ear doesn't perceive sound pressure levels uniformly across the frequency spectrum, several correction schemes have been devised to produce a more realistic scale The most common is the A-weighted scale with units of dBA From a reference point of 1000 Hz, this scale rolls off strongly for lower frequencies, has a modest gain in the 2-4 kHz region and rolls off slightly at very high frequencies Other scales are dBB and dBC Most standards, regulations and inexpensive sound level meters employ the A-weighted scale

ACOUSTICAL SOURCES MONOPOLE (7.1) The monopole source is a basic theoretical acoustic

source consisting of a small (small ka) pulsating

r = radial distance from the center of the source [m]

c = dx dt is the phase speed (speed of sound) [m/s]

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The dipole source is a basic theoretical acoustic

source consisting of two adjacent monopoles 180° out

of phase Mathematically this approximates a single

source in translational vibration, which is what we

really want to model

r = radial distance from the center of the source [m]

c = dx dt is the phase speed (speed of sound) [m/s]

sin

kL D

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DIRECTIVITY FUNCTION

The directivity function is responsible for the lobes of

the dispersion pattern The function is normalized to

have a maximum value of 1 at θ = 0 Different

directivity functions are used for different elements;

the following is the directivity function for the line

source

1 2

sin

kL D

J1(x) = first order Bessel J function

see also Half Power Beamwidth p16

CIRCULAR SOURCE (7.4, 7.5a)

A speaker in an enclosure may be modeled as a

circular source of radius a in a rigid infinite baffle

vibrating with velocity µ0ejωt For the far field pressure

3.83 sin

ka

  First null

r = radial distance from the source [m]

θ = angle with the normal from the circular source [radians]

t = time [s]

ρ0c = impedance of the medium [rayls] (415 for air)

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2θθHP HALF-POWER BEAMWIDTH

The angular width of the main lobe to the points where

power drops off by 1/2; this is the point at which the

directivity function equals 1/ 2

For a circular source:

HP

1

sin 2

J ka D

For a line source:

from the Directivity function: ( )HP 1(1 )

sin sin 1

sin 2

kL D

1.391558 sin 1.391558 2 2sin

The dispersion pattern of a focused source is

measured at the focal plane, a plane passing through

the focal point and perpendicular to the central axis

Focal Plane pressure p r ( ) = ρ G c D r0 ( )

where

22

ka G d

r = radial distance from the central axis [m]

G = constant [radians]

a = radius of the source [m]

d = focal length [m]

z0 RAYLEIGH NUMBER [rad. ·m]

The Rayleigh number or Rayleigh length is the distance along the central axis from a circular piston

element to the beginning of the far field Beyond this

point, complicated pressure patterns of the near field can be ignored

2

2 1

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MOVING COIL SPEAKER (14.3b, 14.5)

Model for the moving coil

u = velocity of the voice coil [m/s]

I = electrical current [A]

R0 = electrical resistance of the voice coil [Ω]

L0 = electrical inductance of the voice coil [H]

s = spring stiffness due to flexible cone suspension material

[N/m]

R m = mechanical resistance, a small frictional force [(N· s)/m

or kg/s]

R M = effective electrical resistance due to the mechanical

resistance of the system [Ω]

C M = effective electrical capacitance due to the mechanical

stiffness [F]

L M = effective electrical inductance due to the mechanical

inertia [H]

V = voltage applied to the voice coil [V]

Z E = electrical impedance due to electrical components [Ω]

Z A = effective electrical impedance due to mechanical air

loading [Ω]

Z M = effective electrical impedance due to the mechanical

effects of spring stiffness, mass, and (mechanical)

resistance [Ω]

F = force on the voice coil [V]

φ = Bl coupling coefficient [N/A]

B = magnetic field [Tesla (an SI unit)]

l = length of wire in the voice coil [m]

Zm MECHANICAL IMPEDANCE [(N ·s)/m]

(1.7) The mechanical impedance is analogous to electrical impedance but does not have the same units Where electrical impedance is voltage divided by current, mechanical impedance is force divided by speed,

sometimes called mechanical ohms.

m

F Z u

jj

m mo

due to spring mass effect

m = mass of the speaker cone and voice coil [kg]

x = distance in the direction of motion [m]

s = spring stiffness due to flexible cone suspension material [N/m]

R m = mechanical resistance, a small frictional force [(N·s)/m

or kg/s]

F = force on the speaker mass [N]

ω = frequency in radians

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ZM ELECTRICAL IMPEDANCE DUE TO

R M = effective electrical resistance due to the mechanical

resistance of the system [Ω]

C M = effective electrical capacitance due to the mechanical

m = mass of the speaker cone and voice coil [kg]

s = spring stiffness due to flexible cone suspension material

[N/m]

φ = Bl coupling coefficient [N/A]

Z mo = mechanical impedance, open-circuit condition

[(N·s)/m or kg/s]

ZA ELECTRICAL IMPEDANCE DUE TO

AIR [ Ω]

The factor of two in the denominator is due to loading

on both sides of the speaker cone

22

A r

Z Z

φ

=

Zr RADIATION IMPEDANCE [(N ·s)/m]

(7.5) This is the mechanical impedance due to air resistance For a circular piston:

S = surface area of the piston [m2]

J1 = first order Bessel J function

R1 = a function describing the real part of Z r

X1 = a function describing the imaginary part of Z r

x = just a placeholder here for 2ka

a = radius of the source [m]

H1 = first order Struve function

ω = frequency in radians

mr RADIATION MASS [kg] (7.5) The effective increase in mass due to the loading of the fluid (radiation impedance)

r r

X

m = ω

The effect of radiation mass is small for light fluids such as air but in a more dense fluid such as water, it can significantly decrease the resonant frequency

The functions R1 and X1 are defined as:

X r = radiation reactance, the imaginary part of the radiation impedance [(N·s)/m]

Tiêu đề	ENGINEERING ACOUSTICS EE 363NINDEX (p,q,r) modes
Trường học	University of Texas at Austin
Chuyên ngành	Engineering Acoustics
Thể loại	lecture notes
Thành phố	Austin

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