On waveguide modeling of stiff piano string

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Eric DUCASSE - On waveguide modeling of stiff piano string - J. Acoust. Soc. Am. - Vol. 118,

n°3, p.1776-1781 - 2005

On waveguide modeling of sti piano strings

Éri Du asse

É ole NationaleSupérieured'Arts etMétiers, C.E.R. de Bordeaux-Talen e,33405 Talen e edex, Fran e

eri .du asseensam.eu

ABSTRACT

Bensa et al. [J. A oust. So . Am. 114(2) 10951107 (2003), Se tion IV℄ re ently proposed a waveguide

model for the transverse displa ement of a sti piano string. The study des ribed here is an attempt to ast

a omplementary light on this topi , based on a ommon wave approa h instead of a modal approa h. A pair

of weakly attenuated traveling waves and a pair of fast-de aying waves both satisfy the one-dimensional wave

equation developed by Bensa et al.. These solutions have to be arefully onsidered, however, for portions of

string intera ting with the hammer felt,the bridge,or the apo d'astro bar.

PACS number: 43.75.Mn

I. INTRODUCTION

The rst attempt for synthesizing musi al sounds using physi al models was made more than 30 years ago

by Hiller and Ruiz

1

. Starting from the one-dimensional wave equation

2

of the transverse displa ement of a

string, three main approa hes are generally used for buildinga omputationalmodel: either the use of a nite

dieren e s heme (e.g. Refs. 3, 4), or a modal synthesis (e.g. Ref. 5), or the building of a Digital Waveguide

model(e.g. Refs.6,7,3). Awaveguide is onsideredhereasalterlikestru ture modelingone-dimensionalwave

propagationas purely losslessthroughoutthe length of thestring, withlossand dispersionlumped in terminating

lters

3

. The hammera tionand theree tion onditions atbothendsof thestringare alsomodeledaslumped

lters. Portions ofstring are distributed elements

6

represented by two-portnetworks. Inthis paperwe fo us on

this latter approa h.

Bensa et al. propose an improved one-dimensional wave equation [Ref. 3, Eq. (6)℄ orresponding to a well

posed modelofsti pianostrings. In Se . IV.Bthey use amodalapproa h toextra tthe waveguide parameters

from the partial dierential equation (PDE). This approa h requires the knowledge of both the length of the

string and the boundary onditions [Ref. 3, Eq. (10)℄ for the al ulation of the vibration modes asso iated

with standing waves. The waveguide parameters are then dedu ed from the hara teristi s of the rst mode

in luding the fundamental frequen y of the ideal string. The present paperis an attemptto show that a wave

approa h using omplexwave-numbers appearsasmore onvenientbe auseea hdispersiveattenuatedtraveling

wave isdire tly obtained fromthe one-dimensional wave equation,apartfrom the ree tionsat the endsof the

string. Assumingthattheone-dimensionalwaveequationislinear,time-andspa e-invariant,atwo-dimensional

Fourier-Lapla e transformation gives an algebrai equation relating the transverse displa ement of the string

(output)tothespa edistributionofexternalfor es(input)inthewave-number/frequen y domain. Thesolution

of this equation an beseen as the superposition of traveling waves whi ho ur ina waveguide model.

Somegeneralpointsaboutwaveguides, travelingwaves, and omplexwave-numbers areoutlinedinSe tionII

tohelp the readertounderstandthe followingse tions. In Se tionIII, afterthe one-dimensionalwave equation

advan edbyBensaetal.

3

isre alled,thisequationissolvedtoobtaintheresponseofaninnitestringtoapoint

impulse inthe wave-number/frequen y domain,showing that two fast-de aying traveling waves are omittedby

the modal approa h in Bensa et al.

3

. These fast-de aying waves an be negle ted only for portions of string

without sour es whi h are long enough. They should, however, be taken into a ount at the neighborhood of

II. WAVEGUIDE MODELING AND COMPLEX WAVE-NUMBERS

The question is this: howto pass from the wave equationof a one-dimensional system to awaveguide model?

A. From the one-dimensional wave equation to the transfer fun tion

1. A general one-dimensional wave equation

The vibration state of a one-dimensional system (e.g. the sti piano string) is assumed to be dened by a

fun tion

y

(e.g. the transverse displa ement of the string) of time

t

and position

x

along the system.

Ageneralone-dimensionalwaveequation,assumingthatitislinear,time-andspa e-invariant, anbewritten

as:

dx

X

m=0

dt

X

n=0

α

m,n

∂

m+n

_y

∂ x

m

_{∂ t}

n

(x, t) = f (x, t) ,

(1)

where

f

is the spa e distribution of external for es (e.g. exerted by the hammer felt, the bridge and the apo d'astro bar).

2. Time/frequen y and spa e/wave-number onversions

In the spa e/frequen y domain, ea h signal

s(x, t)

is onverted into

bs(x, ω)

by Fourier transformation:

bs(x, ω) =

Z

+∞

−∞

s(x, t) e

−i ω t

dt ,

(2)

where

ω

is the angular frequen y and

i

2

_{= −1}

.

In the wave-number/frequen y domain,ea h signal

s(x, t)

be omes

S(k, ω)

by FourierLapla e transforma-tion:

S(k, ω) =

Z

+∞

−∞

bs(x, ω) e

−i k x

_{dx =}

Z

+∞

−∞

Z

+∞

−∞

s(x, t) e

−i (ω t+k x)

dt dx ,

(3)

where

k

is the omplex wave-number.

3. An ordinary dierential equation in the spa e/frequen y domain

The Fourier transformationof the waveequation (1)leads toanordinary linear onstant- oe ient dierential

equation with respe t to

x

:

dx

X

m=0

"

_dt

X

n=0

α

m,n

(i ω)

n

#

∂

m

_by

∂ x

m

(x, ω) = b

f (x, ω) .

(4)

4. The transfer fun tion

In the wave-number/frequen y domain,we obtain:

"

_dx

X

m=0

dt

X

n=0

α

m,n

i

m+n

ω

n

k

m

#

Y (k, ω) = F (k, ω) ⇐⇒ Y (k, ω) = H(k, ω) F (k, ω) ,

(5)

where

H(k, ω)

is the transfer fun tion of the system. The denominator of

H(k, ω)

is a polynomial of order

d

x

in

k

and order

d

t

in

ω

. The poles of

H(k, ω)

(where

H(k, ω)

is onsidered as afun tion of

k

)are the roots of a polynomialequation ommonly alled the dispersion equation.

B. From the transfer fun tion to the blo k-diagram model

1. The impulse response as the sum of pairs of symmetri al traveling waves

Assuming both that the propagation model orresponds to a well-posed physi al problem ( f. e.g. Ref. 3 for

more details) and that the problem is symmetri al (i.e., un hanged by substituting

−x

for

x

), the oe ients

α

2k+1,n

are zero, whi h impliesthat

d

x

= 2 J

is an even number, and the transfer fun tion

H(k, ω)

isrewritten as follows:

H(k, ω) =

a(ω)

J

Y

j=1

[ k

2

_{− k}

j

(ω)

2

]

=

J

X

j=1

a

j

(ω)

k

2

_{− k}

j

(ω)

2

=

J

X

j=1

g

j

(ω)



1

k − k

j

(ω)

−

1

k + k

j

(ω)



,

(6) where

a

j

(ω) = a(ω)/

Q

J

m=1, m6=j

[ k

m

(ω)

2

− k

j

(ω)

2

]

and

g

j

(ω) = a

j

(ω)/[2 k

j

(ω)]

, assuming

k

j

(ω)6=0

and

k

j

(ω)6=k

m

(ω)

if

j6=m

.

For an innite system, the assumed boundary (no sour es at innity) and initial onditions (rest initial

ondition) are:

y|

x→+∞

= y|

x→−∞

= 0 ; y|

t<0

= 0 ; ∀n , 1 6 n 6 d

t

− 1 ,

∂

n

_y

∂ t

n









t<0

= 0 .

(7)

Comingba k tothe spa e/frequen y domain, Eqs.(6) and (7) lead tothe onvolution produ twith respe t

to the spatial variable:

by(x, ω) =

Z

+∞

−∞

b

f (x

0

, ω) b

h(x − x

0

, ω) d x

0

,

(8)

bh(x, ω)

satisfying

bh(x, ω) =

J

X

j=1

g

j

(ω)



e

−i kj(ω) x

u(x) + e

+i kj(ω) x

_u(−x)



,

(9)

where

u

isthe Heavisideunit step fun tion:

∀ x < 0 , u(x) = 0 and ∀ x > 0 , u(x) = 1 .

Thefun tion

bh(x, ω)

orrespondstotheFouriertransformoftheimpulseresponse

h(x, t)

(alsonamedGreen's fun tion)to

f (x, t) = δ(x) δ(t)

where

δ

isthe Dira impulse. Thisresponse

h(x, t)

isthe ee t ofatimeimpulse (at

t = 0

)exerted onasingle point(at

x = 0

)of the system. Theunit of

h

isthe unit of

y

perNewton and per se ond. An inverse Fouriertransformation gives:

h(x, t) =

1

2 π

J

X

j=1

Z

+∞

−∞

g

j

(ω)



e

i [ ω t−kj(ω) x ]

u(x) + e

i [ ω t+kj(ω) x ]

_u(−x)

d ω .

(10)

Note that

h(x, t) = h(−x, t)

: the symmetry of the problem issatised. Ea h

k

j

(ω)

is a omplex wave-number su hthat:

k

j

(ω) = ω τ

j

(ω) − i α

j

(ω) =

ω

c

j

(ω)

− i α

j

(ω) ,

(11)

where

τ

j

(ω) > 0

isthe propagationdelay perunit length (orslowness),

c

j

(ω)

the phase velo ity,and

α

j

(ω) > 0

the attenuation perunit length.

The impulseresponse an berewritten as:

h(x, t) =

1

2 π

J

X

j=1

Z

+∞

−∞

g

j

(ω)



e

−αj(ω) x

e

i ω [ t−x/cj(ω)]

u(x) + e

+αj(ω) x

e

i ω [ t+x/cj(ω)]

_u(−x)

d ω .

(12)

This impulseresponse is onsequently the superpositionof

J

pairs of symmetri alde aying traveling waves starting fromthe ex itation point and propagatingin opposite dire tions.

Thus, the general solution

y

is given by the two-dimensional onvolution:

y(x, t) =

Z

+∞

−∞

Z

+∞

−∞

f (x

0

, t

0

) h(x − x

0

, t − t

0

) d x

0

d t

0

.

(13)

2. Waveguide modeling of a region without sour es

Under the assumption that no sour es exist in a region between

x = x

min

and

x = x

max

, one an demonstrate that the vibration state is the sum of

J

de aying traveling waves

y

+

j

in the in reasing

x

dire tion and of

J

de ayingtraveling waves

y

−

j

inthe de reasing

x

dire tion:

∀ x , x

min

6

x 6 x

max

, y(x, t) =

J

X

j=1

y

+

_j

(x, t) +

J

X

j=1

y

−

j

(x, t) ,

where c

y

_j

+

(x, ω) = e

−i kj(ω) (x−xmin)

_y

c

+

j

(x

min

, ω)

and c

y

_j

−

(x, ω) = e

+i kj(ω) (x−x

max

)

c

y

−

_j

(x

max

, ω) .

(14)

This region an bemodeled as

J

waveguidesin parallelwhi hbe ome

J

digital waveguides

e.g. 6, 7

indis rete

time. Ea htransfer fun tion

e

−i kj(ω) ∆x

isgenerally designed asadelay lineinseries with adigital ausal lter.

After suitabledigital lters havebeen found, e ient time-domainsimulations an be made.

Inthe aseofpianostrings,the sour esarelo atedinthesegmentofhammerfelt/string onta tand atboth

ends. A blo k-diagrammodelis drawn inFig. 1,in ludinganobservationpoint. This gure isageneralization

of the model involved in Bensa et al. (Ref. 3, Fig. 2). Note that the boundary onditions hara terizing the

stringterminations an be hangedby onlymodifyingonelumped lteratea h end,apartfromthe waveguides

modeling the portions of string withoutsour es. PSfragrepla ements

x

0

x

1

x

0

− δx

x

0

+ δx

by(x

1

, ω)

x

0

L

c

y

+

_{(x, ω)}

c

y

−

_{(x, ω)}

Left termination Righ t termination

H

am

m

er

/s

tr

in

g

in

te

ra

ct

io

n

H

1

(ω)

H

1

(ω)

H

2

(ω)

H

2

(ω)

H

3

(ω)

H

3

(ω)

Figure1: Blo k-diagramofthewaveguidemodelofapianostring. Thestringsegmentintera tingwiththehammerfelt isassumedtobe

[ x

0

−δx ; x

0

+ δx ]

.

x

1

isthepositionofobservation.

x

= 0

isthe apod'astrobarpositionand

x

= L

isthebridgeposition. Ea h

H

n

(ω)

isequivalentto

J

delayline/lterblo ksinparallel:

H

1

(ω) =

J

X

j=1

e

−

i k

j

(ω) (x

0

−

δx)

,

H

2

(ω) =

J

X

j=1

e

−

i k

j

(ω) (x

1

−

x

0

−

δx)

and

H

3

(ω) =

J

X

j=1

e

−

i k

j

(ω) (L−x

1

)

.

III. THE FOUR WAVES IN A STIFF PIANO STRING

In this se tion and the next one, we apply the prin iples given in the previous se tion to the one-dimensional

wave equation introdu ed by Bensa et al. [Ref. 3, Eq. (6)℄. The dispersion equation provides two pairs of

de ayingtraveling waves whi hneed a detailedanalysis.

A. The transfer fun tion of a sti piano string

The transverse displa ement

y

is assumed to satisfy [Ref.3, Eq. (6)℄:

∂

2

_y

∂ t

2

+ 2 b

1

∂ y

∂ t

− 2 b

2

∂

3

_y

∂ x

2

_{∂ t}

− c

2

∂

2

y

∂ x

2

+ κ

2

∂

4

y

∂ x

4

=

1

µ

f ,

(15)

where

b

1

isthe rst oe ientof damping(duetovis osity of theair),

b

2

the se onddamping oe ient (inner losses),

c =

p

T /µ

the transverse wave velo ity of string

4

,

T

the string tension,

µ

the linear mass density of string,

κ = c

p

E I/T

a stiness oe ient,

E

the Young modulus,

I = S r

2

the moment of inertia of the

ross-se tion,

S

the ross-se tion,and

r

the radius of gyration.

Inthe wave-number/frequen y domain,Eqs. (5)and (15) lead to the transfer fun tion:

H(k, ω) =

Y (k, ω)

F (k, ω)

=

1

µ [(−ω

2

_{+ 2 i ω b}

1

) + (c

2

+ 2 i ω b

2

) k

2

+ κ

2

k

4

]

.

(16) B. Complex wave-numbers

The two (

J = 2

) wave-numbers

k

s

and

k

d

satisfy:

k

2

_s

(ω) =

c

2

2 κ

2

"

−1 − i

2 ω

c

2

b

2

+

s

1 +

4 (κ

2

_{− b}

2

2

) ω

2

c

4

+ 4 i ω



1

c

2

b

2

− 2

κ

2

c

4

b

1

#

,

(17)

k

2

_d

(ω) =

c

2

2 κ

2

"

−1 − i

2 ω

_c

₂

b

2

−

s

1 +

4 (κ

2

_{− b}

2

2

) ω

2

c

4

+ 4 i ω



1

c

2

b

2

− 2

κ

2

c

4

b

1

#

.

(18)

TheseequationsaresimilartoEq.(11)inRef.3. Theapproa hes,however,aredierent. Inthemodalapproa h

3

β

+

and

β

−

are real-valued fun tions of the omplex frequen y

s

. In the present wave approa h

k

s

and

k

d

are the rootsof the dispersion equation and are omplex-valuedfun tionsof the ommonangularfrequen y

ω

(real number).

The assumptions[Ref. 3 Eqs. (24) and (25)℄

b

1

b

2

≪ c

2

; b

2

2

≪ κ

2

and b

2

1

≪ ω

2

(19) lead to

k

s

(ω) ≃

c

κ

r

ξ

2

− i

κ ω

c

3

_{(1 + ξ)}

r

2

ξ



b

1

+

c

2

_ξ

2 κ

2

b

2



(20) and

k

d

(ω) ≃

1

c

3

√

_{2 ξ (1 + ξ)}

(2 ω

2

_b

2

− c

2

ξ b

1

) − i

r

2

ξ

ω

c

,

(21) with

ξ = −1 +

r

1 +

4 κ

2

c

4

ω

2

.

(22)

C. Waveguide modeling of portions of string without sour es

Therstwave-number

k

s

hara terizesaweaklyattenuateddispersivetravelingwavesimilartotheonedes ribed by Bensa et al. [Ref. 3, Eq. (34)℄. At low frequen ies [

b

2

1

≪ ω

2

≪ c

4

/(4 κ

2

)

℄ the phase velo ity is lose to

c

. Its numeri alvalues are between

160

and

420 m · s

−1

for piano tones C2, C4, and C7 (Ref. 3, Table I). The

attenuationperunit length

α

s

≃ b

1

/c

is small: itsnumeri alvalues [

10 α

s

/ log(10)

℄are less than

0.1 dB · m

−1

.

These ondwave-number

k

d

representsafast-de ayingwavewithaveryhighphasevelo ity. Atlow frequen- ies the phase velo ity is approximately

c κ/(b

2

− κ

2

_b

1

/c

2

)

. Its numeri al values are in the range

2.6 × 10

5

to

1.62 × 10

6

_{m · s}

−1

. The attenuationperunit lengthis

α

d

≃ c/κ

. The numeri alvalues [

10 α

d

/ log(10)

℄of

α

d

are greaterthan

1.1 × 10

3

_{dB · m}

−1

. Forany length

∆x

greaterthan

1.8 cm

,

e

−αd

∆x

(the modulusof

e

−i kd

∆x

)isless than

10

−2

. This se ond wave is quasi-evanes ent i.e., its phase velo ity is almost innite and its attenuation

perunitlength ishigh. It ould represent the fa tthat apart ofthe energy of ahammerstrikewouldinstantly

propagate along the sti string around the onta t region su h as to avoidthe formation of a sharp orner, as

shown belowand in a ordan e with Cremer

8

.

Consequently, the waveguide model of a portion of string without sour es will ontain only a single delay

line/lter(wave-number

k

s

)forea hdire tionof propagation,provided that theportionofstringislongerthan a few entimeters. But the other wave hara terized by

k

d

annot be negle ted at the neighborhood of the ex itation region,as shown in the following se tion.

IV. The impulse response of the string

A. Spa e/frequen y domain

In the spa e/frequen y domain Eqs. (1618), (6), and (9) imply that the Fourier transform of the impulse

response is:

bh(x, ω) =



















c

h

+

s

(0, ω) e

−i ks(ω) x

+ c

h

+

d

(0, ω) e

−i kd(ω) x

x > 0

c

h

+

s

(0, ω) + c

h

−

s

(0, ω) + c

h

+

d

(0, ω) + c

h

−

d

(0, ω) x = 0

c

h

−

s

(0, ω) e

+i ks(ω) x

+ c

h

−

d

(0, ω) e

+i kd(ω) x

_{x < 0}

,

(23) where

c

h

+

s

(0, ω) = c

h

−

s

(0, ω) = g

s

(ω) =

−i

2 µ κ

2

_[k

s

(ω)

2

− k

d

(ω)

2

] k

s

(ω)

c

h

+

_d

(0, ω) = c

h

−

_d

(0, ω) = g

d

(ω) =

i

2 µ κ

2

_[k

s

(ω)

2

− k

d

(ω)

2

] k

d

(ω)

.

(24)

The omplex omplian es

g

s

(ω)

and

g

d

(ω)

(unit:

m · N

−1

) hara terize the onversion of for e tomotion.

B. Example of a C2 piano string

The response

y

of a C2 piano string (Ref. 3, Table I) to a downward Gaussian time impulse at a single point

x = 0

isgiven by Fig.2. This response isvery losetothe negativeof theimpulse response

h

inthe bandwidth 0

20 kHz

. Thetransversedispla ement

y

isthesumoftwoweaklyattenuatedtravelingwaves

y

±

s

(wave-numbers

±k

s

)andoftwofast-de ayingwaves

y

±

d

(wave-numbers

±k

d

)goingawaysymmetri allyfromtheex itationpoint. In agreement with Refs. 8 and 2 (Se tion 2.18), the bend is rounded appre iably by the stiness of the string.

The fast-de ayingwaves inhibit the sharp orner whi h ould begenerated by the weakly attenuated traveling

waves, ifthe formerwere not onsidered. Asshown inthepreviousse tion,onlythe weaklyattenuatedtraveling

waves exist far from the ex itation point (more than

2 cm

), as well as in the neighborhood of the ex itation pointafter

0.1 ms

.

-

2

0

2

4

6

8

10

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0

-

1

-

2

-

3

-

2

-

1

0

1

-

3

-

2

-

1

0

1

-

3

-

2

-

1

0

1

PSfragrepla ements

t

[m

s]

x

[cm]

ϕ(t)

[×10

5

N ]

y

(x

,t

)

[m

m

]

Figure 2: Response of a C2 piano string to a downward Gaussian impulse

f

.

f

(x, t) = f

0

δ(x) ϕ(t)

where

ϕ(t) =

−λ e

−

π λ

2

(t−τ )

2

.

f

0

= 1 N

,

λ

= 2 × 10

5

_s

−1

, and

τ

= 0.02 ms

. The parametersof the C2 piano stringare (Ref. 3, TableI):

c

= 160.9 m · s

−1

,

κ

= 0.58 m

2

· s

−1

,

b

1

= 0.25 s

−1

,and

b

2

= 7.5 × 10

−5

_m

2

· s

−1

. Theresponse

y

(plainline) isthesumofaweaklyattenuatedwave

y

s

(dashedline)andofaquasi-evanes entwave

y

d

(dottedline).

C. Non- ausal omponents of the ausal impulse response

An important additional point is that

y

±

s

and

y

±

d

are not ausal responses whereas their sum is ausal. A mathemati al explanation an be found for this by fo using on the impulse response at the ex itation point

[Eq. (24)℄. The signals

h

±

s

(0, t)

and

h

±

d

(0, t)

and the modulus of their Fourier transforms

g

s

(ω) = c

h

±

s

(0, ω)

and

g

d

(ω) = c

h

±

d

(0, ω)

are plotted inFig. 3 (inthe ase of a C2 piano string). The latter signal

h

±

d

(0, t)

is nearly an even fun tion (with respe t to

t

). The former signal

h

±

s

(0, t)

is nearly the sum of a onstant [

1/(4 µ c)

℄and of an odd fun tion.

(a)

-

0.02

-

0.01

0

0.01

0.02

0.03

-

0.25

0.

0.25

0.5

0.75

1.

-

0.02

-

0.01

0

0.01

0.02

0.03

-

0.25

0.

0.25

0.5

0.75

1.

PSfragrepla ements

t

[ms]

t

[ms]

c

µ

h

(0

,t

)

;

c

µ

h

±

s/

d

(0

,t

)

c

µ

h

(0

,t

)

;

c

µ

h

±

s/

d

(0

,t

)

(b)

0

5

10

15

20

0

0.2

0.4

0.6

0.8

1

cg

PSfragrepla ements

f = ω/(2 π)

[kHz]

Figure 3: (a) Impulse responses of a C2 piano string and (b) the modulus of their Fourier transforms times the

angularfrequen y. The omplian e

g

s

(ω) = c

h

±

s

(0, ω)

istheFouriertransformofthesignal

h

±

s

(0, t)

(weaklyattenuated waves, dashed line).

g

d

(ω) = c

h

±

d

(0, ω)

is the Fourier transform of

h

±

d

(0, t)

(quasi-evanes ent waves, dotted line). The impulse response

h(0, t)

(plain line) is two times the sum of

h

±

s

(0, t)

and

h

±

d

(0, t)

. Its Fourier transform is

bh(0, ω) = 2 [g

s

(ω) + g

d

(ω)]

.

Thisis naturalbe ause, onassumingthat

b

1

and

b

2

are zero, and

ω

isgreaterthan zero[see Eqs. (17), (18), (22), and (24)℄,the omplian es be ome:

g

s

(ω) ≃

i κ

µ c

3

√

_{2 ξ (1 + ξ)}

and g

d

(ω) ≃

−

√

ξ

2 µ

√

2 c (1 + ξ) ω

.

(25)

Consequently, the omplian e

g

s

is an imaginary-valued fun tion and

g

d

is a real-valued fun tion. Be ause the signals

h

±

s

(0, t)

and

h

±

d

(0, t)

are real-valued, the former should be an odd fun tion and the latter an even fun tion. ButtheFouriertransformof

h

±

s

(0, t)

has asingularityat

ω = 0

andthe limitof

h

±

s

(0, t)

,when

t

tends to

−∞

, has to be zero. This implies that a onstant has to be added to

h

±

s

(0, t)

, giving an additional term

π δ(ω)/(2 µ c)

inits Fouriertransform.

Applying the nal value theorem to the signal

h

±

s

(0, t)

, the limit of

h

±

s

(0, t)

with

t

tending to innity is

1/(2 µ c)

(see Fig. 3). The nal value of the impulse response

h(0, t)

issimilarly

1/(µ c)

. This lastproperty is false if

b

1

> 0

: all the signals tend slowly to zero when

t

tends to innity. Indeed,

h

±

d

(0, t)

is lose to zero for great values of

t

and

h(0, t) ≃ h

+

s

(0, t) + h

−

s

(0, t)

. In summary, the two waves

h

±

s

and

h

±

non-Y

PSfragrepla ements

x = x

0

x

by(x

0

, ω)

e

−i k

s

(ω) L

e

−i k

s

(ω) L

e

−i k

s

(ω) L

′

e

−i k

s

(ω) L

′

bh(0, ω)

g

s

(ω)

b

ϕ(ω)

Figure4: Blo k-diagramofaportionofstringex itedbyapointfor e. Thisfor e

ϕ(t)

isexertedbythehammerfelt attheposition

x

= x

0

. Thefor e distributionisassumedto be

f

(x, t) = ϕ(t) δ(x − x

0

)

for

x

in therange

x

0

− L

to

x

0

+ L

′

(

L

and

L

′

arebothgreaterthanafew entimeters).

This implies that if the for e distribution

f (x, t) = δ(x − x

0

) ϕ(t)

is assumed to represent a point-for e exerted by the hammer felt at the position

x = x

0

, then the response

y(x

0

, t)

at the ex itation point satises

by(x

0

, ω) = 2 [ g

s

(ω) + g

d

(ω) ]

ϕ(ω)

b

. This ase isrepresented inFig. 4asablo k-diagram. Similarblo k-diagrams are used forbowed strings( f. e.g. the review arti le by Smith

9

, Fig. 14).

V. CONCLUSION AND PROSPECTS

The on lusion of this paperis that a omplete mathemati al treatment of the one-dimensional wave equation

(15) arefully takes into a ount all the waves provided by the dispersion equation. Fast-de aying waves an

benegle ted inthe waveguidemodelingof portions ofstring withoutsour esif they are long enough. However,

they annota urately be negle ted atthe neighborhoodof the ex itationregion (hammer) and of the xation

points(bridge, apo d'astro bar).

In parti ular if a point-for e is exerted by the hammer felt on the string, the waveguide model proposed

by Bensa et al. (Ref. 3, Fig. 2) needs to be omplemented at the ex itation point, as shown in Fig. 4 whi h

summarizes the resultsof the lastse tion. Notethat arealisti ex itationby the hammeris distributed over a

ertainwidth

4

. This extension ould bemade by aspatial onvolution. A future study may determinewhether

or not this orre tion would noti eably improvethe quality of synthesized sti-stringsounds.

This paper is an attempt toextra t as mu h information as possible from the physi almodeldened by a

one-dimensional wave equation, in a waveguide modeling ontextand froma mathemati alpoint of view. It is

hoped that this study will help to improvefuture waveguide models of strings and toenhan e the a ura y of

e ient omputationalmodelsusing Digital Waveguides

6, 7, 3

,afterasubstantial eortto designsuitabledigital

REFERENCES

1

L. Hiller and P. Ruiz: Synthesizing musi al sounds by solving the wave equation for vibrating obje ts.

J. Audio Eng. So . 19 462472(Part I) and 542551(Part II) (1971).

2

N. Flet her and T. Rossing, The Physi s of Musi al Instruments (2nd ed., ISBN 0387983740,

Springer-Verlag, New York,1999).

3

J. Bensa, S. Bilbao, R. Kronland-Martinet, and J. O. Smith III:  The simulation of piano string vibration:

from physi al model to nite dieren e s hemes and digital waveguides. J. A oust. So . Am. 114(2)

10951107 (2003).

4

A. Chaigne and A. Askenfelt: Numeri al simulations of stru k strings. I. A physi al model for string using

nite dieren e methods. J.A oust. So .Am. 95(2)11121118 (1994).

5

J.-M.Adrien: TheMissing Link: ModalSynthesis. inRepresentations of Musi alSignals. MITPress, 1991,

269297.

6

J. O.Smith III: Physi almodeling using digitalwaveguides. Computer Musi J. 16(4) 7491(1992).

7

J. O.Smith III: Physi almodeling synthesis update. Computer Musi J. 20(2) 4456(1996).

8

L. Cremer, The Physi s of Violin (translation by J.S. Allen,MIT Press, Cambridge,Massa husetts, 1984).

9

J. O. Smith III: Virtual a ousti musi al instruments: Review and update, J. New Musi Resear h 33(3)