Lecture21

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```					ACOUSTIC RESONATORS, WAVEGUIDES, AND ANTENNAS
TEM Resonances:
TEM0 Current I(z,t): Voltage V(z,t): TEM1 TEM2 TEM3

0

0

0

TEM Resonators:
ωo = 2πfo = 2πc/λo λo = 2d ωo = πc/d ωo = πc/d ωo = πc/2d

u(z,t) I(z,t): p(z,t) V(z,t): 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:

p(z,t) u(z,t) f1 = cs/λ1 = cs/4d = 340/(4 × 0.16) = 531 Hz vocal chords z

0

Second Resonance:
f2 = 1594 Hz

p(z,t) u(z,t) 0

z

say ~16 cm

Third Resonance:
f3 = 2655 Hz

p(z,t) u(z,t)

0

z
L21-2

ACOUSTIC WAVEGUIDES
Acoustic Waveguides—Parallel Plate:
x n (λ x 2 ) = b k x = 2π λ x p = poe -jk x x - jkz z λz Power u y z

b

k 2 = ω2 c 2 = k 2 + k 2 = 4π2 (λ −2 + λ −2 ) s z x z x

(∇ 2 + k 2 ) p = 0

Acoustic Waveguides—Rectangular Waveguide:
x m (λ y 2 ) = a n (λ x 2 ) = b p = po e x y z k 2 = ω2 c 2 = k 2 + k 2 + k 2 s x y z
-jk x - jk y - jk z

a

b y z
L21-3

ACOUSTIC RESONATORS
Amnp Resonances of a Box:
x m(ly/2) = a n(lx/2) = b p(lz/2) = d λz

b u y d z

(∇ 2 + k 2 ) p = 0
e.g., p = po e
− jk x x − jk y y − jk z z

− k 2 = ω2 c 2 = k 2 + k 2 + k 2 = 4π2 λ −2 + λ −2 + λ z2 s x y z x y

(

)

Resonant Frequencies of the Amnp Mode in a Box:
fmnp2 fmnp f000 f001 = cs2(λx-2 + λy-2 + λz-2) = cs(m2a-2 + n2b-2 + p2d-2)0.5/2 [Hz] = 0 Hz (constant pressure) = 340/2d ⇒ 170 Hz for a one-meter closed pipe
L21-4

MORE ACOUSTIC RESONATORS
Modal Density in Rectangular Resonators:
Recall: fmnp = cs(m2a-2 + n2b-2 + p2d-2)0.5/2 [Hz] Each cube has volume = cs3/8V where 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
p=2

pcs/2d

∆f

fmnp
m=2 3

mcs/2a

Example:

ncs/2b

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/Hz How can we select just one when we sing (a single note)?
L21-5

EXCITATION OF TEM RESONATORS

Emission from TEM Resonators:
I(t) Zo,c V(z,t) = Vo cos(ωot) sin(πz/d)
z Escaping wave: Ve = [sin(πδ/d)Vo] cos(ωot – kz)
(If escaping to a matched load)
Zo,c δ d Vo V(z,t)

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 << Zo Note:	 Cannot excite TEMm modes by driving current into voltage nulls! (Or by placing voltage across terminals at current nulls.) The power fed to the resonator is zero in both these cases.

L21-6

EXCITATION OF ACOUSTIC RESONATORS
Acoustic Intensity I [Wm-2]:
ˆ I[Wm−2 ] = pu • n

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

Wave equation: (∇2 + k2)p = 0 where k = ω/cs Spherically (radially) vibrating sound source: ∂/∂θ = ∂/∂φ = 0 Yields: Equivalent to: d2p/dr2 + (2/r)dp/dr + k2p = 0 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/jωρo = (A/ηsr)(1 + [jkr]-1)e-jkr
kr = 2πr λ >> 1 if r >> λ 2π

(Recall

∇p = − jωρo u(r), ∇ • u = − jωp γpo ) ∇ = r ∂ + θ1 ∂ + φ 1 ∂  ˆ ˆ ˆ   ∂r r ∂θ r sin θ ∂φ  

L21-8

ACOUSTIC ANTENNAS (2)

Radiation outward: p(r) = (A/r)e-jkr Velocity field u: u(r) = -∇p/jωρo = (A/ηsr)(1 + [jkr]-1)e-jkr

Far-Field—Spherical Waves Become Plane Waves (r>>λ/2π):
λ π
p(r) = (A/r)e-jkr u(r) = (A/rηs)e-jkr = p(r)/ηs

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

ACOUSTIC ANTENNAS (3)
Antenna Gain G(θ,φ) and Effective Area A(θ,φ) [m2]: θφ θφ
G(θ,φ) = Pr(θ,φ)/[PT/4πr2] Preceived = I(θ,φ) A(θ,φ) [W]

Antenna (Loudspeaker, Microphone) Configurations:
Monopole Baffled monopole Dipole Array (end-fire or broadside) Lense Horn Parabolic dish + ⇒

θnull

λ/2

L21-10

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