The forced sound transmission of finite periodic plates using a variational approch Jonas Brunskog 1 Acoustic Technology, DTU Electro Technical University of Denmark (DTU), DK-2800 Kgs. Lyngby, Denmark

ABSTRACT Many engineering structures consist of plates being periodically stiffened or loaded. Examples can be wooden joist floors, dry walls with wooden or metal beams, or metamaterials for increased sound insulation. Even though some of these structures are well known and thoroughly studied in the past, there lack prediction models that can handle both the finiteness and periodicity in a simple and accurate manner. In two previous attempts, the periodicity of the structure have been tried to be incorporated in a variational technique based on integral-differential equations of the fluid loaded plate, based on [J. Brunskog, JASA 132, 1482–1493, 2012]. These attempts have not worked well: they show a poor fit with measured data. In this paper, this issue is investigated again, and a different tactics is used to incorporate the periodicity, using space harmonics. In this way, the paper try to explain the previous misfit and improve the model. 1. INTRODUCTION

In a couple of papers [1, 2] the variational approach of deriving expressions for the sound insulation of single homogenous walls of finite area [3] has been tried to be extended to spatially periodic walls, such as walls reinforced by periodically spaced beams. Another potential use for such a theory would be for metamaterials for increased sound insulation [4, 5]. The first result of these investigations [1], using a line connection between beams and plate, was a disappointment; some results are reproduced in Fig. 1 below. When compared to measured results found in the literature, Northwood [6], the match was poor; the theory showed a too strong influence of the beams, and a too low transmission loss in a large frequency range. In [2] the theory was thus developed further to see if a better agreement with the measured data is found is assuming a resilient point connection instead of a line connection. However, even if the point connection theory results in a better fit to the experimental data, it also includes a sharp transition from the low frequency theory to the high frequency theory, which is not seen in the experimental data. Inspecting the theoretical results in [1,2] it can be noted that periodic effects are not included in the present form of the theory – the influence of the periodic location of the beams is canceled out by the adjoint field when forming the variational functional. This fact can be essential explaining the discrepancies between the present theory and the experimental results.

1 jbr@elektro.dtu.dk

In this paper this issue is investigated again, using a different tactics to incorporate the periodicity, using space harmonics. In this way, the paper try to explain the previous misfit and improve the model. The paper presents work in progress, and final results of the new theory is still missing. Some problems with the new approach will be discussed.

Figure 1: Examples from the studies in [1] and [2]. Transmission loss 𝑅𝑅 for plaster board wall with line connection theory (red line diamond symbol) as compared to measured results by Northwood [6] (blue line + symbol), theory including no beams (cyan line triangle symbol), and theory with only mass impedance of the beams (green line circle symbol). 2. THEORY

The theory in [2] is the starting point for the theory developed here. The impedance operator for a plate reinforced with a periodic array of beams

∞

𝒵𝒵= 𝒵𝒵 𝑝𝑝 + 𝒵𝒵 𝑏𝑏 ෍𝛿𝛿(𝑥𝑥−𝑚𝑚𝑙𝑙 𝑥𝑥 )

, (1)

𝑚𝑚=−∞

where 𝒵𝒵 𝑝𝑝 is the impedance operator for the plate and 𝒵𝒵 𝑏𝑏 is the impedance operator for the beams. Using the combinations of the pressures acting on the plate, all evaluated at 𝑧𝑧= 0 , the governing equation for the vibrations of the finite wall, see Fig. 2, can be written

𝒵𝒵𝑣𝑣(𝑥𝑥, 𝑦𝑦) = 2𝑝𝑝̂ 𝑖𝑖 𝑒𝑒 −𝑖𝑖(𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦)

+ 2𝑖𝑖𝑖𝑖𝑖𝑖න𝑣𝑣(𝑥𝑥 ′ , 𝑦𝑦 ′ )𝐺𝐺(𝑥𝑥, 𝑦𝑦, 0|𝑥𝑥 ′ , 𝑦𝑦 ′ , 0

)𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (2)

𝑆𝑆 𝑤𝑤

= Theory ink rib —©— Theory mass rib. 9 Theory no rib —+— NRCC 100 200 400 800 Frequency f [Hz] 1600 3150

Figure 2: The model of a finite wall, located in an infinite rigid baffle in the x-y-plane, at 𝑧𝑧= 0 . The wall consist of a plate and ribs located with a distance 𝑙𝑙 𝑥𝑥 apart. On the left side of the wall

is the source side ( 𝑧𝑧< 0 ), the right side is the receiver side ( 𝑧𝑧> 0 ). The present author [3] have developed a variatonal formulation to achieve approximate solutions of eq. (2). The theory is based on a method developed in Morse and Ingard [7], and deeply inspired by the corresponding theory for sound absorbers of finite size, as developed by Thomasson [8]. Following this procedure the variational functional of the problem can be shown to be

𝑉𝑉= 2𝑝𝑝̂ 𝑖𝑖 ∫ 𝑣𝑣(𝑆𝑆)𝑒𝑒 𝑖𝑖(𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦) 𝑑𝑑𝑑𝑑 𝑆𝑆 𝑤𝑤 + 2𝑝𝑝 𝑖𝑖 ∫ 𝑣𝑣 𝑎𝑎 (𝑆𝑆)𝑒𝑒 −𝑖𝑖(𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦) 𝑑𝑑𝑑𝑑 𝑆𝑆 𝑤𝑤 −

∫ 𝑣𝑣 𝑎𝑎 (𝑥𝑥, 𝑦𝑦)𝒵𝒵𝑣𝑣(𝑥𝑥, 𝑦𝑦) 𝑆𝑆 𝑤𝑤 𝑑𝑑𝑑𝑑+ 2𝑖𝑖𝑖𝑖𝑖𝑖∫ ∫ 𝑣𝑣 𝑎𝑎 (𝑆𝑆)𝑣𝑣(𝑆𝑆 ′ )𝐺𝐺(𝑆𝑆|𝑆𝑆 ′ )𝑑𝑑𝑆𝑆 ′ 𝑑𝑑𝑑𝑑 𝑆𝑆 𝑤𝑤 𝑆𝑆 𝑤𝑤 .

(3)

In [1,2,3] the test function used was 𝑣𝑣(𝑥𝑥, 𝑦𝑦) = 𝑣𝑣ො𝑒𝑒 −𝑖𝑖(𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦) and adjoint function 𝑣𝑣 𝑎𝑎 (𝑥𝑥, 𝑦𝑦) = 𝑣𝑣ො𝑒𝑒 𝑖𝑖(𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦) . That did not work well in [1,2], most likely due to the periodic effects missing. Therefore, let’s instead try to use an expansion with the space harmonic series,

∞

𝑣𝑣(𝑥𝑥, 𝑦𝑦) = 𝑒𝑒 −𝑖𝑖𝑘𝑘 𝑦𝑦 𝑦𝑦 ෍𝑣𝑣ො 𝑛𝑛

𝑒𝑒 −𝑖𝑖(𝑘𝑘 𝑥𝑥 +2𝑛𝑛𝑛𝑛/𝑙𝑙 𝑥𝑥 )𝑥𝑥 , (4)

𝑛𝑛=−∞

where 𝑙𝑙 𝑥𝑥 is the periodic length. And the adjoint function

∞

𝑣𝑣 𝑎𝑎 (𝑥𝑥, 𝑦𝑦) = 𝑒𝑒 𝑖𝑖𝑘𝑘 𝑦𝑦 𝑦𝑦 ෍𝑣𝑣ො 𝑜𝑜

𝑒𝑒 𝑖𝑖(𝑘𝑘 𝑥𝑥 +2𝑜𝑜𝑜𝑜/𝑙𝑙 𝑥𝑥 )𝑥𝑥 . (5)

𝑜𝑜=−∞

Applying the impedance operator (1) to eq. (4)

∞

−𝑖𝑖ቀ𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛

𝒵𝒵𝑣𝑣(𝑥𝑥, 𝑦𝑦) = 𝒵𝒵 𝑝𝑝 𝑒𝑒 −𝑖𝑖𝑘𝑘 𝑦𝑦 𝑦𝑦 ෍𝑣𝑣ො 𝑛𝑛

𝑒𝑒

𝑙 𝑙 𝑥 𝑥 ቁ𝑥𝑥

𝑛𝑛=−∞

∞

∞

𝑛𝑛=−∞ 𝑒 −𝑖𝑖ቀ𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝜋

+ 𝒵𝒵 𝑏𝑏 ෍𝛿𝛿(𝑥𝑥 −𝑚 𝑚 𝑙 𝑙 𝑥 𝑥 )

𝑚𝑚=−∞ 𝑒𝑒 −𝑖𝑖𝑘𝑘 𝑦𝑦 𝑦𝑦 ෍ 𝑣 𝑣 ො 𝑛

𝑙 𝑙 𝑥 𝑥 ቁ𝑥𝑥

= ෍𝑍𝑍 𝑝𝑝 ൬𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛 𝑙 𝑙 𝑥

∞

, 𝑘𝑘 𝑦𝑦 ൰𝑣𝑣ො 𝑛𝑛 𝑒𝑒 −𝑖𝑖ቀ𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛 𝑙 𝑙 𝑥 𝑥 ቁ𝑥𝑥 𝑒𝑒 −𝑖𝑖𝑘𝑘 𝑦𝑦 𝑦𝑦

(6)

𝑛𝑛=−∞

∞

∞

−𝑖𝑖ቀ𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛

+ 𝑍𝑍 𝑏𝑏 ൫𝑘𝑘 𝑦𝑦 ൯𝑒𝑒 −𝑖𝑖𝑘𝑘 𝑦𝑦 𝑦𝑦 ෍𝛿𝛿(𝑥𝑥−𝑚𝑚𝑙𝑙 𝑥𝑥 )

෍𝑣𝑣ො 𝑛𝑛

𝑒𝑒

𝑙 𝑙 𝑥 𝑥 ቁ𝑥𝑥 .

𝑚𝑚=−∞

𝑛𝑛=−∞

Next, one of the crucial terms in the variational formulation (3) is

න𝑣𝑣 𝑎𝑎 (𝑥𝑥, 𝑦𝑦)𝒵𝒵𝑣𝑣(𝑥𝑥, 𝑦𝑦)

𝑑𝑑𝑑𝑑=

𝑆𝑆 𝑤𝑤

= ෍ ෍𝑍𝑍 𝑝𝑝 ൬𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛 𝑙 𝑙 𝑥

∞

∞

, 𝑘𝑘 𝑦𝑦 ൰𝑣𝑣ො 𝑛𝑛 𝑣𝑣ො 𝑜𝑜 න𝑒𝑒 𝑖𝑖ቀ𝑘𝑘 𝑥𝑥 + 2𝑜 𝑜 𝑜

𝑙 𝑙 𝑥 𝑥 ቁ𝑥𝑥 𝑒𝑒 −𝑖𝑖ቀ𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛 𝑙 𝑙 𝑥 𝑥 ቁ𝑥𝑥

𝑛𝑛=−∞ 𝑑𝑑𝑑𝑑

𝑜𝑜=−∞

𝑆𝑆 𝑤𝑤

(7)

∞

∞

∞

+ 𝑍𝑍 𝑏𝑏 (𝑘𝑘 𝑦𝑦 ) ෍ ෍ ෍ 𝑣 𝑣 ො 𝑛 𝑛 𝑣𝑣ො 𝑜𝑜 න𝛿𝛿(𝑥𝑥

𝑛𝑛=− ∞

𝑜 𝑜=−∞

𝑚𝑚= −∞

𝑆𝑆 𝑤𝑤

𝑖𝑖ቀ𝑘𝑘 𝑥𝑥 + 2𝑜 𝑜 𝑜

−𝑖𝑖ቀ𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛

−𝑚𝑚𝑙𝑙 𝑥𝑥 )𝑒𝑒

𝑙 𝑙 𝑥 𝑥 ቁ𝑥𝑥 𝑒𝑒

𝑙 𝑙 𝑥 𝑥 ቁ𝑥𝑥 𝑑𝑑𝑑𝑑

where 𝑆𝑆 𝑤𝑤 is the area of the plate. In the second term, we can see that

𝑑𝑑𝑑𝑑= ൜ 𝑒𝑒 𝑖𝑖2𝜋𝜋(𝑜𝑜−𝑛𝑛)𝑚𝑚 if 𝑚𝑚𝑙𝑙 𝑥𝑥 ∈𝑆𝑆 𝑤𝑤

𝑖𝑖ቀ𝑘𝑘 𝑥𝑥 + 2𝑜 𝑜 𝑜

−𝑖𝑖ቀ𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛

න𝛿𝛿(𝑥𝑥−𝑚𝑚𝑙𝑙 𝑥𝑥 )𝑒𝑒

𝑙 𝑙 𝑥 𝑥 ቁ𝑥𝑥 𝑒𝑒

𝑙 𝑙 𝑥 𝑥 ቁ𝑥𝑥

0 otherwise (8)

𝑆𝑆 𝑤𝑤

So eq. (6) can be expressed as

න𝑣𝑣 𝑎𝑎 (𝑥𝑥, 𝑦𝑦)𝒵𝒵𝑣𝑣(𝑥𝑥, 𝑦𝑦)

𝑑𝑑𝑑𝑑

𝑆𝑆 𝑤𝑤

= ෍ ෍𝑍𝑍 𝑝𝑝 ൬𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛 𝑙 𝑙 𝑥

∞

∞

, 𝑘𝑘 𝑦𝑦 ൰𝑣𝑣ො 𝑛𝑛 𝑣𝑣ො 𝑜𝑜 න𝑒𝑒 𝑖𝑖 2(𝑜 𝑜 −𝑛 𝑛 )𝜋

𝑑𝑑𝑑𝑑

𝑙 𝑙 𝑥 𝑥𝑥

(9)

𝑜𝑜=−∞

𝑛𝑛=−∞

𝑆𝑆 𝑤𝑤

∞

∞

+ 𝑍𝑍 𝑏𝑏 ൫𝑘𝑘 𝑦𝑦 ൯෍ ෍𝑣𝑣ො 𝑛𝑛 𝑣𝑣ො 𝑜𝑜 ෍ 𝑒𝑒 𝑖𝑖2𝜋𝜋(𝑜𝑜−𝑛𝑛)𝑚𝑚

.

𝑚𝑚𝑙𝑙 𝑥𝑥 ∈𝑆𝑆 𝑤𝑤

𝑛𝑛=−∞

𝑜𝑜=−∞

The variational formulation term related to the injected power is

2𝑝𝑝̂ 𝑖𝑖 ∫ 𝑣𝑣(𝑥𝑥, 𝑦𝑦)𝑒𝑒 𝑖𝑖(𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦) 𝑑𝑑𝑑𝑑 𝑆𝑆 𝑤𝑤 = ∑ 𝑣𝑣ො 𝑛𝑛 ∞ 𝑛𝑛=−∞ ∫ 𝑒𝑒

𝑙 𝑙 𝑥 𝑥 𝑥𝑥 𝑆𝑆 𝑤𝑤 𝑑𝑑𝑑𝑑 . (10)

−𝑖𝑖 2𝑛 𝑛 𝑛

And for the corresponding term for the adjoint expression,

∞

𝑖𝑖 2𝑜 𝑜 𝑜

2𝑝𝑝̂ 𝑖𝑖 න𝑣𝑣 𝑎𝑎 (𝑥𝑥, 𝑦𝑦)𝑒𝑒 −𝑖𝑖(𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦) 𝑑𝑑𝑑𝑑

= ෍𝑣𝑣ො 𝑜𝑜

න 𝑒𝑒

𝑑𝑑𝑑𝑑.

𝑙 𝑙 𝑥 𝑥 𝑥𝑥

𝑆𝑆 𝑤𝑤

𝑆𝑆 𝑤𝑤

𝑜𝑜=−∞

(11)

Together (10) and (11) forms

2𝑝𝑝̂ 𝑖𝑖 ቆන𝑣𝑣(𝑥𝑥, 𝑦𝑦)𝑒𝑒 𝑖𝑖(𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦) 𝑑𝑑𝑑𝑑

+ න𝑣𝑣 𝑎 𝑎 (𝑥 𝑥 , 𝑦 𝑦)𝑒𝑒 −𝑖𝑖(𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦) 𝑑𝑑𝑑𝑑

ቇ

𝑆𝑆 𝑤𝑤

𝑆𝑆 𝑤𝑤

න cos ൬ 2𝑛 𝑛 𝑛

∞

= 4𝑝𝑝 𝑖𝑖 ෍𝑣𝑣ො 𝑛𝑛

𝑥𝑥൰ 𝑆𝑆 𝑤𝑤

𝑑𝑑𝑑𝑑

𝑙 𝑙 𝑥

(12)

𝑛𝑛=−∞

Lastly, the radiation term in (3) will be

2𝑖𝑖𝑖𝑖𝑖𝑖න න𝑣𝑣 𝑎𝑎 (𝑆𝑆)𝑣𝑣(𝑆𝑆 ′ )𝐺𝐺(𝑆𝑆|𝑆𝑆 ′ )𝑑𝑑𝑑𝑑 ′𝑑 𝑑 𝑑

𝑆𝑆 𝑤𝑤 𝑆𝑆 𝑤𝑤

∞

∞

𝑖𝑖ቀ 2𝑜 𝑜 𝑜

−𝑖𝑖ቀ 2𝑛 𝑛 𝑛

𝐺𝐺(𝑆𝑆|𝑆𝑆 ′ )𝑑𝑑𝑑𝑑′𝑑𝑑𝑑𝑑

= 2𝑖𝑖𝑖𝑖𝑖𝑖෍ ෍𝑣𝑣ො 𝑛𝑛 𝑣 𝑣 ො 𝑜𝑜 න න 𝑒𝑒

𝑙 𝑙 𝑥 𝑥 +𝑘𝑘 𝑥𝑥 ቁ𝑥𝑥 𝑒𝑒

𝑒𝑒 𝑖𝑖𝑘𝑘 𝑦𝑦 (𝑦𝑦−𝑦𝑦 ′ )

𝑙 𝑙 𝑥 𝑥 +𝑘𝑘 𝑥𝑥 ቁ𝑥𝑥′

(13)

𝑆𝑆 𝑤𝑤 𝑆𝑆 𝑤𝑤

𝑛𝑛=−∞

𝑜𝑜=−∞

𝑧𝑧 𝑓𝑓 = − 𝑖 𝑖 𝑖

In the previous papers [1,2,3] a radiation impedance was identified as

𝑒𝑒 𝑖𝑖𝑘𝑘 𝑦𝑦 ൫𝑦𝑦−𝑦𝑦 ′ ൯ 𝐺𝐺(𝑆𝑆|𝑆𝑆 ′ )𝑑𝑑𝑆𝑆 ′ 𝑑𝑑𝑑𝑑.

𝑆 𝑆 න න𝑒𝑒 𝑖𝑖𝑘𝑘 𝑥𝑥 ൫𝑥𝑥−𝑥𝑥 ′ ൯

𝑆𝑆 𝑤𝑤 𝑆𝑆 𝑤𝑤

(14)

𝑧𝑧 𝑓𝑓,𝑛𝑛𝑛𝑛 = − 𝑖 𝑖 𝑖

In parallel with (14) defined the following radiation impedance

𝑖𝑖ቀ 2𝑜 𝑜 𝑜

−𝑖𝑖ቀ 2𝑛 𝑛 𝑛

𝑒𝑒 𝑖𝑖𝑘𝑘 𝑦𝑦 (𝑦𝑦−𝑦𝑦 ′ ) 𝐺𝐺(𝑆𝑆|𝑆𝑆 ′ )𝑑𝑑𝑆𝑆 ′ 𝑑𝑑𝑑𝑑

𝑆 𝑆 න න 𝑒𝑒

𝑙 𝑙 𝑥 𝑥 +𝑘𝑘 𝑥𝑥 ቁ𝑥𝑥 𝑒𝑒

𝑙 𝑙 𝑥 𝑥 +𝑘𝑘 𝑥𝑥 ቁ𝑥𝑥′

𝑆𝑆 𝑤𝑤 𝑆𝑆 𝑤𝑤

(15)

∞

∞

Thus, (13) can be expressed as

2𝑖𝑖𝑖𝑖𝑖𝑖න න𝑣𝑣 𝑎𝑎 (𝑆𝑆)𝑣𝑣(𝑆𝑆 ′ )𝐺𝐺(𝑆𝑆|𝑆𝑆 ′ )𝑑𝑑𝑑𝑑′𝑑𝑑𝑑𝑑

= −𝑆𝑆2𝜌𝜌𝜌𝜌෍ ෍𝑣𝑣ො 𝑛𝑛 𝑣𝑣ො 𝑜𝑜 𝑧𝑧 𝑓𝑓,𝑛𝑛𝑛𝑛

𝑆𝑆 𝑤𝑤 𝑆𝑆 𝑤𝑤

𝑛𝑛=−∞

𝑜𝑜=−∞

(16)

All terms in the variational formulation (3) has now been expressed with the space harmonic expansion. In summary the variational functional is

න cos ൬ 2𝑛 𝑛 𝑛

∞

𝑉𝑉= 4𝑝𝑝̂ 𝑖𝑖 ෍𝑣𝑣ො 𝑛𝑛

𝑥𝑥൰ 𝑆𝑆 𝑤𝑤

𝑑𝑑𝑑𝑑

𝑙 𝑙 𝑥

𝑛𝑛=−∞

−෍ ෍𝑍𝑍 𝑝𝑝 ൬𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛 𝑙 𝑙 𝑥

∞

∞

, 𝑘𝑘 𝑦𝑦 ൰𝑣𝑣ො 𝑛𝑛 𝑣𝑣ො 𝑜𝑜 න𝑒𝑒 𝑖𝑖 2(𝑜 𝑜 −𝑛 𝑛 )𝜋

𝑑𝑑𝑑𝑑

𝑙 𝑙 𝑥 𝑥𝑥

𝑜𝑜=−∞

𝑛𝑛=−∞

𝑆𝑆 𝑤𝑤

∞

∞

−𝑍𝑍 𝑏𝑏 (𝑘𝑘 𝑦𝑦 ) ෍ ෍𝑣𝑣ො 𝑛𝑛 𝑣𝑣ො 𝑜𝑜 ෍ 𝑒𝑒 𝑖𝑖2𝜋𝜋(𝑜𝑜−𝑛𝑛)𝑚𝑚

(17)

𝑛𝑛=−∞

𝑜𝑜=−∞

𝑚𝑚𝑙𝑙 𝑥𝑥 ∈𝑆𝑆 𝑤𝑤

∞

∞

−𝑆𝑆2𝜌𝜌𝜌𝜌෍ ෍𝑣𝑣ො 𝑛𝑛 𝑣𝑣ො 𝑜𝑜 𝑧𝑧 𝑓𝑓,𝑛𝑛𝑛𝑛

𝑛𝑛=−∞

𝑜𝑜=−∞

Appling the variational procedure, the functional (17) is derived with respect to the amplitude 𝑣𝑣 𝑛𝑛 and equaled to zero, so that a stationary point is found,

𝜕 𝜕 𝜕 𝛿 𝛿 𝑣 𝑣 𝑛

= 4𝑝𝑝̂ 𝑖𝑖 න cos ൬ 2𝑛 𝑛 𝑛

𝑥𝑥൰ 𝑆𝑆 𝑤𝑤

𝑑𝑑𝑑𝑑

𝑙 𝑙 𝑥

−෍𝑍𝑍 𝑝𝑝 ൬𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛 𝑙 𝑙 𝑥

∞

, 𝑘𝑘 𝑦𝑦 ൰𝑣𝑣ො 𝑜𝑜 න𝑒𝑒 𝑖𝑖 2(𝑜 𝑜 −𝑛 𝑛 )𝜋

𝑑𝑑𝑑𝑑

𝑙 𝑙 𝑥 𝑥𝑥

𝑜𝑜=−∞

𝑆𝑆 𝑤𝑤

(18)

∞

∞

−𝑍𝑍 𝑏𝑏 ൫𝑘𝑘 𝑦𝑦 ൯෍𝑣𝑣ො 𝑜𝑜 ෍ 𝑒𝑒 𝑖𝑖2𝜋𝜋(𝑜𝑜−𝑛𝑛)𝑚𝑚

−𝑆𝑆2𝜌𝜌𝜌𝜌෍𝑣𝑣ො 𝑜𝑜 𝑧𝑧 𝑓𝑓,𝑛𝑛𝑛𝑛

𝑚𝑚𝑙𝑙 𝑥𝑥 ∈𝑆𝑆 𝑤𝑤

𝑜𝑜=−∞

𝑜𝑜=−∞ = 0

This will form a system of equations with elements

𝐴𝐴 𝑛𝑛𝑛𝑛 = 𝑍𝑍 𝑝𝑝 ൬𝑘𝑘 𝑥𝑥 + 2𝑛 𝑛 𝑛

𝑖𝑖 2(𝑜 𝑜 −𝑛 𝑛 )𝜋

, 𝑘𝑘 𝑦𝑦 ൰න𝑒𝑒

𝑑𝑑𝑑𝑑+ 𝑍𝑍 𝑏𝑏 ൫𝑘𝑘 𝑦𝑦 ൯ ෍ 𝑒𝑒 𝑖𝑖2𝜋𝜋(𝑜𝑜−𝑛𝑛)𝑚𝑚

𝑙 𝑙 𝑥 𝑥𝑥

𝑙 𝑙 𝑥

𝑚𝑚𝑙𝑙 𝑥𝑥 ∈𝑆𝑆 𝑤𝑤 + 𝑆𝑆2𝜌𝜌𝜌𝜌𝑧𝑧 𝑓𝑓,𝑛𝑛𝑛𝑛

𝑆𝑆 𝑤𝑤

(19)

𝑝𝑝 𝑛𝑛 = 4𝑝𝑝̂ 𝑖𝑖 න cos ൬ 2𝑛 𝑛 𝑛

and

𝑥𝑥൰ 𝑆𝑆 𝑤𝑤

𝑑𝑑𝑑𝑑

𝑙 𝑙 𝑥

(20)

The remaining integrals in (18,19,20), assuming a rectangular wall with length 𝑎𝑎 in the x-direction and 𝑏𝑏 in the y-direction, can be found to be

𝑖 𝑖 2𝜋 𝜋 (𝑜 𝑜 −𝑛 𝑛 ) 𝑎

𝑒

𝑙 𝑙 𝑥 𝑥 −1 𝑖 𝑖 2𝜋 𝜋 (𝑜 𝑜 −𝑛 𝑛 ) if 𝑛𝑛≠𝑜𝑜

𝑖𝑖 2(𝑜 𝑜 −𝑛 𝑛 )𝜋

𝑑𝑑𝑑𝑑= ቐ 𝑏𝑏𝑙𝑙 𝑥𝑥

න𝑒𝑒

𝑙 𝑙 𝑥 𝑥𝑥

𝑎𝑎𝑎𝑎 if 𝑛𝑛= 𝑜𝑜

𝑆𝑆 𝑤𝑤

(21)

and

න cos ൬ 2𝑛 𝑛 𝑛

= 𝑙 𝑙 𝑥

2𝑛 𝑛 𝑛 𝑛 sin 2𝑛 𝑛 𝑛

𝑥𝑥൰𝑑𝑑𝑑𝑑 𝑆𝑆 𝑤𝑤

𝑎𝑎 (22)

𝑙 𝑙 𝑥

𝑙 𝑙 𝑥

It can be noted that if 𝑙𝑙 𝑥𝑥 = 𝑎𝑎 eq. (22) is equal to zero, and the system is not excited. The system of equations in (18) is now expressed as a matrix equation

𝐀𝐀𝐀𝐀= 𝐩𝐩 (23)

with elements found by (19) and (20). Here, the matrix 𝐀𝐀 and vectors 𝐯𝐯 and 𝐩𝐩 needs to be truncated. The vector 𝐯𝐯 contains the space harmonic amplitudes and is found as 𝐯𝐯= 𝐀𝐀 −1 𝐩𝐩 . 3. DISCUSSIONS

The derivations of these expressions are quite straight forward. However, what is still problematic is the radiation impedance (15), that contains a double integration over the surface with a singular function, the Green’s function 𝐺𝐺(𝑆𝑆|𝑆𝑆 ′ ) . In [3] the corresponding integral was rewritten in a form containing only one integration over the surface, which makes it possible to efficiently integrate numerically. There are also available good approximate expressions for this integral. These steps is still to be added to this research.

4. CONCLUSIONS

It has been shown that it is possible to form expressions for the vibration of a periodically supported plate with the radiation load of a finite opening, using a variational approach. However, efficient numerical integration of the involved radiation impedance is still missing to having numerical results. 5. REFERENCES

1. Brunskog, J. The sound transmission of finite ribbed plates using a variational technique.

Proceedings of Inter-Noise 2012 , New York City, USA, 2012. 2. Brunskog, J. The forced sound transmission of finite ribbed plates, investigating the influence of point connections and periodicity, NOVEM 2015, Dubrovnik, Croatia, 2015. 3. Brunskog, J. The forced sound transmission of finite single walls using a variational technique.

J. Acoust. Soc. Am. , 132(3) ,1482–1493, 2012. 4. Vazquez Torre, J. H., Brunskog, J., Cutanda Henriquez, V. An analytical model for 5. broadband sound transmission loss of a finite single leaf wall using a metamaterial, J. Acoust. Soc. Am. , 147 , 1697-1708, 2020. DOI: https://doi.org/10.1121/10.0000923 6. Vazquez Torre, J. H., Brunskog, J., Cutanda Henriquez, V., Jung, J. Hybrid analytical- numerical optimization design methodology of acoustic metamaterials for sound insulation, J. Acoust. Soc. Am. , 149 , 4398–4409, 2021. DOI: 10.1121/10.0005316 7. Northwood, T. D. Transmission loss of plasterboard walls. Building Research Note BRN-66 ,

National Research Council of Canada, 1968. 8. Morse, P. M. and Ingard, K. U. Theoretical acoustics . Princeton university press, Princeton, New Jersey, 1968. 9. Thomasson, S.-I. On the absorption coefficient. Acustica , 44 , 266–273, 1980.