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ACOUSTIC RESONATORS, WAVEGUIDES, AND ANTENNAS
TEM Resonances: TEM0 TEM1 TEM2 TEM3
Current I(z,t): Voltage V(z,t): 0 0 0
TEM Resonators:
ωo = 2πfo = 2πc/λo λo = 2d
u(z,t) I(z,t): p(z,t) V(z,t):
ωo = πc/d ωo = πc/d ωo = πc/2d
0 0 0 Acoustic Resonators:
A001 A001 A001 Boundary Conditions:
u = 0 at closed end (of course); p ≠ 0
p ≅ 0 at open end (abrupt opening for narrow pipe reduces p)
L21-1
HUMAN ACOUSTIC RESONATORS
Human Vocal Tract:
f1 = cs/λ1 = cs/4d = 340/(4 × 0.16) = 531 Hz
Second Resonance: f2 = 1594 Hz
Third Resonance:
f3 = 2655 Hz
0 z
vocal chords
z say ~16 cm 0
0 z
p(z,t) u(z,t)
p(z,t) u(z,t)
p(z,t) u(z,t)
L21-2
L21-3
ACOUSTIC WAVEGUIDESAcoustic Waveguides—Parallel Plate:
Acoustic Waveguides—Rectangular Waveguide:
Power
x
b
y z
λz
u
(
(( −
λ= πλ
=∇ + =
= ω = π λ + λ
x
x
o2
2 2 2 2 2 2 2s z x z
n bkp p
k p 0k k k 4
-jk x- jk zxe
((
y
x
o2 2 2 2
s y z
m an bp pk c k
λλ
== ω = + +
-jk x- jk y - jk zx ze b
a
y
x
z
)
))−
=
=+
x
2
2x
22
c
z
))
22x
22
kk
==
y
L21-4
ACOUSTIC RESONATORSAmnp Resonances of a Box:
Resonant Frequencies of the Amnp Mode in a Box:
m(ly/2) = an(lx/2) = bp(lz/2) = d
(
(
2
jk x jk y jk zx zo
2 2 2 2 2 2 2 2 2s y z x y z
k p 0e.g., p ek k k k 4
− −
− −
∇ + ==
= ω = + + = π λ + λ + λ
fmnp2 = cs2(λx-2 + λy-2 + λz-2)fmnp = cs(m2a-2 + n2b-2 + p2d-2)0.5/2 [Hz]f000 = 0 Hz (constant pressure)f001 = 340/2d ⇒ 170 Hz for a one-meter closed pipe
d
λz
u z
x
b
y)
)
2
y
2x
pc
−
−
L21-5
MORE ACOUSTIC RESONATORSModal Density in Rectangular Resonators:
Example:
Recall: fmnp = cs(m2a-2 + n2b-2 + p2d-2)0.5/2 ]
Each cube has volume = cs3/8Vwhere V = abd (volume of resonator)
Number of modes in ∆f ≅(Volume of shell)/(vol. of cell) ≅4πf2 ∆f/[8(cs3/8V)] ≅
4πf2V ∆f/cs3 modes in ∆f
Bathroom 3×3×3 meters ⇒ lowest f100 =cs/2a ≅ 340/6 ≅ 57 Hz Modal density at 1 kHz ≅ 4π × 10002 × 33 × 1/3403 ≅ 9 modes/HzHow can we select just one when we sing (a single note)?
ncs/2b
pcs/2d
mcs/2a
∆f
fmnpp=2
m=2 3
[Hz
EXCITATION OF TEM RESONATORS
Emission from TEM Resonators:
z
Zo,c
I(t) Vo V(z,t)
Zo,c
d δ
V(z,t) = Vo cos(ωot) sin(πz/d)
Escaping wave: Ve = [sin(πδ/d)Vo] cos(ωot – kz)
(If escaping to a matched load)
Excitation of TEM Resonators:
If an external source provides I(t) = Io cos ωot to the terminals where V(z,t) = Vo cos(ωot ) sin(πδ/d), then the power input Pi(t) = v(t)i(t) to the resonator is:
Pi(t) ≅ IoVo cos2(ωot) sin(πδ/d) [W]
Note: Resonator impedance
EXCITATION OF ACOUSTIC RESONATORS Acoustic Intensity I [Wm-2]:
[I Wm−2 ] = pu • n̂
Excitation of TEM Resonators:
Pk(t) = v(t) i(t) = power into the resonator
= 0 if vk or ik = 0
Excitation of Acoustic Resonators:
For the same reasons we cannot excite acoustic modes:
with velocity sources at pressure nulls (pk = 0), or with pressure sources at velocity nulls (vk = 0)
Loudspeakers are roughly velocity sources, so put them at pressure maxima of modes (e.g. corners are pressure maxima for all modes)
Bathroom Opera:
Mouth is approximately a velocity source, so place it near pressure maximum of desired mode, with u in right direction.
L21-7
ACOUSTIC ANTENNAS
Monopole Radiators:
Wave equation: (∇ 2 + k2)p = 0 where k = ω/cs Spherically (radially) vibrating sound source: ∂/∂θ = ∂/∂φ = 0
Yields: d2p/dr2 + (2/r)dp/dr + k2p = 0 Equivalent to: d2(rp)/dr2 + k2(rp) = 0
General solution: rp ∝ e±jkr
Radiation outward: p(r) = (A/r)e-jkr
Velocity field u: u(r) = -∇p/j ωρo = (A/ηsr)(1 + [jkr]-1)e-jkr
kr 2 r= π λ >> 1 if r 2>> λ π
j(Recall ∇ p = − ωρou(r), u j p ∇ • = − ω γpo ) ∇ = r̂
∂ + θ̂1 ∂ + φ̂ 1 ∂ ∂r r ∂θ r sin θ ∂φ L21-8
λλλ πππ
ACOUSTIC ANTENNAS (2)
Monopole Radiators:
Radiation outward: p(r) = (A/r)e-jkr
Velocity field u: u(r) = -∇p/j ωρo = (A/ηsr)(1 + [jkr]-1)e-jkr
Far-Field—Spherical Waves Become Plane Waves (r>>λ/2π):
p(r) = (A/r)e-jkr u(r) = (A/rηs)e-jkr = p(r)/ηs
Near-Field Radiation—Velocity Boosted and Reactive:
p(r) = (A/r)e-jkr u(r) = (-jA/r2kηs)e-jkr = -jp(r)/ρoωr
Therefore a velocity microphone held close to the lips (the monopole radiator) will boost low frequencies and need compensation.
(Recall ηs = ρocs = ρoω/k) L21-9
L21-10
ACOUSTIC ANTENNAS (3)Antenna Gain G(θθθθ,φφφφ) and Effective Area A(θθθθ,φφφφ) [m2]:
Antenna (Loudspeaker, Microphone) Configurations:
G(θ,φ) = Pr(θ,φ)/[PT/4πr2]Preceived = I(θ,φ) A(θ,φ) [W]
MonopoleBaffled monopoleDipoleArray (end-fire or broadside)LenseHornParabolic dish
- + ⇒
θnullλ/2