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