Observing the Effects of Waveguide Model Elements in Acoustic Tube Measurements

Tamara Smyth

Jonathan Abel

School of Computing Science, Simon Fraser University

Universal Audio Inc. and Stanford University (CCRMA)

Meeting of the Acoustical Society of America, Honolulu, Hawaii

December 2, 2006

tamaras@cs.sfu.ca

abel@batnet.net

Outline

• Digital waveguide theory• Technique for measuring an impulse

response from acoustic tube structures• Observing waveguide theory in

measured responses• Comparing model and measurement

A Digital Waveguide Section

()

• Both the plane waves of a cylinder and the spherical waves a cone can be modeled using a digital waveguide.

()Z-L

Z-L

• The effects of viscous drag and thermal conduction along the bore walls, lead to an attenuation in the propagating waves, determined by

() = 2 x 10-5 / a

Theoretical Wall Loss

• The round trip attenuation for a tube of L is given by,

2() = e-2L

valid for diameters seen in most musical instruments.

R2()R1()

Termination and ScatteringA change of impedance, such as a termination or connection to another waveguide section, will require additional filters to account for reflection and possibly transmission.

Termination Scattering

R() R()

T()

+

+T1()

T2()

Open End Reflection Filter

• The reflection filter for an open is given by

Rop() = ZL () / Z0 - 1

ZL () / Z0 +1

Z0 = cS

,

where

and ZL() is the complex terminating impedance at the open end of a cylinder, given by the expression by Levine and Schwinger.

is the wave impedance,

j + c/x

Z2 / Z1 +1

Theoretical Junction ReflectionThe reflection at the junction is given by

R() = Z2 / Z1 - 1 ,

*

The impedance for the spherical waves is given by

Zn = cS

j

This leads to a first-order, one-pole, one-zero, filter.

.

The impedance for plane waves is given by

Zy = cS

Cylinder

Cylicone

Cylinder Scattering Cone

Example Waveguide Models

Four Measured Tube StructuresCylinder, speaker-closed Cylinder, speaker-open

Cylicone, speaker-closed Cylicone, speaker-open

Obtaining an Impulse Response from an LTI System

The impulse is limited in amplitude and has poor noise rejection

Measurement noise

Measured responseTest

signalLTI system

h(t) +s(t) r(t)

n(t)

Impulse Response Using a Swept Sine

• The sine is swept over a frequency trajectory (t) effectively smearing the impulse over a longer period of time.

• Since higher frequencies go into the system at later times, they must be realigned to recover the impulse response.

Our Measurement System

Cylinder, Speaker-Closed

From the first measurement we observe:

• The speaker transfer function, ()• The speaker reflection, ()• The round trip wall losses for a cylinder, 2()

Arrival Responses for a Closed Cylinder

L1 = ()

L2 = ()2()(1+())

L3 = ()4() ()(1+())

Closed Cylinder Arrival Spectra

L1

L2

L3

Speaker Reflection Transfer Function• Given the arrival responses:

L1 = ()

L2 = ()2()(1+())

L3 = ()4()()(1+()),

() = ^ 1 -

= L1L3

(L2)2

()

1 + ()=

• We are able to estimate the speaker reflection transfer function

Cylinder Wall Loss Transfer Function

• Given the arrival responses:

L1 = ()

L2 = ()2()(1+())

L3 = ()4()()(1+()),

and the estimate for the speaker reflection, we are able to estimate the wall loss transfer function

2() =L3

() L2^

^

Estimated and Theoretical Propagation Losses

() ^

Cylinder, Speaker-Open

From this measurement we observethe reflection from an open end, Rop().

Arrival Responses for an Open Cylinder

Y1 = ()

Y2 = ()2()Rop()(1+())

Y3 = ()4()R2op() ()(1+())

Open Cylinder Arrival Spectra

Y1

Y2

Y3

Open End Reflection

• Given the second arrival for the closed tube:

Y2 = ()2()Rop()(1+()),

• We are able to estimate the reflection from an open end

Rop() = Y2

L2

L2 = ()2()(1+())

• and the second arrival for the open tube:

^

Cylinder Open End Reflection

Cylicone, Speaker-Closed

We consolidate this measurement with the theoretical reflection and transmission filters at the junction:

• the cylinder: Ry() and Ty(), • the cone: Rn() and Tn()

Arrival Responses for Closed Cylicone

A1 = ()

A2 = …

A3 = …

Second Arrival, Closed Cylicone

A(2,1) = ()y()Ry()(1+())2

A(2,2) = ()y() n()Ty()Tn()(1+())2 2

Measured and Modeled Closed Cone Arrival

Cylicone, Speaker-Open

From this measurement, we observe the behaviourof the reflection filter from the cone’s open end.

Arrival Responses for Open Cylicone

N1 = ()

N2 = …

Second Arrival, Open Cylicone

Closed Cylinder Comparison

Open Cylinder Comparison

Closed Cylicone Comparison

Open Cylicone Comparison

Summary• We observed the behaviour of the

following waveguide filter elements, from measured impulse responses:– Open end reflection (cylinder and cone)– Propagation losses (cylinder and cone)– Junction reflection and transmission

(cylicone)

• We confirmed that the impulse response measurements matched the responses of the waveguide models.

Conclusions

• We observed and verified theoretical waveguide filter elements using our measurements.

• The measurement system yields very good data at relatively low cost.

• The validation of the measurement system implies it can be extended to any tube structure.

Acknowlegements

• We would like to thank Theresa Leonard and the Banff Centre for Performing Arts.

• Natural Sciences and Engineering Research Council (NSERC).