Theoretical Investigation on the Resonance Effect in the Acoustics

DU Chuan-mei ZHANG Ming-xu School of Civil Engineering and Architecture School of Civil Engineering and Architecture Anhui University of Science and Technology Anhui University of Science and Technology Huainan Anhui 232001，China Huainan Anhui 232001，China [email protected]

XU Ying NI Xiu-quan ZHOU Heng-linag School of Civil Engineering and School of Civil Engineering and School of Civil Engineering and Architecture Architecture Architecture Anhui University of Science and Anhui University of Science and Anhui University of Science and Technology Technology Technology Huainan Anhui 232001，China Huainan Anhui 232001，China Huainan Anhui 232001，China [email protected]

Abstract—In this paper, the resonance interaction at the acoustic vibration caused by the inharmonic vibration is discussed. Based on the perturbation theory, the influences of the resonance interaction between the two and three vibrant levels on the level energy, wave-function, and the spectral line intensity are investigated in detail respectively. The analytical expressions of the vibration level energy, wave-function, and spectral line intensity with the consideration of the resonance interaction are also obtained under the third order inharmonic approximation.

Keyword-Resonance; Vibration level; Wave-function; Spectral line intensity Fig.1. The actually measured resonance effect

I. INTRODUCTION Consequently, the theory study of the effect on acoustic The analysis and application of acoustic resonance spectrum structure with the consideration of the resonance phenomenon had been researched. For example ,Sun Guang- interaction between the vibration levels have important rong in the university of Nanking carried on a research instructional function for the correct analysis of the test of responding to Steady State Frequency Response in rooms— acoustic resonance spectrum .In this paper ,the basic model of Acoustics of Room and Small Room to indoor steady state the resonance interaction between the three vibration levels on frequency [1] . Vera Markovi Etc has carried on theory analysis the level energy, wave-function and the spectral line intensity to the acoustic resonance [2-4]. The intensity and structural of are investigated in theory respectively and are also obtained vibrant acoustic spectrum take place to change according to the express of the theory investigated in detail . the resonance interaction between the difference vibrant levels. The resonance interaction of the different vibrant levels causes II. THE BASIC THEORY OF RESONANCE INTERACTION spectral structure complicated and bring for difficulty for analysis of acoustic vibration structure. Such as figure 1 show: It is the most roughly approximation regarding vibration as harmonic vibration .It belongs to zero classes approximation .Actually, vibration is inharmonic . Therefore in motive Hamiltonian operator, higher order term of normal

coordinate Qi should be considered for potential energy operator. In order to simplify the theory calculation , suppose

in these harmonic vibration model Q1 、 Q2 and their overtones vibration have homological symmetry and vibrant

frequency satisfy 1  22 . The potential energy operator is

evolved to three cubed of normal coordinate under the three 0 level inharmonic approximation .Its expression: det H mn  E  E m  mn  0 (10) 1 2 (1) Among which ˆ  is the resonance interaction matrix V  ue  k Qk   i, j,k QiQ j Qk H mn 2 kki, j, element , from(9) and(10) can get the energy of vibration

Then at this time vibration Hamiltonian can write for: levels E1 , E2 , E3  and state function  1 , 2 , 3 ,...... ˆ ˆ ˆ H  H 0  H (2) III. RESONANCE INTERACTIONS BETWEEN TWO VIBRATION STATE Among which 2  2 1 A energy level and state function after the interaction Hˆ       Q 2 0  2 e  k k (3) Suppose the state function of vibration energy level 2 Qk 2 k V1 ,V2 ,V3  is  1 , the state function of vibration energy

level V1 1,V2  2,V3  is  2 , then these two vibrations ˆ  energy levels are almost degenerate, by equation (10) can get : H   ijk QiQ j Qk ijk 0 E1  H11  E H12 3 2 2 3  0 (11)   Q  Q Q  Q Q  Q (4) 0 1,1,1 1 1,1,2 1 2 1,2,2 1 2 2,2,2 2 H12 E2  H 22  E

 ijk is three ranks inharmonic coefficient , the higher rank Under three ranks inharmonic approximation there have: ' ˆ number is ,the more small inharmonic coefficient is. The H is H11  H 22  0 (12) a resonance function item, can be regarded as a perturbation item, be adopted as perturbation theory processing , We adopt H   V V V H  V 1,V  2,V three ranks inharmonic approximation in concrete calculation. 12 1 2 3 1 2 3 Considering the resonance interaction, state function is the 2 linear addition of the zero function with interaction of  122 V1V2V3 Q1Q2 V1 1,V2  2,V3 vibration state and keep same alike symmetry with the zero 3 function, but make interaction of level energy take place to 2  1  (13) change, for Hˆ there have  122   V1 1 V2 1V2 0  2a 

0 0 0 2 Hˆ E 4 ve  0 n  n  n (5) In equation a  , ve is vibrant frequency,  is  Then, considering resonance interaction there have 1   3U  reduced mass,     is three ranks inharmonic 122  2  ˆ ˆ 6  Q1Q2  H 0  H   E (6) coefficient, from equation (11-13) can get: For solving the equation (6), will launch to the liner 1 1 combining of zero class state function, then E  E 0  E 0  E 0  E 0  4 | H |2 (14)  2 1 2 2 1 2 12 (0)   an n (7) Thus get the abruption value of energy level: n 1  2 2  got by the (5, 6, 7) W   4 H      ,   1,2 (15)  2  12 0 0  0 (0) ˆ (0) E  E a   H  a   0 (8) o o  n n n  n n Among which 0 Is difference between E1 and E2 n n without considering perturbation, From equation (15) we can 0 see the relation between energy level’s splitting with The equation (8) both sides is left to multiply  m and resonance matrix element. The change of level energy is integral calculus, can get shown as below diagram considering interaction, among them  is difference between E and E for considering  0   [H mn  E  Em  mn ]an  0 (9) perturbation. n

0 Among them 0 is lower state for transition, k is upper state for transition, k  1、2 , considering interaction spectral line intensity is:

2 I    d k  0 k

2   a  0  b  0 d  0 k 1 k 2 2 0 2 0 0 0  ak I1  bk I 2  2ak bk I1 I 2 Fig.2. The influence of the resonance interaction on the energy of the two vibration levels  2   0 2   0  H12  0  Ek  E1   0 2H 12 Ek  E1  0 0  2 I1  2 I 2  2 I1 I 2  E  E 0  H 2   E  E 0  H 2  E  E 0  H 2 The state function of interaction can empress as linear  k 1 12   k 1 12  k 1 12 combination of originally state function . (23) o 0 IV. THE INTERACTION OF THREE RESONANCE VIBRATION  k  ak 1  bk 2 ,（ k  1,2 ，） (16) STATE Substitute the value of equation (14) into equation (9) can get the combination coefficient of state function respectively A Energy levels and state function after considering after interaction State function of vibration energy level considering interaction . 1/ 2 2 1/ 2 V1 1,V2  2,V3  is  1 , state function of vibration energy  H 2   E  E 0  （k=1,2） a   12   k 1  k 2 bk  2  0 2   0 2  level V1 ,V2 ,V3 is  3 , these three vibration energy levels are  Ek  E1  H12   Ek  E1  H12  (17) approximately combined and belong to identical homology of the symmetry ,from equation (10) there have Then there have E 0  H   E H  H  a H  1 11 12 13 k 12 (18) 0 (24)  0 H12 E2  H 22  E H 23  0 bk ` Ek  E1 0 H13 H 23 E3  H 33  E Through reasonable approximation there have Under three ranks inharmonic approximation there have a (E 0  E 0 )H  H   H   H   0 (25) k 1 2 12 (19) 11 22 33  0 0 0 2 bk Ek (E2  E1 )  H12 H12  V1 1,V2  2,V3 H  V1V2V3 Under the general situation has   V 1,V  2,V Q Q 2 V V V 2 0 0 0 122 1 2 3 1 2 1 2 3 H 12  Ek (E2  E1 ) （k=1、2） (20) 3 2 Can prove  1  (26)  122   V1 V2  2 V2  2a  d ak ( )  0 (21) H 23  V1 ,V2 ,V3 H  V1 1,V2  2,V3 dH12 bk 2 (0)  122 V1V2V3 Q1Q2 V1 1,V2  2,V3 Therefore, the more H12 is, the proportion of  1 in  1 and 3  increase . 2 2  1  (27)  122   V1 1 V2 1 V2 B spectral line intensity considering interaction  2a  Spectral line intensity taking no account of interaction is: H   0 (28) 2 13 I o    0 d k  0 k (22) After considering interaction from equation (24-28), we canget energy of the three approximately combined vibration

energy level respectively: 0 ； 0 ； Among which mk  Ek  E3 nk  E1  Ek  P    ； E1   cos  3 sin  k  1,2,3 3  3  B spectral line intensity considering interaction  P    Spectral line intensity taking no account of interaction is: E2  2  cos  ； 3  3  2 I o    0 d k  0 k  P    E   cos  3 sin  (29) 3 3 3 0   Among them 0 is lower state for transition,  k is upper Among which state for transition , k  1,2,3 , considering interaction   q    cos1   spectral line intensity is:  2  p3 / 27  2   I    d k  0 k 0 0 0 0 0 0 2 0 2 0 2 0 2 2 E1 E2  E2 E3  E1 E3  E 1  E 2  E 3  3H12  3H 23 p  2 3 =  a  0  b  0  c  0 d   0  k 1 k 2 k 3  (30) 2 0 2 0 2 0 0 0 0 0 0 0  ak I1  bk I 2  ck I 3  2ak bk I 1 I 2  2ak ck I 1 I 3  2bk ck I 2 I 3  1 2 0 0 2 0 0 2 0 0 0 2 0 2 0 0 2 0 0 0 0 0 q  3E 1 E 2  3E 1 E3  3E 2 E3  3E1 E 2  3E 3 E1  3E 3 E2 12E1 E 2 E3 27 2 2 2 2  mk H12  0  mk nk  0 0 0 0 2 2 0 0 0 0 2 0 2 3 3 3  2 2 2 2 2 2 I1   2 2 2 2 2 2 I 2  9H12  H 23 E1  E2  E3  27E1 H 23  27E3 H12  2E 1  2E 2  2E 3  mk H12  nk mk  nk H 23  mk H12  nk mk  nk H 23  (31) 2 2 4 2 2  nk H 23  0 2mk nk H12 0 0   2 2 2 2 2 2 I3  2 2 2 2 2 2 I1 I 2 The state function of interaction can empress as m H   n m  n H  m H   m n  n H  linercombination of originally state function  k 12 k k k 23  k 12 k k k 23   a  0  b  0  c  0 . Here, k  1,2,3 , Substitute the k k 1 k 2 k 3 2m 2n 2 H 2 H 2 2m 2n 4 H 2 value of equation (29) into equation (9) can get the  k k 12 23 I 0 I 0  k k 23 I 0 I 0 2 2 2 2 2 2 1 3 2 2 2 2 2 2 2 3 combination coefficient of state function respectively after mk H12  mk nk  nk H 23 mk H12  mk nk  nk H 23 considering interaction . (34) 1 0 0 2 2  0 2  Among which mk  Ek  E3 ， nk  E1  Ek Ek  E3 H12 ak    0 2 2 0 2 0 2 0 2 2 （ k  1,2,3 ） Ek  E3 H12  E1  Ek E3  Ek  E1  Ek H 23 

1 ACKNOWLEDGMENT  0 2 0 2  2 E1  Ek E3  Ek bk    The work described in this paper was fully supported by 0 2 2 0 2 0 2 0 2 2 Ek  E3 H12  E1  Ek E3  Ek  E1  Ek H 23  the Civil engineering provincial pilot project, Constructional （ (k=1,2,3) (32) industry plan project in Anhui in 2011 Project No.2011YF- 04 ）， financially supported by National Basic Research 1  0 2 2  2 Program of China under Grant No. 2012CB214900, China E1  Ek H 23 ck    national coal association project(No. MTK2011-450). 0 2 2 0 2 0 2 0 2 2 Ek  E3 H12  E1  Ek E3  Ek  E1  Ek H 23  A series of formula of above can simply write: REFERENCE 1  m 2 H 2  2 [1] SUN Guang-rong. Steady State Frequency Response in rooms—— k 12 Acoustics of Room and Small Room[J]. Electricity voice technique， ak  ；  2 2 2 2 2 2  2010,34（4）：4-6. mk H12  nk mk  nk H 23  [2] Vera Markovi, Bratislav Milovanovi and Olivera Proni. Determination 1 of complex resonant frequencies in rectangular and circular cylindrical  m 2 n 2  2 rooms[J] .Applied Acoustics, 2000, 59(3): 265-274. k k ； (33) bk   2 2 2 2 2 2  [3] L. M. Dorojkine, S. V. Boritko, G. N. Dorojkina, A. V. Fokin and I. A. mk H12  nk mk  nk H 23  Rozanov. The resonant character of the sensitivity of the acoustic wave sensors [J]. Sensors and Actuators B: Chemical, 2003,89(1-2) :15-20. 1 2 2 2 [4] Neves e Sousa and B.M. Gibbs. Low frequency impact sound  nk H 23  transmission in dwellings through homogeneous concrete floors and ck   2 2 2 2 2 2  floating floors[J]. Applied Acoustics, 2011, 72(4) :177-189 mk H12  nk mk  nk H 23 