Brunskog, Jonas

Many lightweight structures consist of plates being stiffened by ribs. The rib stiffeners can significantly change the vibration field and the radiation behavior of the structure. These type of structures has thus often been studied in the past. However, there is a lack of simplified expressions for the sound transmission of these structures. Therefore, simplified expressions for the sound transmission of finite single leaf ribbed plates are derived, using a variational technique based on integral equations of the fluid loaded plate

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Downloaded from orbit.dtu.dk on: Dec 20, 2017The sound transmission of finite ribbed plates using a variationalBrunskog, JonasPublished in:Internoise 2012Publication date:2012Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Brunskog, J. (2012). The sound transmission of finite ribbed plates using a variational. In Internoise 2012 The sound transmission of finite ribbed plates using a variational technique    Jonas Brunskoga) Acoustic Technology, DTU Electrical Engineering, Technical University of Denmark,  Kgs. Lyngby DK-2800, Denmark  Many lightweight structures consist of plates being stiffened by ribs. The rib stiffeners can significantly change the vibration field and the radiation behavior of the structure. These type of structures has thus often been studied in the past. However, there is a lack of simplified expressions for the sound transmission of these structures. Therefore, simplified expressions for the sound transmission of finite single leaf ribbed plates are derived, using a variational technique based on integral equations of the fluid loaded plate. 1 INTRODUCTION   The purpose of the present paper is to derive simple analytical formulas for the airborne sound insulation of a single ribbed wall of finite size, using a variational technique based on the integral-differential equation of the fluid loaded wall. The so derived formulas are valid in the entire audible frequency range. The paper is an extension of a previous work concerned with the same problem of a single wall without rib stiffeners1,2.  2 THEORY   Small amplitudes and linear theory are assumed. The time dependence is of the form e௜ఠ௧, where ߱ ൌ 2ߨ݂ is the angular frequency and ݐ is time, which is suppressed throughout the paper.  2.1 Formulation of the problem   Consider Fig. 1, where a single ribbed wall of finite size is located in an infinite baffle, an acoustically hard plane, at ݖ ൌ 0. The ribbed wall consist of a plate with beams attached to it. The beams are located with a distance ݈ inbetween. As a simplification, it is assumed that the presence of the beams in the sound field will not influence the sound field directly. Thus, the beams are approximated to be sound transparent, only affecting the vibration pattern of the wall plate. On the source side, the total acoustic field will consist of one plane incident wave ݌௜, one                                                  a) email:  jbr@elektro.dtu.dk  plane geometrically reflected wave ݌௥ and one scattered field ݌௦ that is due to the motion of the finite wall, ݌ ൌ ݌௜ ൅ ݌௥ ൅ ݌௦. On the receiver side, only the transmitted wave is present, ݌ ൌ ݌௧.    l pt z x pi pr ps   Fig. 1 - The model of a finite wall, located in an infinite rigid baffle in the x-y-plane, at z = 0. The wall consist of a plate and ribbs located with a distanse l apart. On the left side of the wall is the source side (z < 0), the right side is the receiver side (z > 0). On the source side the total acoustic field consists of one plane incident wave, one plane geometrically reflected wave and one scattered field due to the motion of the finite wall. On the receiver side only the radiated field will be present, it being the same as the scattered field but with opposite z-direction propagation.  With the coordinate system defined as the wall being in the ݔ-ݕ-plane and ݖ pointing into the receiver room in Fig. 1, the incident and the reflected plane waves are  ݌௜ሺݔ, ݕ, ݖሻ ൌ ݌̂௜݁ି௜൫௞ೣ௫ା௞೤௬ା௞೥௭൯, ݌௥ሺݔ, ݕ, ݖሻ ൌ ݌̂௜݁ି௜൫௞ೣ௫ା௞೤௬ି௞೥௭൯,   (1)  where ݌̂௜ is the complex amplitude of the incident wave. The wavenumbers are ݇௫ ൌ݇ sin ߠ cos ߮, ݇௬ ൌ ݇ sin ߠ sin ߮, and ݇௭ ൌ ݇ cos ߠ, where ݇ ൌ ߱/ܿ. The scattered and transmitted wave will be described by means of the Rayleigh integral, for the scattered wave  ݌௦ሺݔ, ݕ, ݖሻ ൌ ݅߱ߩ ׬ ݒሺܵᇱሻܩሺܵ, ݖ|ܵᇱ, 0ሻ݀ܵԢௌೢ       (2)  and for the transmitted wave  ݌௧ሺݔ, ݕ, ݖሻ ൌ െ݅߱ߩ ׬ ݒሺܵᇱሻܩሺܵ, ݖ|ܵᇱ, 0ሻ݀ܵԢௌೢ ,     (3)  where ݔԢ, ݕԢ and ܵԢ ൌ  ሺݔԢ, ݕԢሻ refers to the integration point, ݒ is the vibration velocity of the wall and ܩሺܵ, ݖ|ܵԢ, 0ሻ is the free field Greens function,  ܩሺܵ, ݖ|ܵᇱ, 0ሻ ൌ  െ ௘ష೔ೖೃଶగோ .        (4)  In eqs. (2-3), the integral is a surface integral over the wall area ܵ௪. The distance ܴ between the observation point and the integration point is  ܴ ൌ ඥ ሺݔ െ  ݔԢሻଶ  ൅ ሺݕ െ  ݕԢሻଶ  ൅  ݖଶ.      (5)  It should be noted that due to the symmetry, the transmitted field is the same as the scattered field, but with opposite phase due to the direction of the vibration velocity of the wall ݒ.  The vibrations of the wall can in general terms be describe with a differential operator  ࣴݒሺݔ, ݕሻ ൌ  ݌௜ሺݔ, ݕ, 0ሻ ൅ ݌௥ሺݔ, ݕ, 0ሻ ൅ ݌௦ሺݔ, ݕ, 0ሻ– ݌௧ሺݔ, ݕ, 0ሻ    (6)  where the wall impedance operator ࣴ, including both the plate and the beams, will be described in more detail below. Using the equations above,  ࣴݒሺܵሻ  ൌ  2݌̂௜݁ି௜ሺ௞ೣ௫ା௞೤௬ሻ  ൅  2݅߱ߩ ׬ ݒሺܵԢሻܩሺܵ, 0|ܵԢ, 0ሻ݀ܵԢௌೢ     (7)  valid for ሺݔ, ݕሻ  א  ܵ௪. This is an integral-differential equation with the vibration velocity of the wall ݒ as unknown.   The variational formulation for finite single walls by Brunskog1,2, based on Morse and Ingard3, can now directly be adopted,  ܸሺݒ, ݒ௔ሻ ൌ 2݌̂௜ ׬ ݒሺܵሻ݁௜ሺ௞ೣ௫ା௞೤௬ሻௌೢ ݀ܵ ൅ 2݌̂௜ ׬ ݒ௔ሺܵሻ݁ି௜൫௞ೣ௫ା௞೤௬൯ௌೢ ݀ܵ െ ׬ ݒ௔ሺܵሻࣴݒሺܵሻௌೢ ݀ܵ ൅2݅߱ߩ ׬ ׬ ݒ௔ሺܵሻݒሺܵᇱሻௌೢ ܩሺܵ, 0|ܵᇱ, 0ሻ݀ܵௌೢᇱ ݀ܵ,            (8)  where ݒ௔ is the vibration velocity of the adjoint problem, with the incident wave in the opposite direction. This is a symmetric functional in ݒ and ݒ௔, if ࣴ is a symmetric operator (meaning that ݒ௔ࣴݒ ൌ  ݒࣴݒ௔). The functional ܸሺݒ, ݒ௔ሻ will be used to find approximate variational solutions to the problem by means of minimizing it.  2.2 The impedance operators for the wall plate and the beam stiffeners   The wall impedance operator ࣴ is used as a general description of wall. If the wall is a thin plate, the Kirchhoff plate equation is the governing equation for the wave motion in the plate,  ࣴ௣ ൌ ஻ᇱ௜ఠ ቀ డమ డ௫మ ൅ డమ డ௬మቁଶ ൅ ݅߱݉ᇱᇱ.       (9)      To this impedance operator will now be added the contribution from the rib stiffeners. The impedance operator for one Euler beam located in the y-direction is,  ݂ࣴ ൌ ஻೑௜ఠ డర డ௬మ ൅ ݅߱݉௙ᇱ .         (10)   Assume the beams to be located at ݔ ൌ ݈݊, ݊ being an integer and ݈ the distance between the rib stiffeners, see again Fig. 1. Also assume the connection between the beams and the plate to be in the form of a line force and equal displacement of the beam and the plate at ݔ ൌ ݈݊. It should be noted that this is a rough simplification of the actual situation, ignoring effects as moment coupling, in-plane wave motion and eccentricity of the beams, and point vice coupling. Equation (7) can with these assumptions be written   ൫ࣴ௣ ൅ ∑ ௙ࣴߜሺݔ െ ݈݊ሻஶ௡ୀିஶ ൯ݒሺݔ, ݕሻ ൌ ݌ሺݔ, ݕሻ      (11)   where  ݌ሺݔ, ݕሻ is the left hand side of eq. (7). The wall impedance operator ࣴ can be identified as  ࣴ ൌ ࣴ௣ ൅ ∑ ௙ࣴߜሺݔ െ ݈݊ሻஶ௡ୀିஶ .       (12)   2.3 Forced variational solution   As a first use of the variation formulation, assume a forced vibration velocity, i.e., the vibration of the wall to be of the same form as the incident wave  ݒሺݔ, ݕሻ  ൌ  ݒො݁ି௜ሺ௞ೣ௫ା௞೤௬ሻ,       (13)  and for the adjoint field  ݒ௔ሺݔ, ݕሻ ൌ  ݒො݁௜൫௞ೣ௫ା௞೤௬൯.       (14)  Then, if applying the wall impedance operator to (13),   ࣴݒሺݔ, ݕሻ ൌ ݒොܼ௣݁ି௜൫௞ೣ௫ା௞೤௬൯ ൅ ݒො ௙ܼ݁ି௜൫௞ೣ௫ା௞೤௬൯ ∑ ߜሺݔ െ ݈݊ሻஶ௡ୀିஶ ,   (15)  where use have been made of  ܼ݌ ൌ ஻ᇲ௜ఠ ൫݇௫ଶ ൅ ݇௬ଶ൯ଶ ൅ ݅߱݉ᇱᇱ, ܼ݂ ൌ  ஻೑௜ఠ ݇௬ସ ൅ ݅߱݉௙ᇱ ,      (16)  being the algebraic expressions for the plate and beam impedance (the same apply for ݒ௔). These expressions is inserted in eq. (8). The result is then the same as found by Brunskog1,2, with the  exception for the term including the wall impedance operator   ׬ ݒ௔ሺܵሻࣴݒሺܵሻௌೢ ݀ܵ ൌ ݒො2 ׬ ܼ௣ௌೢ ݀ܵ ൅   ݒො2 ׬ ∑ ௙ܼߜሺݔ െ ݈݊ሻஶ௡ୀିஶௌೢ ݀ܵ ൌ ݒො2൫ܼ௣ܵ ൅ ݊௙ ௙ܼ൯,   (17) where ݊௙ is the number of beams א ܵ௪. Thus,  ܸሺݒොሻ ൌ 4݌̂௜ݒොܵ െ ൫ܼ݌ܵ ൅ ݂ܼ݂݊൯ݒොଶܵ ൅ 2݅߱ߩݒොଶ ׬ ׬ ݁௜ቀ௞ೣ൫௫ି௫ᇲ൯ା௞೤൫௬ି௬ᇲ൯ቁܩሺܵ, 0|ܵᇱ, 0ሻ݀ܵ݀ܵԢௌೢௌೢ .  (18)  Differentiate equation (18) with respect to ݒො and equal to zero to find the optimum, and then finally  ݒො ൌ ଶ௣ො೔௓೛ା௡೑௓೑/ௌାଶఘ௖ ௭ೌ,          (21)  where the integral over the Green’s function in (18) have be identified as the normalized radiation impedance, as defined by Thomasson4.  ݖ௔ ൌ െ ௜௞ௌ ׬ ׬ ݁௜ቀ௞ೣ൫௫ି௫ᇲ൯ା௞೤൫௬ି௬ᇲ൯ቁܩሺܵ, 0|ܵᇱ, 0ሻ݀ܵ݀ܵԢௌೢௌೢ .    (22)  The result in (21) corresponds to the corresponding result by Brunskog1,2, with the addition of the additional contribution ݊௙ ௙ܼ/ܵ to the impedance, being the average beam impedance.  2.4 Variational solution including periodicity   The solution the corresponding problem of the vibration of an infinite plate without radiation load excited by a pressure ݌̂݁ି௜ሺ௞ೣ௫ା௞೤௬ሻ is found by Rumerman5 (we are here ignoring the moment coupling, for simplicity)  ݒሺݔ, ݕሻ ൌ ௣ො௓೛൫௞ೣ,௞೤൯ ݁ି௜ሺ௞ೣ௫ା௞೤௬ሻ െ ௩బ௓೑ሺ௞೤ሻ௟ ݁ି௜ሺ௞ೣ௫ା௞೤௬ሻ ∑௘మഏ೔೙ೣ/೗௓೛൫௞ೣିଶగ௡/௟,௞೤൯ஶ௡ୀିஶ          (23)  where ݒ଴݁ି௜௞೤௬ ൌ ݒሺ0, ݕሻ. An adjoint field, with reverse direction of the incident field, would be  ݒ௔ሺݔ, ݕሻ ൌ ௣ො௓೛൫௞ೣ,௞೤൯ ݁௜ሺ௞ೣ௫ା௞೤௬ሻ െ ௩బ௓೑ሺ௞೤ሻ௟ ݁௜ሺ௞ೣ௫ା௞೤௬ሻ ∑௘షమഏ೔೙ೣ/೗௓೛൫௞ೣିଶగ௡/௟,௞೤൯ஶ௡ୀିஶ             (24)  Define a non-dimensional auxiliary function to be  ܶሺݔሻ ൌ ௓೑ሺ௞೤ሻ௟ ∑௘మഏ೔೙ೣ/೗௓೛൫௞ೣିଶగ௡/௟,௞೤൯ஶ௡ୀିஶ ,                                           (25)  and for the adjoint solution  ௔ܶሺݔሻ ൌ ܶכሺݔሻ ൌ ௓೑ሺ௞೤ሻ௟ ∑௘షమഏ೔೙ೣ/೗௓೛൫௞ೣିଶగ௡/௟,௞೤൯ஶ௡ୀିஶ .                                          (26)  The Rumerman solution (23) will be use as a basis for an approximate variational solution of the finite fluid loaded transmission problem. Thus, using two unknown variable ݒොଵ and ݒොଶ,  ݒሺݔ, ݕሻ ൌ ݒොଵ݁ି௜ሺ௞ೣ௫ା௞೤௬ሻ ൅ ݒොଶܶሺݔሻ݁ି௜ሺ௞ೣ௫ା௞೤௬ሻ                                 (27)  ݒ௔ሺݔ, ݕሻ ൌ ݒොଵ݁௜ሺ௞ೣ௫ା௞೤௬ሻ ൅ ݒොଶ ௔ܶሺݔሻ݁௜ሺ௞ೣ௫ା௞೤௬ሻ.                                  (28)  Appling the wall operator on equation (27) yields  ࣴݒሺݔ, ݕሻ ൌ ݒොଵࣴ൛݁ି௜ሺ௞ೣ௫ା௞೤௬ሻൟ ൅ ݒොଶࣴ൛ܶሺݔሻ݁ି௜ሺ௞ೣ௫ା௞೤௬ሻൟ,                           (29)  where the first term is given in eq. (15), and the last term is  ࣴ൛ܶሺݔሻ݁ି௜൫௞ೣ௫ା௞೤௬൯ൟ ൌ ௙ܼሺ݇௬ሻ݁ି௜൫௞ೣ௫ା௞೤௬൯ ൬ଵ௟ ∑ ݁మഏ೔೙ೣ೗ஶ௡ୀିஶ ൅ ܶሺݔሻ ∑ ߜሺݔ െ ݈݉ሻஶ௠ୀିஶ ൰  (30)  With the use of the Poission’s formula  ଵ௟ ∑ ݁మഏ೔೙ೣ೗ஶ௡ୀିஶ ൌ ∑ ߜሺݔ െ ݈݉ሻஶ௠ୀିஶ ,      (31)  can (30) be written  ࣴ൛ܶሺݔሻ݁ି௜൫௞ೣ௫ା௞೤௬൯ൟ ൌ ௙ܼሺ݇௬ሻ݁ି௜൫௞ೣ௫ା௞೤௬൯൫1 ൅ ܶሺݔሻ൯ ∑ ߜሺݔ െ ݈݉ሻஶ௠ୀିஶ   (32)  Inserted in the integral in eq. (8) containing the wall impedance operator, we have  ׬ ݒ௔ሺܵሻࣴݒሺܵሻௌೢ ݀ܵ ൌ ݒොଵଶ൫ܼ௣ܵ ൅ ௙ܼ݊௙൯ ൅ ݒොଶݒොଵ ቀܼ௣ ׬ ௔ܶሺݔሻ݀ܵௌೢ ൅ ௙ܼ݊௙ ൅ ௙ܼ ∑ ௔ܶሺ݈݉ሻ௠אௌ ൅௙ܼ ∑ ܶሺ݈݉ሻ௠אௌ ቁ ൅ ݒොଶଶ ௙ܼ൫∑ ௔ܶሺ݈݉ሻ௠אௌ ൅ ∑ ௔ܶሺ݈݉ሻܶሺ݈݉ሻ௠אௌ ൯ .                (33)  The periodic properties of ܶሺݔሻ can now be made use of  ܶሺ݈݉ሻ ൌ ௓೑ሺ௞೤ሻ௟ ∑௘మഏ೔೙೘௓೛൫௞ೣିଶగ௡/௟,௞೤൯ ൌஶ௡ୀିஶ ܶሺ0ሻ,                   (34)  and therefore  ∑ ܶሺ݈݉ሻ௠ୀאௌ ൌ ∑ ܶሺ0ሻ ൌ ܶሺ0ሻ݊௙௠ୀאௌ ,              (35)  and the same apply for ௔ܶሺݔሻ. Moreover,  ∑ ௔ܶሺ݈݉ሻܶሺ݈݉ሻ௠ୀאௌ ൌ ∑ ௔ܶሺ0ሻܶሺ0ሻ௠ୀאௌ ൌ ௔ܶሺ0ሻܶሺ0ሻ݊௙.        (36)  For the radiation term in (8), we have  2݅߱ߩ ׬ ׬ ݒ௔ሺܵሻݒሺܵᇱሻௌೢ ܩሺܵ, 0|ܵᇱ, 0ሻ݀ܵௌೢᇱ ݀ܵ ൌ2݅߱ߩݒොଵଶ ׬ ׬ ݁௜ቀ௞ೣሺ௫ି௫ᇱሻା௞೤ሺ௬ି௬ᇱሻቁௌೢ ܩሺܵ, 0|ܵᇱ, 0ሻ݀ܵௌೢᇱ݀ܵ ൅ 2݅߱ߩݒොଵݒොଶ ׬ ׬ ൫ܶሺݔԢሻ ൅ௌೢௌೢ௔ܶሺݔሻ൯݁௜ቀ௞ೣሺ௫ି௫ᇱሻା௞೤ሺ௬ି௬ᇱሻቁ ܩሺܵ, 0|ܵᇱ, 0ሻ݀ܵᇱ݀ܵ ൅2݅߱ߩݒොଶଶ ׬ ׬ ௔ܶሺݔሻܶሺݔԢሻ݁௜ቀ௞ೣሺ௫ି௫ᇱሻା௞೤ሺ௬ି௬ᇱሻቁௌೢ ܩሺܵ, 0|ܵᇱ, 0ሻ݀ܵௌೢᇱ݀ܵ    (37)  The first integral is the same as found in the forced case, eq. (18), related to the forced radiation impedance ݖ௔, equation (22). In a similar manner as for forced radiation impedance,   ݖ௔ ൌ െ ݅݇ܵ න න ݁௜ቀ௞ೣ൫௫ି௫ᇲ൯ା௞೤൫௬ି௬ᇲ൯ቁܩሺܵ, 0|ܵᇱ, 0ሻ݀ܵ݀ܵԢௌೢௌೢ  ݖ௖ ൌ െ ݅݇ܵ න න ൫ܶሺݔሻ ൅ ௔ܶሺݔሻ൯݁௜ቀ௞ೣ൫௫ି௫ᇲ൯ା௞೤൫௬ି௬ᇲ൯ቁௌೢܩሺܵ, 0|ܵᇱ, 0ሻௌೢ݀ܵᇱ݀ܵ ݖ௕ ൌ െ ݅݇ܵ න න ௔ܶሺݔሻܶሺݔሻ݁௜ቀ௞ೣሺ௫ି௫ᇱሻା௞೤ሺ௬ି௬ᇱሻቁௌೢܩሺܵ, 0|ܵᇱ, 0ሻௌೢ݀ܵԢ݀ܵ   (38)  So, using (38) in (37), we have  2݅߱ߩ ׬ ׬ ݒ௔ሺܵሻݒሺܵᇱሻௌೢ ܩሺܵ, 0|ܵᇱ, 0ሻ݀ܵԢ݀ܵௌೢ ൌ െ2ߩܿݒොଵଶܵݖ௔ െ 2ߩܿݒොଵݒොଶܵݖ௖ െ 2ߩܿݒොଶଶܵݖ௕. (39)  Moreover, define the following integrals  ܫ ൌ ଵௌ ׬ ܶሺݔሻ݀ܵௌೢ ൌ௓೑ሺ௞೤ሻ௟ௌ ∑׬ ௘మഏ೔೙ೣ/೗ௗௌೄೢ௓೛൫௞ೣିଶగ௡/௟,௞೤൯ஶ௡ୀିஶ ,         ܫ௔ ൌ ଵௌ ׬ ௔ܶሺݔሻ݀ܵௌೢ ൌ௓೑ሺ௞೤ሻ௟ௌ ∑׬ ௘షమഏ೔೙ೣ/೗ௗௌೄೢ௓೛൫௞ೣିଶగ௡/௟,௞೤൯ஶ௡ୀିஶ      (40)  Finally, inserting eq. (23-24) in the variational formulation (8), making use of the results above, we have  ܸ ൌ 2݌̂௜ሺݒොଵܵ ൅ ݒොଶܫܵሻ ൅ 2݌̂௜ሺݒොଵܵ ൅ ݒොଶܫ௔ܵሻ െ ݒොଵଶ൫ܼ௣ܵ ൅ ௙ܼ݊௙൯െ ݒොଶݒොଵ൫ܼ௣ܫ௔ܵ ൅ ௙ܼ݊௙ ൅ ௙ܼ ௔ܶሺ0ሻ݊௙ ൅ ௙ܼܶሺ0ሻ݊௙൯െ ݒොଶଶ ௙ܼ൫ ௔ܶሺ0ሻ ൅ ௔ܶሺ0ሻܶሺ0ሻ݊௙൯ െ 2ߩܿݒොଵଶܵݖ௔ െ 2ߩܿݒොଵݒොଶܵݖ௖ െ 2ߩܿݒොଶଶܵݖ௕  (41)  Differentiate equation (41) with respect to ݒොଵ and ݒොଶ, and equal to zero to find the optimum, we finally gets  ൤ܼଵଵ ܼଵଶܼଶଵ ܼଶଶ൨ ൤ݒොଵݒොଶ൨ ൌ ൤2݌̂௜݌̂௜ሺܫ௔ ൅ ܫሻ൨,    ൤ݒොଵݒොଶ൨ ൌଵ஽ ൤ܼଶଶ െܼଵଶെܼଶଵ ܼଵଵ ൨ ൤2݌̂௜݌̂௜ሺܫ௔ ൅ ܫሻ൨  (42)  where ܼଵଵ ൌ ܼ௣ ൅ ௙ܼ݊௙ܵ ൅ ߩܿ2ݖ௔, ܼଶଶ ൌ௙ܼ݊௙ܵ ௔ܶሺ0ሻ൫1 ൅ ܶሺ0ሻ൯ ൅ 2ߩܿݖ௕ ܼଵଶ ൌ ܼଶଵ ൌ ܼ௣ܫ௔2 ൅௙ܼ݊௙2ܵ ൫1 ൅ ௔ܶሺ0ሻ ൅ ܶሺ0ሻ൯ ൅ ߩܿݖ௖  and the determinate ܦ ൌ ܼଵଵܼଶଶ െ ܼଵଶଶ .  2.5 Transmission loss  The transmission coefficient is defined as ߬ ൌ Π௧/Π௜,which with the present notations becomes  ߬ ൌ െ ఠఘమ௖ௌ ୡ୭ୱ ఏ|௣ො೔|మ Ը ቄ݅ ׬ ׬ ݒሺܵᇱሻݒכሺܵሻܩሺܵ, ݖ|ܵᇱ, 0ሻ݀ܵԢௌೢ ݀ܵௌೢ ቅ                       (43)  For a diffuse field result, an integration over al angels of propagation have to be preformed.  For the forced case, section 2.3, the result reads  ߬ ൌ ఠఘమ௖మԸሼݖܽሽୡ୭ୱ ఏ หܼ݌൅݂ܼ݂݊/ܵ൅2ߩܿ ݖܽหమ.                            (44)  For the solution including periodicity, section 2.4, the result if found by inserting (27) and (42) in (43).  4 DISCUSSION AND CONCLUDING REMARKS  Unfortunate no numerical results are yet ready. However, one can clearly see from the results in sections 2.3 to 2.4 that the present approach can provide a theoretically solid base for the simplified formulas used in building acoustics. 6 REFERENCES  1. J. Brunskog, ‘Sound transmission of finite single walls using a variational technique’, Proceedings of Euronoise 2009, Edingburgh, GB, (2009).  2. J. Brunskog, ‘The forced sound transmission of finite single walls using a variational technique’, submitted to J. Acoust. Soc. Am., (2012).  3. P. M. Morse, K. U. Ingard: Theoretical acoustics, Prinston university press, Prinston, New Jersey, (1968).  4. S.-I. Thomasson, “On the absorption coefficient”, Acustica, 44, 266–273, (1980).  5. M. L. Rumerman, “Vibration and wave propagation in ribbed plates”, J. Acoust. Soc. Am., 57, 370-373, (1975).    

The sound transmission of finite ribbed plates using a variational

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The sound transmission of finite ribbed plates using a variational

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