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TWO-DIMENSIONAL FINITE AMPLITUDE ACOUSTIC WAVES RADIATING FROM A FLAT PLATE IN ARBITRARY PERIODIC VIBRATION.
J. Ginsberg
To cite this version:
J. Ginsberg. TWO-DIMENSIONAL FINITE AMPLITUDE ACOUSTIC WAVES RADIATING
FROM A FLAT PLATE IN ARBITRARY PERIODIC VIBRATION.. Journal de Physique Col-
loques, 1979, 40 (C8), pp.C8-35-C8-38. �10.1051/jphyscol:1979807�. �jpa-00219512�
JOURNAL DE PHYSIQUE Colloque C8, supplement au n°ll, tome 40, novembre 1979, page C8- 35
TWO-DIMENSIONAL FINITE AMPLITUDE ACOUSTIC WAVES RADIATING FROM A FLAT PLATE IN ARBITRARY PERIODIC VIBRA- TION
J.H. Ginsberg
School of Mechanical Engineering, Purdue University, West Lafayette, Indiana, 47907, USA.
Résumé.- Les ondes acoustiques rayonnées par une plaque plane périodiquement soutenue subissant une vibration arbitraire sont déterminées par des méthodes asymptotiques. Une équation non linéaire d'onde est résolue pour le potentiel de vitesse. Puis, les expressions dérivées pour la vitesse et la pression sont rendues uniformément précises en employant la méthode de renormalisation pour dila- ter les coordonnées physiques. La réponse prédite par la théorie linéaire consiste en une série d'ondes sinusoïdales, dont chacune a une vitesse de phase qui dépend de la fréquence et de la lon- gueur d'onde comptée sur la plaque. Cependant, certains harmoniques engendrés par les effets non li- néaires ont la même vitesse de phase que les ondes prévues par la théorie linéaire. En conséquence, le rayonnement est constitué de groupes d'onde non dispersifs entre eux, mais avec une équation de dispersion affectant chaque groupe d'onde. Les transformations de coordonnées, et donc les phénomè- nes de distorsion, pour chaque groupe non dispersif sont proportionnelles aux contributions du grou- pe aux composantes de vitesse totale, tandis qu'elles sont indépendantes du rayonnement associé aux autres groupes.
Abstract.- The two-dimensional acoustic waves that radiate from a periodically supported plate under- going an arbitrary periodic vibration are determined by asymptotic methods. A nonlinear wave equa- tion is solved for the velocity potential. The derived expressions for the particle velocity compo- nents and pressure are then rendered uniformly accurate by employing the method of renormalization to strain the physical coordinates. The response predicted by linear theory consists of a series of sinusoidal waves, each of whose phase velocity is dependent on its frequency content and wavelength tangent to the plate. However, some of the harmonics generated by nonlinear effects have the same phase velocity as the sinusoidal waves predicted by linear theqry. As a result the wave motion con- sists of groups of waves that are nondispersive relative to each other, but there is a dispersion relation for the separate wave groups. The coordinate strainings, and thus the distortion phenomena, for each group of nondispersive waves are found to be proportional to the contribution of that group to the components of velocity, whereas they are independent of the wave motion contained in the other groups.
1. Introduction.- A convenient method for evalua- ting the distortion of small, but finite, amplitude acoustic waves has been derived in a series of in- vestigations, beginning with /1//2/. The method ex- pands the velocity potential in a perturbation se- ries, from which non-uniformly accurate expressions for the velocity components and pressure are deri- ved. Uniform accuracy is then obtained by employing the method of renormalization / 3 / , which involves a coordinate straining transformation.
The initial application of this method to a multi-dimensional system / 4 / treated the acoustic waves that radiate from an infinite flat plate whose transverse displacement has a harmonic spatial distribution in one direction, and is also harmonic in time. Subsequent investigations /5//6/ of the same problem recast the initial formulation into a more direct form, and also corrected certain inac- curacies pertaining to the coordinate straining.
Thus far, the technique has only been employ- ed in multi-dimensional situations where the
linearized version of the wave motion is sinusoidal.
The following investigation describes the response of the plate system described above for a more ge- neral excitation.
2. Velocity Potential.- An infinite flat plate, pinned to line supports that are spaced at a cons- tant interval L, is the boundary for an acoustic fluid contained within the upper half-space. The transverse displacement of the plate is considered to be sinusoidal in space, but merely periodic in time, having a dimensional period of 2-rr/fi and a zero mean value. Cartesian coordinates defining position relative to one support, tangent and normal to the plate, are selected as Lx and Lz, respectively, and time is measured by t/Q. The transverse displace- ment of the plate may then be written as sLw(x,t).
The value of e is small in most practical situa- tions, so it will be used as the perturbation para- meter. A suitable Fourier series for the displace- ment is :
Article published online by EDP Sciences and available at
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979807
C8- 36 JOURNAL DE PHYSIQUE
w(x,t) =
izl
wi cos ( i t+
Qi) s i n (NTx) ( 1 ) where N i s an i n t e g e r , and t h e values o f wi and $i are s p e c i f i e d parameters.The p a r t i c l e v e l o c i t y components o f t h e f l u i d are denoted as LRvX and LRv,, and a nondimensional v e l o ~ i t y p o t e n t i a l f u n c t i o n @(x,z,t) i s d e f i n e d such t h a t :
-
am
-a@
v X - x
,
v z - - az ( 2 )The c o n t i n u i t y c o n d i t i o n f o r t h e v e l o c i t y compo- nents normal t o t h e s u r f a c e r e q u i r e s t h a t :
The n o n l i n e a r wave equation governing t h e po- t e n t i a l f u n c t i o n f o r an O(E) a c o u s t i c wave i s /7/ :
where V ( ) denotes t h e two-dimensional s p a t i a l gra- d i e n t i n terms o f t h e nondimensional s p a t i a l coor- d i n a t e s x and z, and y i s the r a t i o o f s p e c i f i c heats. Also, the parameter K i s a reduced frequency i n v o l v i n g t h e phase v e l o c i t y co o f an i n f i n i t e s i m a l p l a n a r wave i n the reference s t a t e .
A s o l u t i o n o f eq.(4) s a t i s f y i n g boundary con- d i t i o n ( 3 ) , as w e l l as t h e r a d i a t i o n c o n d i t i o n a t l a r g e distances from t h e p l a t e , i s obtained by ex- panding @ i n the f o l l o w i n g p e r t u r b a t i o n s e r i e s .
The equations corresponding t o t h e E and E~
terms t h a t a r i s e from s u b s t i t u t i o n o f eq.(6) i n t o eq.(4) are :
"
2
2 2
a
V
m2 -
K 2 = K2{(y-1) -v
$ 1+
-([email protected]@l))a t 2 a t a t
(76) Only t h e O(E) term r e s u l t i n g from s u b s t i t u t i o n o f eq.(6) i n t o eq.(3) i s needed. This i s :
t h e fundamental frequency K exceeds t h e c u t o f f f r e - quency separating h i g h frequency s i n u s o i d a l waves from low frequency e x p o n e n t i a l l y decaying waves.
This means t h a t :
I t i s r e a d i l y v e r i f i e d t h a t t h e corresponding solu- t i o n f o r i s :
where the wave numbers hi are :
With a view toward s o l v i n g eq. (7b), consider t h e general r e s u l t of s u b s t i t u t i o n o f eq.(lO). Af- t e r t h e t r i g o n o m e t r i c i d e n t i t i e s f o r the s i n e and cosine o f the sum o f two angles i s employed, eq.(7b) wi 11 c o n t a i n t h e f o l 1 owing types o f nonhomogeneous terms :
The p a r t i c u l a r s o l u t i o n s corresponding t o these terms are unbounded when t h e nonhomogeneous terms a r e n o t orthogonal t o the complementary s o l u t i o n Such a c o n d i t i o n occurs whenever
K 2 ( i
*
j j 2 = (hi l j ) 2 o r2 2 2
K ( i
*
j ) ' = (hi'
hj12+
4N n (12) Upon s u b s t i t u t i o n o f e q . ( l l ) , b o t h o f t h e f o r e g o i n g c o n d i t i o n s are found t o occur o n l y when :i = j (13)
Thus, t h e s e c u l a r terms i n the equation f o r @2 a r i s e from q u a d r a t i c products o f each term forming
m1
i n eq.(lO). Aside from t h e p a r t i c u l a r s o l u t i o n s a r i s i n g from t h e s e c u l a r terms, a l l o t h e r terms forming @2 a r e bounded. The r e s u l t i n g expression f o r . m 2 s a t i s f y i n g p e r i o d i c i t y i n x and t i s :a
-
2ti) cos (2N71x)+
0 ( 1 )-
a'1- -
-z
i w i s i n ( i t+
ti) s i n ( ~ = x ) ( 8 ) (14) 33 lz=o i = l3. V e l o c i t y Components and Pressure.- Expressions I n view o f eq.(8), t h e s o l u t i o n o f eq.(7a)
f o r t h e p a r t i c l e v e l o c i t y components a r e r e a d i l y c o n s i s t s o f an i n f i n i t e s e r i e s , each o f whose terms
obtained from the f o l l o w i n g formulae : i s p r o p o r t i o n a l t o sin(Nnx). It i s now assumed t h a t
J.H. Ginsberg c8- 3 7
Also, l e t ( p
+
l ) p o be the absolute pressure, where po i s t h e ambient pressure i n t h e reference s t a t e corresponding t o co. A r e l a t i o n s h i p between p and @ i s /7/.2
a@
1=
-
yK {= + ;?[email protected]@}
( I 6 ) A binomial expansion, coupled w i t h t h e f a c t t h a t both p and @ a r e O(E) q u a n t i t i e s , y i e l d s :The expressions obtained from e q s . ( l 5 ) and (17) have t h e common form :
2 3
u j = i = l
E
{ ~ f ~ ~ ( x , z , t ) t zgji(x,z,t))+
O(E ) (18) where u . i s a generic symbol f o r v,, v,, o r p. TheJ
f u n c t i o n s fij and g
'?
a r e s i n u s o i d a l , so t h e presert ce o f a f a c t o r z i n d i c a t e s t h a t t h e expressions lack u n i f o r m accuracy. This f a u l t may be c o r r e c t e d by t h e method o f renormal i z a t i o n /3/.The conventional technique i s t o d e f i n e a s i n g l e s t r a i n i n g o f t h e independent v a r i a b l e s .
Hm-
ever, such a procedure does n o t succeed i n t h e p r e sent problem, because o f t h e d i s p a r a t e s p a t i a l and
transformed t o :
(15a) 2
u j =
igl
cfji(ai , B ~ , t ) + O ( E )( I " ) The s p e c i f i c f u n c t i o n and (ZIli S a t i s f y i n g
temporal harmonic c o n t e n t o f the f u n c t i o n s f . . . J 1
Thus, a d i f f e r e n t s e t o f s t r a i n e d coordinates i s used f o r each term. Because z and t o n l y occur i n t h e combination Xi,
-
it, the s t r a i n i n g o f t may a r b i t r a r i l y be allowed t o vanish. Thus, f o r each i i n eq.(18) d e f i n e :This c o o r d i n a t e t r a n s f o r m a t i o n i s s u b s t i t u t e d i n t o eq. (18) and t h e r e s u l t expanded i n a T a y l o r s e r i e s i n ascending powers o f E. The c r i t e r i o n f o r removal o f t h e non-uniformly accurate O(E 2 ) term i n eq.(18) i s then :
=
-
6.9.. (ai ,Bi , t ) + O(E 2 )1 J 1 (20)
When t h i s requirement i s f u l f i l l e d , eq.(19) i s
eq.(20) a r e obtained by using t h e method o f harmon- i c balance t o generate l i k e terms on e i t h e r s i d e o f t h e equation. Note t h a t t h e r e n o r m a l i z a t i ~ n s o f v,, v,, and p must a l l y i e l d t h e same coordinate s t r a i n i n g s . The r e s u l t o f t h i s procedure i s :
The expressions f o r v e l o c i t y and pressure ob- t a i n e d from eq.(21) may be w r i t t e n as :
CO m m
vx =
igl
(vXli,
vz=igl
(vzli,
P =iil
( P ) ~ (23a)where : n i
(v,)~ =
-
E-
Xi W . 1 cos(XiBi-
i t ) cos(Nnai) ( v ~ ) ~ = €iwi sin(XiBi-
i t ) sin(Nnai)S i g n i f i c a n t l y , t h e coordinate s t r a i n i n g f u n c t i o n s i n eq.(22) a r e p r o p o r t i o n a l t o t h e v e l o c i t y terms i n eq.(23b). This observation a l l o w s c o o r d i n a t e t r a n s f o r m a t i o n (19) t o be w r i t t e n as :
Eqs. (23) and (24) a r e uniformly v a l i d f i r s t a p p r a - imations t o t h e n o n l i n e a r response. Simultaneous s o l u t i o n o f eq.(23b) and (24) by a numerical tech- nique, such as Newton's method, y i e l d s t h e values o f ai, pi, and t h e s t a t e v a r i a b l e s f o r s e l e c t e d va- l u e s o f x, z, and t.
4.0iscussion and Conclusion.- The n o n l i n e a r waves described by eqs. (23b) and (24) f o r any one value o f i a r e i d e n t i c a l t o the e a r l i e r r e s u l t s f o r s i n - g l e frequency s i n u s o i d a l e x c i t a t i o n s o f t h e p l a t e /5//6/. The i n d i v i d u a l n o n l i n e a r waves combine i n 1 in e a r superposition, as described by eq. (23a).
Q u a n t i t a t i v e examples o f t h e f i n i t e amp1 i tude wave corresponding t o a s p e c i f i c value o f i
,
i n c l u d i n g t h e f o r m a t i o n o f shocks, a r e described i n /6/.JOURNAL DE PHYSIQUE
Therefore, numerical r e s u l t s w i l l n o t be presented here.
The i t h i n d i v i d u a l wave reduces, i n l i n e a r theory, t o a s i n u s o i d a l wave t h a t propagates i n the z d i r e c t i o n a t t h e phase speed :
C l e a r l y , t h e f o r e g o i n g i s a d i s p e r s i o n r e l a t i o n s f o r t h e v a r i o u s waves. Now r e c a l l eq.(14), which describes the terms i n
a2
t h a t l e a d t o s i g n i f i c a n t d i s t o r t i o n . I t can be seen t h e r e t h a t t h e i t h har- moni c i n t h e sum a1 so has a phase speed o f i / X i.
Thus, t h e f i r s t approximation o f t h e f i n i t e - amplitude wave c o n s i s t s o f separate groups. Each group i s composed o f a primary term ( t h e l i n e a r ap- proximation), p l u s a1 1 n o n l i n e a r harmonics having the same phase v e l o c i t y . As these groups propagate, energy i s c o n t i n u a l l y t r a n s f e r r e d from t h e primary wave t o t h e harmonics, w i t h t h e r e s u l t t h a t t h e h a r monics grow. This growth leads t o a non-uniformly v a l i d s i t u a t i o n , which was c o r r e c t e d w i t h i n t h e an-
a l y s i s by the method o f r e n o r m a l i z a t i o n .
I n t e r m o d u l a t i o n between t h e various groups i s n o t i m p o r t a n t i n t h i s approximation. The groups d i ? perse according t o eq.(25), thereby removing t h e o p p o r t u n i t y f o r continuous energy t r a n s f e r .
I t i s s i g n i f i c a n t t h a t , according t o eqs. (23) and (24), vx = 0 along x = j/(2Na), j odd. Thus, the present r e s u l t s may a l t e r n a t i v e l y be considered t o be a two-dimensional response w i t h i n a semi-infi- n i t e hard-walled duct ; t h e w a l l s o f t h i s duct co- i n c i d e w i t h any two o f the aforementioned values o f x. The r a m i f i c a t i o n s o f the present method a n a l y s i s f o r waves i n ducts a r e t r e a t e d i n g r e a t e r general- i t y i n a f u r t h e r study /8/.
References
/1/ Ginsberg, J.H., J. Sound and Vib. - 40 (1975) 351.
/2/ Nayfeh, A.H. and Kluwick, A., J . Sound and Vib.
48 (1976) 293.
-
/3/ Nayfeh, A.H., P e r t u r b a t i o n Methods ( W i l e y - I n t e r science, New York) 1973.
/4/ Ginsberg, J.H., J . Sound and Vib.
40
(1975) 359./5/ Nayfeh, A.H. and K e l l y , S.G., "Nonlinear I n t e r - a c t i o n s o f Acoustic F i e l d s w i t h P l a t e s under Resonant E x c i t a t i o n s " , J . Sound and Vib.
-
60 (1978) 371./6/ Ginsberg, J.H., "A Re-Examination o f t h e N o n l i - near I n t e r a c t i o n between an Acoustic F l u i d and a F l a t P l a t e undergoing Harmonic E x c i t a t i o n " , J. Sound and Vib. 60 (1978) 449.
/7/ Goldstein, S., Lectures on F l u i d Mechanics (Wi l e y I n t e r s c i e n c e , New York) 1960.
/8/ Ginsberg, J.H., J. Acoust. Soc. Am. 65 (1979) 1127.
