Curved cylindrical duct at resonance (1st, 5th, 10th and 12th)

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A non-plane-wave resonance of a curved cylindrical duct

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Linear vs nonlinear for a curved cylindrical duct

0 5 10 15 20 25 30

kL

10-2

100

102

|Z|

Linear

M=0.20

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Helical vs equiv. length equiv. curvature non-helical duct

0 5 10 15 20 25 30

kL

10-2

100

102

|Z|

Curved

Helical

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Resonances of a closed helical duct (1st, 5th and 10th)

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13th resonance of a closed helical duct

Binormal at exit into page Normal at exit out of page

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Linear vs nonlinear impedance for a helical duct

0 5 10 15 20 25 30

kL

10-2

100

102

|Z|

Linear

M=0.20

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5th resonance of a closed helical duct (M = 0.1)

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Impedance for a simplified trombone

0 200 400 600 800 1000

Frequnecy (Hz)

102

103

104

105

|Z| (P

a s

m-3

)

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Trombone impedance with and without mouthpiece

0 200 400 600 800 1000

Frequency (Hz)

101

102

103

104

105

|Z| (P

a s

m-3

)

Mouthpiece

No mouthpice

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Simplified trombone at resonance (2nd, 4th and 8th)

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Nonlinearity in a simplified trombone

0 200 400 600 800 1000

Frequency (Hz)

101

102

103

104

105

|Z| (P

a s

m-3

)

Linear

M=0.01

M=0.05

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Pressure within a simplified trombone

0 1 2 3 4

time (s) 10-3

-3

-2

-1

0

1

2

3

pre

ssure

(P

a)

104

0 1 2 3 4

time (s) 10-3

-2

-1

0

1

2

3

pre

ssure

(P

a)

104

Pressure in the mouthpiece Pressure in the pipe just before the second bend

0 1 2 3 4

time (s) 10-3

-1000

0

1000

2000

pre

ssu

re (

Pa

)

Pressure radiated at the bell

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Simplified trombone at M = 0 and M = 0.1

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Conclusion

Generalization of Félix & Pagneux (2001 JASA; 2002 WM) to weak nonlinearity.

Generalization of Fernando, Druon, Coulouvrat & Marchiano (2011 JASA) to

non-straight ducts.

New concept of weakly nonlinear reflectance, impedance, admittance, transmission

coefficients, etc.

New algebra (e.g. Ψ[α]β [r], R[p, p]) to handle complicated expressions.

Expand in a basis of straight duct modes (from Félix & Pagneux):

Calculate linear and nonlinear admittance Y and Y at duct exit.Solve ODE to find Y and Y throughout duct.Use pressure at duct inlet to find pressure everywhere.

Applied to curved and helical cylindrical ducts of varying width. Other duct shapes are

also possible.

Possible implications for brass and woodwind instruments.

For details of 2D, see McTavish & Brambley (2019 JFM).

For details of 3D, see future JFM paper (McTavish & Brambley).

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Work by Cambridge PhD student James McTavishAim of this workWave steepeningMusical ShocksGoverning equations IGoverning equations IIGoverning equations IIIGeometryGeometry IIGeometry IIIGeometry IVExpansion in terms of straight duct modesAlgebra IAlgebra IICan calculate explicitly coefficients, e.g.:Solution using impedance/admittanceSolution using impedance/admittance

Exit admittance into a straight uniform ductOther nonlinear generalizationsCurved cylinder (linear)Curved cylinder ($M=0.05$)Curved cylinder ($M=0.10$)Curved cylinder ($M=0.15$)Exponential HornHelix Linear $au h=0.14$Helix $M=0.05$ $au h=0.14$Helix $M=0.10$ $au h=0.14$Helix $M=0.15$ $au h=0.14$Helix Linear $au h=0.20$Helix $M=0.05$ $au h=0.20$Helix $M=0.10$ $au h=0.20$Helix $M=0.15$ $au h=0.20$Helix Linear $au h=1.00$Helix $M=0.025$ $au h=1.00$Helix $M=0.05$ $au h=1.00$Helix $M=0.10$ $au h=1.00$What we don't yet so farAdmittance at a baffled opening IAdmittance at a baffled opening I

Admittance at a baffled opening IIDipole pressure source IDipole pressure source IIOpen ended ducts IOpen ended ducts I

Open ended ducts IIComparison of duct exit with Wiener--HopfLinear resonances of an open ductLinear resonances (1, 5 and 10)Linear vs nonlinear impedance for $L=8h$Modal amplitudes at 5th resonanceDuct exit at resonance (5th and 10th),$L=8h$, $M=0.1$Exponential horn vs equivalent length straight ductThe closed horn at resonances (1, 5 and 10)Linear vs Nonlinear impedance in an exponential hornClosed horn at resonance (5th and 10th),$M=0.05$Conical vs straight duct impedanceClosed conical horn at resonance (1st, 5th and 10th)Linear vs nonlinear for a conical ductClosed conical duct at resonance (5th and 10th),$M=0.1$Curved vs straight cylindrical ductCurved cylindrical duct at resonance (1st, 5th, 10th and 12th)A non-plane-wave resonance of a curved cylindrical ductLinear vs nonlinear for a curved cylindrical ductHelical vs equiv. length equiv. curvature non-helical ductResonances of a closed helical duct (1st, 5th and 10th)13th resonance of a closed helical ductLinear vs nonlinear impedance for a helical duct5th resonance of a closed helical duct ($M=0.1$)Impedance for a simplified tromboneTrombone impedance with and without mouthpieceSimplified trombone at resonance (2nd, 4th and 8th)Nonlinearity in a simplified trombonePressure within a simplified tromboneSimplified trombone at $M=0$ and $M=0.1$Conclusion