EE1.el3 (EEE1023): Electronics III

Acoustics lecture 17

Resonators and waveguides

Dr Philip Jackson

www.ee.surrey.ac.uk/Teaching/Courses/ee1.el3

Acoustic devices

Objectives:

• derive mode shapes of standing waves

• identify the modes of propagating waves

• relate resonances to properties of acoustical artifacts

Topics:

• resonators

• waveguides

Q.1

Preparation for acoustic devices

• What is an acoustic resonator?

– find a definition

– draw a sketch of an example

• What is an acoustic waveguide?

– look up a definition

– give an example

Q.2

Cavities and waveguides

Cavity: accrual of acoustic

energy in an enclosed space

e.g., a room, a silencer, or

a loudspeaker cabinet

Waveguide: transmission

of acoustic energy along a

bounded channel

e.g., a system of ducting,

a bass reflex port, sound in

the atmosphere or oceans

xy

z H

WL

yx

z

L

H

Q.3

Resonators

Modes in a cavity

Resonance frequencies of a cavity are

defined by their mode number (l,m,n):

f(l,m, n)

=c

2

√(l

L

)2+(m

W

)2+(n

H

)2

Q.4

Pressure modes in a closed pipe

The 1-D wave eq. provides general solutions of the form

p(t, x) = g

(t−

x

c

)+ h

(t+

x

c

)(1)

At one frequency, ω = 2πf , we have

p(t, x) = p(+) cos(ω

(t−

x

c

)+ φ(+)

)+ p(−) cos

(ω

(t+

x

c

)+ φ(−)

)= Re

{p(+)ej(ωt−kx) + p(−)ej(ωt+kx)

}(2)

where p(+) = p(+)ejφ(+)

and p(−) = p(−)ejφ(−)

are the com-plex amplitudes of the positive and negative travelling waves,k = ω/c is the wavenumber, and p(t, x) = Re {p(t, x)} is thereal part of the complex pressure field in the pipe.

At resonance, wave interference creates a standing wave

p(t, x) = X(x)ejωt (3)

To get the mode shape, X(x), we must use the pipe’sboundary conditions, which specify how waves are reflectedat the ends. Q.5

Mode shapes in a closed pipe

In complex form, we have general sinusoidal soundfields

p(t, x) = p(+)ej(ωt−kx) + p(−)ej(ωt+kx) (4)

The boundary conditions specify the pressure gradient

∂p(t, x)

∂x= −jk

(p(+)ej(ωt−kx) − p(−)ej(ωt+kx)

)= −jk

(p(+)e−jkx − p(−)e+jkx

)ejωt (5)

At x = 0, the first boundary condition:

−jk(p(+) − p(−)

)ejωt = 0

p(+) = p(−) (6)

This result gives us soundfields that are even about x = 0,

p(t, x) = p(+)(e−jkx + e+jkx

)ejωt

= 2p(+) cos (kx) ejωt (7)

Q.6

Mode shapes in a closed pipe

At x = L, the second boundary condition:

−jk(p(+)e−jkL − p(+)e+jkL

)ejωt = 0(

e−jkL − e+jkL)

= 0

sin (kL) = 0 (8)

kL = nπ for n = {0,1,2, . . .} (9)

which can be expressed in various ways,

k =nπ

Land ω =

nπc

Land f =

nc

2L

At resonance, we obtain pressure mode shapes of the form

p(t, x) = 2p(+)X(x) ejωt

where X(x) = cos(nπx

L

)(10)

Q.7

Pressure and displacement/velocity modes

0 0.2 0.4 0.6 0.8 1−1

0

1l =

1Pressure modes

0 0.2 0.4 0.6 0.8 1−1

0

1

l = 2

0 0.2 0.4 0.6 0.8 1−1

0

1

l = 3

0 0.2 0.4 0.6 0.8 1−1

0

1

l = 1

Displacement modes

0 0.2 0.4 0.6 0.8 1−1

0

1

l = 2

0 0.2 0.4 0.6 0.8 1−1

0

1

l = 3

Considering number of wavelengths that fit the return path,

we get:

f(l) =l c

2LC(11)

Q.8

Modes in open and semi-closed pipes

f(n) =n c

2LO

f(m) =(2m− 1)c

4LS

Q.9

Rectangular waveguide

Given four rigid walls in x and y directions, and open endsin the z direction that allow acoustic energy to propagatedown the waveguide, we obtain solutions of the form:

p(x, y, z, t) = A cos kxx cos kyy ej(ωt−kzz) (12)

with kx = lπL , ky = mπ

W , and kz =

√(ωc

)2−(k2x + k2

y

)The longitudinal wavenumber kz depends on the transversewavenumbers, but is only real above the cut-on frequency,f ≥ fcut-on,

fcut-on =c

2π

√k2x + k2

y (13)

otherwise with imaginary roots, tranverse mode becomesevanescent and does not propagate, i.e, waveguide onlysupports axial waves along z.

Q.10

Transverse modes in a rectangular waveguide

Plane wave propagation

zero transverse mode, for all f

L

z

Transverse wave propagation

non-zero cross mode, for f ≥ fcut-on

L

zQ.11

Summary of acoustic devices

• cavities

- rectangular

• resonators

- open, closed & semi-closed pipes

• waveguides

- rectangular

Reference

L. E. Kinsler, A. R. Frey, A. B. Coppens and J. V. Sanders,

“Fundamentals of Acoustics”, 4th ed., Wiley, 2000.

Chapters 9 & 10, [shelf 534 KIN]Q.12