dicate thatincreasingr.

Physics Letters A 327 (2004) 91–94

www.elsevier.com/locate/pla

Sound generated by rubbing objects

Zhen Ye

Wave Phenomena Lab, Department of Physics, National Central University, Chungli, Taiwan, ROC

Received 3 February 2004; received in revised form 5 May 2004; accepted 6 May 2004

Available online 18 May 2004

Communicated by P.R. Holland

Abstract

Here we present a general discussion of the properties of the sound generated by rubbing two objects. The results intonal features of the sound can be generated due to the finiteness of the rubbing surfaces. It is also shown that withrubbing speed, more and more high frequency tones can be excited and the frequency band gets broader and broade 2004 Elsevier B.V. All rights reserved.

PACS: 43.20.+g

Keywords: Sound generation

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Rubbing objects is an everyday experience. Peowho have gone through cold winters may have rubhands to get warm. Rubbing also occurs in natuThe earthquake rupture is one of the most familiardisastrous ones. A common observation is that rubbcan generate sound or noise. The sound generaterubbing hands is certainly common to nearly evone. The sound generation by ice-floe rubbing inArctic ocean contributes significantly to the oceambient noise[1,2], and may also help sea animain finding appropriate holes or breaking segmentsice to breathe. The sound generated by rubbing spagainst frying pans, however, is likely something mpeople would like to avoid.

E-mail address: zhen@phy.ncu.edu.tw (Z. Ye).

0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2004.05.010

y

Although the sound generated by rubbing objecta common experience, there is not much discussiothe literature. In the present Letter, we wish to consithe sound generated by rubbing two finite objetogether. A model is proposed for the sound generadue to the roughness of the contacting surfacethe two objects. The sound field is calculated andanalyzed for its relation to the size of the objecroughness of the contacting surfaces, and the rubspeed. Some general features are proposed. It is sthat the sound field can be expressed in termsa series of eigen-mode excitations, leading to tofeatures. It is suggested that when the rubbing spis low, only low frequency modes are excited. Wincreasing speed, more and more high frequemodes can be excited, making the frequency specbroader. These features appear to be in accordwith our intuition or daily experience. We must al

.

92 Z. Ye / Physics Letters A 327 (2004) 91–94

er

o-s

im-al;ill

ects

eldd asoughorm.g.,inves

avesannds

ide

ean

thattting

Fig. 1. Conceptual layout of the system.

point out that at this initial stage we will not considthe nonlinear and dissipation effects.

Consider the problem of two object rubbing tgether, as illustrated inFig. 1. One of the surfaces imoving with a constant velocity along the positivey

axis, while the other is assumed to be at rest. For splicity, we assume that the two objects are identicfor different sizes, the effective rubbing surface wequal that of the smaller one. The shape of the objis a plate with thicknessL, width W and heightHalong thex-, y-, andz-th axes, respectively.

When rubbing occurs, a shear displacement fiwill be generated at the surfaces and is revealeshear waves. Such shear waves can leak out thrdefects or radiate from the boundaries, and transfinto the sound we hear or record by machines, emicrophones. A discussion of this can be foundRef.[2]. For example, when the generated shear waencounter a defect at a boundary, the shear wwill be scattered into different directions, and cproduce longitudinal waves which become the souwe record or hear.

The governing equation for the shear waves insthe objects is

(1)

(∂2

∂t2− c2

s ∇2)

u(x, y, z, t) = 0,

wherecs is the shear speed of the material.

The boundary conditions are[3]

(2)∂u

∂z

∣∣∣∣z=0,H

= 0, u|y=0,W = 0,

and

(3)∂u

∂x

∣∣∣∣x=L

= 0, µ∂u

∂x

∣∣∣∣x=0

= F(y, z, t),

whereµ is the shear modulus of the material.The stress can be separated into two parts: the m

and the variation, i.e.,

F(t, y, z) = ⟨F(t, y, z)

⟩ + �F(y, z, t);here〈·〉 denotes the ensemble average. We assume〈F(t, y, z)〉 is independent ofy, z. It can be shown thathe averaged stress can only excite the unintereszero-mode, and will thus be ignored.

By Fourier transformation,

(4)u(x, y, z, t) =∫

dωe−iωt u(x, y, z,ω),

Eq. (1)becomes

(5)(∇2 + k2

s

)u(x, y, z,ω) = 0,

with

ks = ω

cs

.

The general solution toEq. (5)with Eqs. (2) and (3)is

u(x, y, z,ω) =∞∑

n=0,m=0

Amn cos(kmn(x − L)

)

(6)×sin(kmy)cos(knz),

where

kn = nπ

H, km = mπ

W,

kmn =√

k2s − k2

m − k2n,

andm,n are positive integers.The coefficientAmn is determined from∑

m,n

kmnAm,n sin(kmnL)sin(kmy)cos(knz)

(7)= 1

µF(ω,y, z).

ThenAmn is solved as

Z. Ye / Physics Letters A 327 (2004) 91–94 93

d

not

earat

isrt

h-t theom,enbing-

me

ceughn be

ten-

the

e-s thef

x-

t theext,

ing

po-e

Amn = 4

HW [1+ δn,0][1+ δm,0]1

kmn sin(kmnL)

1

µ

(8)

×H∫

0

dz

W∫0

dy F(ω,y, z)sin(kmy)cos(knz),

with

F(ω,y, z) = 1

2π

∫dt eiωtF (t, y, z).

It is easy to verify that〈Amn〉, caused by the averagestress, is proportional toδn,0δm,0. Therefore, only zeromodes are possible for constant stresses. We willdiscuss this situation further.

The purpose is to calculate the intensity of the shfield which is related to the relevant sound field. This, we need calculate the correlation function

D(x,y, z, t;x ′, y ′, z′, t ′)(9)≡ ⟨

u(x, y, z, t)u�(x ′, y ′, z′, t ′)⟩.

From the above derivation, it is clear that the keyto find the correlation function of the fluctuating paof the stress at the surface,〈�F(y, z, t)�F(y ′, z′, t ′)〉.Obviously the fluctuation is caused by the rougness of the contacting surfaces. If we assume tharoughness is homogeneous and completely randi.e. spatially uncorrelated at different points whthe system is at rest. When the surfaces are rubagainst each other along they-axis, the spatial separation along this direction is correlated at a later tiwhich is determined by�t = (y − y ′)/v. This consid-eration leads to⟨�F(y, z, t)�F(y ′, z′, t ′)

⟩(10)= Sδ(z − z′)δ

(y − y ′ − v(t − t ′)

),

whereS is a strength factor related to the rubbing forand the properties of the surfaces. For different rosurfaces, the stress correlation may vary and caincorporated into the formulation.

By using Eq. (10), the intensity field can becalculated. The procedure is to substituteEqs. (6)and (8)into Eq. (4), then calculateEq. (9)by takinginto account ofEq. (10). We finally get⟨u(x, y, z, t)u�(x, y, z, t)

⟩= S

∫dω

∑n

cos2(knz)

×∑m,m′

cos(kmn(x − L)

)sin(kmy)

× cos(km′n(x − L)

)sin(km′y)

× H

2Cmn,m′n

km

(ω/v)2 − k2m

((−1)mei ω

v W − 1)

(11)× km′

(ω/v)2 − k2m′

((−1)m

′ei ω

v W − 1).

This equation can be rewritten as⟨u(x, y, z, t)u�(x, y, z, t)

⟩(12)= S

µ2

∫dω

∑n

cos2(knz)∣∣Qn(x, y,ω)

∣∣2,where

Qn(x, y,ω) =∑m

2

W sin(kmnL)

× cos(kmn(x − L)

)sin(kmy)

(13)

× km

(ω/v)2 − k2m

((−1)mei ω

vW − 1

).

Therefore, the frequency spectrum of the sound insity field generated by rubbing is

(14)

P(x, y, z,ω) = S

µ2

∑n

cos2(knz)∣∣Qn(x, y,ω)

∣∣2,

with Qn being given byEq. (13).A few general features can be observed from

spectral formula. First, the strength factorS controlsthe overall sound intensity level. It is expected to dpend on a few parameters of the surfaces, such afriction coefficients and themechanical properties othe surfaces. Second, due to the factor sin(kmnL) inthe denominator ofQn, the resonance feature is epected to appear when sin(kmnL) = 0, leading to thephenomenon of tonal sounds. This also shows thathickness mainly defines the resonance feature. Nthe rubbing speedv entersQn in the form of ω/v

in the denominator. For a fixed frequency, decreasrubbing speed will reduced the strength ofQn. There-fore, with decreasing speed, high frequency comnents tend to decay accordingly. Furthermore, sinc

kmn =√

(ω/cs)2 − (mπ/W)2 − (nπ/H)2,

the cut-off frequencies are determined by

94 Z. Ye / Physics Letters A 327 (2004) 91–94

n,

ri-ex-

ust-Allportatedalodelob-ob-eedfre-erty

an

orant;e

henge.

ted

o

edednce, the

ben).bandtent

aterub-thatss ise ofthe

m toeichlude,

andthectorer-t thens.

o-ngare

ic

Fig. 2. Frequency spectra for tworubbing speeds. In the simulatiowe scale all lengths by the widthW , and the frequencyω is scalednon-dimensionally asωcs W .

(ω/cs)2 − (mπ/W)2 − (nπ/H)2 � 0.

These features comply with the intuition or expeence. It is apparent that the present model can betended to consider other rubbing surfaces, by adjing the correlation function of the surface stress.the features deduced from the theory tend to supthe experimental observation of the sound generby ice-floe rubbing[2]. We note that the experimentresults have been interpreted by an alternative m[4]. Depending on the geometries of the rubbingjects, the tonal feature may or may not exist or beservable. But the result that increasing rubbing spwould broaden the frequency spectrum in the highquency region is conjectured to be a general propof this process.

To quantify our discussion, let us considerexample. Suppose thatW = H , andL = 0.1W . In thecomputation, we assume that (1) the strength factS

is independent of the rubbing speed and is const(2) the physical conditions will not change in thrubbing process. These assumption may fail wthe rubbing speed or the rubbing force is too larTwo speeds are considered:v/cs = 1/600 and 1/300.As an example, we compute the sound field loca

at (L,W/2,0). The frequency spectra for the twrubbing speeds are presented inFig. 2. The units arearbitrary. Here it is clearly shown that there is indea cut-off frequency at about 200 for the lower specurve. Also for the lower speed case, the resonafeature does appear. For the higher rubbing speedspectrum is broader, but the resonance tends toweaker. The cut-off frequency is larger (not showCompared to the lower speed case, the frequencyis obviously broader. All these results are consiswith the qualitative analysis.

In summary, a model is established to investigthe spectral properties of the sound generated bybing two objects. In the model, we have assumedthe contacting surfaces are rough and the roughnedescribed as completely random. The dependencthe frequency spectrum on the rubbing speed andobject’s geometries is discussed. The results seebe in line with intuition or experience. However, wpoint out that there are a few relevant issues whhave not been addressed here. These issues incfor example, (1) the relation between the soundthe statistical properties and the characteristics ofsurface roughness; these effects may affect the faS; (2) possible variation of the physical properties vsus the rubbing strength and speed. It is hoped thapresent work can stimulate experimental exploratio

Acknowledgements

The work is stimulated by a discussion with ChaHisen Kuo. The communication with Dr. Gang Sheis appreciated. The referee’s careful commentsthanked.

References

[1] I. Dyer, Phys. Today 41 (1) (1988) 5.[2] Y. Xie, D. Farmer, J. Acoust. Soc. Am. 91 (1992) 1423.[3] L.M. Brekhovskikh, Waves in Layered Media, Academ

Press, New York, 1980.[4] Z. Ye, J. Acoust. Soc. Am. 97 (1995) 2191.