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An explanatory model for

sound radiation from

subsonic surface vibrations

IEEE International Ultrasonics Symposium 2022

HÅVARD KJELLMO ARNESTAD

Department of Informatics

University of Oslo

haavaarn@iﬁ.uio.no

ERLEND MAGNUS VIGGEN

Centre for Innovative Ultrasound Solutions

Department of Circulation and Medical Imaging

Norwegian University of Science and Technology

erlend.viggen@ntnu.no

Case: Surface vibration of speed c

v

generates

sound wave in adjacent ﬂuid of sound speed cf

Sound wave pressure

p=p0ei(kxx+kyy−ωt)

Vibrational normal velocity

vy=vy0ei(kxx−ωt)

ky=kq1−(kx/k)2=kq1−(cf/cv)2

λˆ

k

0

0.5

1

1.5

2

Norm. height y/λf

cv/cf=1.1

λˆ

k

0

0.5

1

1.5

2

Norm. height y/λf

cv/cf≈1.0

λˆ

kr

ki

0 0.5 1 1.5 2

0

0.5

1

1.5

2

Norm. surface position x/λf

Norm. height y/λf

cv/cf=0.95

Supersonic domain (cv>cf)

•ky=kp1−(cf/cv)2

•Vibration radiates outgoing wave

Near coincidence (cv≈cf)

•ky≈0

•Wave nearly parallel to surface

Subsonic domain (cv<cf)

•ky=ikp(cf/cv)2−1

•Evanescent wave clings to surface

THE CLASSICAL RADIATION MODEL...

No energy conservation

•Supersonic sound wave radiates power away

•Vibration does not weaken from power loss

Disagrees with exact guided wave results

•In the last 30 years, several articles on leaky

guided waves found subsonic radiation

•Via numerical solutions of characteristic equations:

Correct approach but gives no physical intuition!

Radiated intensity for leaky A0Lamb waves

1170 1180 1190 1200 1210 1220 1230

0 0.5 1 1.5 2

subsonic supersonic

Frequency (Hz)

Rad. intensity Iy(arb.units)

Steel plate in air

Exact IySubsonic radiation Iyfrom classical model

1670168016901700171017201730

subsonic supersonic

Frequency (Hz)

Brass plate in air

...AND ITS DEFICIENCIES

Complex wavenumber: kx=kxr+ikxi

Inhomogeneous sound wave pressure

p=p0ei(kxrx+kyry−ωt)e−kxixe−kyiy

Attenuated vibrational normal velocity

vy=vy0ei(kxrx−ωt)e−kxix

ky=krh1−(kxr/k)2+(kxi/k)2i−i2kxrkxi/k2

Consequences of attenuated surface vibration

•Sound waves become inhomogeneous; amplitude

increases exponentially along wavefront because:

•Constant amplitude along wave propagation direction

•Source amplitude increases exponentially as we go left

•Every attenuated vibration radiates an outgoing wave

•If kxr>0and kxi>0, then kyr>0

λˆ

krki

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

0

Normalised surface position x/λf

Normalised height y/λf

Supersonic and attenuated

cv/cf=1.05

λˆ

kr

ki

0 0.5 1 1.5 2 2.5 3

Normalised surface position x/λf

Subsonic and attenuated

cv/cf=0.98

0

1

2

3

4

5

6

subsonic supersonic

Norm. rad. intensity

kxi/k=0kxi/k=0.02 kxi/k=0.04

kxi/k=0.06 kxi/k=0.08 kxi/k=0.10

40

50

60

70

80

90

subsonic supersonic

Rad. angle θ(°)

0.5 0.75 1 1.25 1.5

0.5

0.6

0.7

0.8

0.9

1

subsonic supersonic

c=cv

c=cf

Norm. vibr. speed cv/cf

Norm. wave speed c/cf

Sound wave properties:

•Radiated intensity

•Iy=kyr/k

|ky/k|2

ρcf|vy0|2

2e−2kxix

•With attenuation: subsonic radiation

•Radiation angle

•θ=arctan kxr/kyr

•With attenuation: never grazing incidence

•Wave speed

•c=ω/|kr|=cf/q(kxr/k)2+ (kyr/k)2

•With attenuation: calways below cv

•More attenuation ⇒more smoothing

ATTENUATED SURFACE VIBRATIONS GENERATE INHOMOGENEOUS SOUND WAVES

Energy conservation requires balancing...

1Radiated intensity per length along surface

2Power loss per length along surface

Power ﬂow of the underlying guided wave

Px(x) = ZSISx(x,y)dy=Px0e−2kxix

SPx(x)Px(x+dx)

I(x)

Iy(x)

Leaky Rayleigh wave

S

Px(x)Px(x+dx)

I+(x)

I+

y(x)

I−(x)

I−

y(x)

Leaky Lamb wave

Power balance equation

Iy(x) = −dPx

dx(x) = 2kxiPx(x)

Radiation model equation

f(kxr,kxi) = kxi−Iy0(kxr,kxi)

2Px0(kxr,kxi)=0

•Implicit equation that considers inhomogeneity

•Literature: Explicit eqs., but no inhomogeneity

Radiation model predicts two modes

Radiating mode (extends into subsonic domain!)

Evanescent mode

(Comparison: Eqs. from literature, homog. waves)

subsonic supersonic

0.5 0.75 1 1.25 1.5 1.75 2

0

0.05

0.1

0.15

0.2

Norm. vibration speed cv/cf=k/kxr

Norm. attenuation kxi/k

f(kxr,kxi)with stronger radiation

subsonic supersonic

0.5 0.75 1 1.25 1.5 1.75 2

Norm. vibration speed cv/cf=k/kxr

f(kxr,kxi)with weaker radiation

THE NEW RADIATION MODEL

Comparison to leaky A0Lamb waves

1Validation: Is the exact leaky solution also a

solution of the radiation model equation?

•Yes! This supports the correctness of the model

2Perturbation method: Use radiation model to

ﬁnd immersed-plate attenuation from free-plate

solution?

(Reference: Exact leaky solution)

Homogeneous waves, free-plate dispersion kfree

xr

Inhomogeneous waves, free-plate dispersion kfree

xr

Inhomogeneous waves, exact leaky dispersion kleaky

xr

Perturbation method results

0 0.05 0.1

Attenuation (Np/m)

Steel in air

kleaky

xikhom

xifrom kfree

xrkxifrom kfree

xrkxifrom kleaky

xrfleaky

cffree

c

0 0.05 0.1

Brass in air

0 5 10 15

Steel in water

0 5 10 15

Brass in water

1180 1200 1220

335 340 345

Frequency (Hz)

Phase speed (m/s)

cleaky

v=ω/kleaky

xrcfree

v=ω/kfree

xrcffleaky

cffree

c

1680 1700 1720

335 340 345

Frequency (Hz)

0 20 40

500 1000 1500

Frequency (kHz)

0 20 40 60 80 100

500 1000 1500

Frequency (kHz)

Homogeneous waves from kfree

xr:

•Like Merkulov (1964), Auld (1973), Watkins et al. (1982)

•Inﬁnite attenuation at coincidence

•No subsonic radiation

Inhomogeneous waves from kfree

xr:

•Finite attenuation at coincidence

•Subsonic radiation, but wrong shape because of extra

dispersion around coincidence due to ﬂuid loading

Inhomogeneous waves from kleaky

xr:

•Very good match for plates in air

•Imperfect match for plates in water when not

considering all internal changes in Lamb waveﬁeld

•Excellent match when including these changes (not shown)

VALIDATION AND RESULTS

Fairly straightforward to derive an enhanced

radiation model

•Conserves energy — balances radiated power

and lost vibrational power

•Considers inhomogeneity of radiated waves

Model validated against leaky A0Lamb waves

•Exact Lamb solution also solves model equation

•Moderately successful as a perturbation method

Model explains subsonic radiation

•Because of inhomogeneity due to attenuation

•Attenuation due to power radiated into the ﬂuid

•Radiated mode exists only as long as energy

conservation is possible

CONCLUSIONS

Subsonic radiation from A0Lamb waves

•Dabirikhah and Turner (1992): "Anomalous behaviour of ﬂexural

waves in very thin immersed plates", presented at IEEE IUS 1992

•Dabirikhah and Turner (1996): "The coupling of the A0and interface

Scholte modes in ﬂuid-loaded plates", J. Acoust. Soc. Am.

•Kiefer et al. (2019): "Calculating the full leaky Lamb wave spectrum

with exact ﬂuid interaction", J. Acoust. Soc. Am.

•Kiefer et al. (2020): "From Lamb waves to quuasiguided waves: On

the wave ﬁeld and radiation of elastic and viscoelastic plates",

preprint on ResearchGate

•Ducasse and Deschamps (2022): "Mode computation of immersed

multilayer plates by solving an eigenvalue problem", Wave Motion

Subsonic radiation from Rayleigh waves; realised inhomogeneity is key

•Mozhaev and Weihnacht (2002): "Subsonic leaky Rayleigh waves at

liquid–solid interfaces", Ultrasonics

Existing perturbation methods

•Merkulov (1964): "Damping of normal modes in a plate immersed in a

liquid", Soviet Phys.: Acoust.

•Auld (1973): Acoustic Fields and Waves in Solids, vol. 2

•Watkins et al. (1982): "The attenuation of Lamb waves in the

presence of a ﬂuid", Ultrasonics

MAIN REFERENCES