The imaginary part of the diffuse field forced normalized radiation impedance of a rectangular panel John Davy 1 School of Science, RMIT University & Infrastructure Technologies, CSIRO GPO Box 2476 Melbourne Victoria 3001 Australia & Private Bag 10 Clayton South Victoria 3169 Australia

ABSTRACT This paper modifies previous approximate formulae for calculating the imaginary part of the azimuthally averaged forced normalized radiation impedance experienced on one side of a rec- tangular panel mounted in an infinite rigid plane baffle by a forced travelling wave on the rec- tangular panel which is excited by an infinite plane sound wave incident at an angle to the normal to the rectangular panel on one side of the rectangular panel. This modification is made so that the formulae can be analytically integrated over all possible solid angles of incidence on one side to obtain formulae for the imaginary part of the diffuse sound field forced normalized radiation impedance of the rectangular panel. The reason for developing approximate formulae for the imaginary part of this diffuse sound field forced normalized radiation impedance is so that they can be used when approximating the equation for the diffuse field sound transmission coefficient of a limp single leaf wall.

1. INTRODUCTION

The imaginary part of the azimuthally averaged forced normalized radiation impedance experi- enced on one side of a rectangular panel mounted in an infinite rigid plane baffle by a forced travel- ling wave on the rectangular panel with sides of length 2 a and 2 b can be calculated by using the ap- proximate formulae given by Davy et al . [1]. The wavenumber of the sound in the acoustic me- dium, which is the same in both the half spaces on each side of the infinite rigid plane baffle and the rectangular panel, is k , and the forcing sound wave is an infinite plane sound wave incident at an angle to the normal to the rectangular panel whose cosine is g . This exciting sound wave is only in- cident from one side of the infinite rigid plane baffle and the rectangular panel, although this inci- dent side may be either side of the infinite rigid plane baffle and the rectangular panel. The radia- tion impedance has been normalized by dividing it by the characteristic impedance of the acoustic medium.

The approximate formulae for the real part of the azimuthally averaged forced normalized radia- tion impedance experienced by a forced travelling wave given in Davy et al . [1] may be averaged over all solid angles of incidence using analytic integration to obtain approximate formulae for the real part of the normalized radiation impedance experienced by the forced travelling wave field ex- cited by a diffuse sound field incident on one side of a limp rectangular panel mounted in an infinite rigid plane baffle [2]. Unfortunately, this is not the case with the current version of the approximate formulae in Davy et al . [1] for the imaginary part of the azimuthally averaged forced normalized radiation impedance experienced by a forced travelling wave.

1 johnldavy@gmail.com

worm 2022

2. THE NORMALIZED FORCED RADIATION IMPEDANCE

The problem is Equation (128) of Davy et al . [1] which for the travelling wave case is

4 4 4 1 1 1 ti tli thi z z z = + . (1)

In this paper, this problem is overcome by replacing Equation (1) with the operation of taking the minimum of z tli and z thi . The maximum difference made by this change will occur when z tli and z thi are equal. In this situation, the operation of taking the minimum of z tli and z thi . will produce a value

4 2 1.19 = which is times or 0.75 dB greater than the value produced by Equation (1). However, to set the limits of integration and determine which function to integrate in each range of integration, it is necessary to determine the value of the cosine of the angle of incidence g when the impedance changes from z tli and z thi . In order to make this paper more self-contained, the some of the necessary equations from Davy et al . [1] are reproduced here.

The half side of the equivalent square s occurs in the theoretical Equations (F4) and (F5) of [3] ( ) 2 s ab a b = + . (2)

The function H is given by the theoretical Equation (C11) in [3]

( ) ( ) ( ) 2 2 H( ) ln 1 1 1 3 x x x x x = + + − + − . (3)

z tli is given by the imaginary part of the theoretical Equation (C12) of [3]

( ) ( ) 2 H H tli z k b a b a b a  = +     . (4)

z thi is given in [1] based on an empirical modification of the second term of the theoretical Equation (7.7) of Leppington et al. [4].

( ) 3 2 thi z ksg  = , (5)

( ) cos g  = where and θ is the angle of incidence relative to the normal to the rectangular panel.

z tli and z thi are equal when g equals g e . From Equations (4) and (5),

( ) ( )   2 3 min 1 H H ,1 e g k s b a b a b a = +     (6)

where g e is restricted have a maximum value of 1 because g cannot be greater than 1 because it is the cosine of the real angle of incidence. The minimum z tiA of z tli and z thi is

z g g z z g g   =   

if if

thi e tiA

. (7)

tli e

The following equations, based on [1,3], can now be used to calculate the imaginary part of the normalized radiation impedance z ti experienced by a forced travelling vibrational wave on the rectan- gular panel which has been excited by an infinite plane acoustic wave incident at an angle of θ to the normal to the rectangle.

worm 2022

( ) 2 r ks  = . (8)

0.9 ti w = (9)

( ) min ,1 ti ti g w r = (10)

0.07 ti = (11)

( )   0 max 0.9616 2 3 ,0.001 t hi ti z r  = +     (12)

( ) 0 0 min , t i tli t hi z z z = (13)

 =  = + −           

if 0 if 0

z g z z g z g z g g g

0

t i

( ) ( )

. (14)

0 0

ti t i tiA ti t i ti ti

if g 1

z g g

tiA ti

Equations (8) to (10) are based on theoretical Equation (19) in [2]. Equation (12) is based on theoret- ical Equation (27) of [2] which has an empirical offset correction and approximate theoretical Equa- tion (B16) of [3]. Equation (14) is based on the empirical linear interpolation Equation (33) of [2]. Equation (13) replaces Equation (1) and combines the appropriate low and high frequency values from Equations (4) and (12).

For a limp panel mounted in an infinite rigid plane baffle, the amplitude of the forced travelling velocity wave in the limp panel excited by an incoming infinite plane acoustic wave is independent of the angle of incidence of the plane acoustic wave because the sound pressure at any point on the surface of the limp panel is independent of the angle of incidence and there is no wave propagation in a limp panel. This means the normalized radiation impedance, experienced by the forced travelling wave field excited by a diffuse sound field incident on one side of a limp rectangular panel mounted in an infinite rigid plane baffle, can be obtained by averaging the normalized radiation impedance over all solid angles of incidence using integration. As noted above this procedure has already been performed for the real part of the normalized radiation impedance [2] to obtain approximate formulae for the real part of the normalized radiation impedance experienced by the diffuse field excited forced travelling wave vibration field using the approximate formulae developed by Davy et al . [1]. In this paper, the same procedure will be performed to obtain the imaginary part of the normalized radiation impedance < z ti > experienced by the forced travelling wave field excited by a diffuse sound field.

Averaging z ti over all solid angles of incidence gives

0 sin ti ti z z d    =  . (15)

( ) 2

( ) cos g  = ( ) sin dg d   = − Because , and

0 ti ti z z g dg =  . (16)

( ) 1

To evaluate this integral, it is first necessary to consider the relative values of the limits g e and g ti . The limit g ti is always greater than g e except when they are both restricted to be equal to 1. The integral on the right hand side of Equation (16) is evaluated from g = 0 to g = 1 along a vertical line

worm 2022

1 ti g g   for a fixed value of k . For , the integrand is z thi given by Equation (5) because g e < g ti when g e < 1.

( ) ( ) ( ) ( ) 1 1 3 2 2 1 1

ti ti ti ti g g z g dg ksg dg g ks   = = −   . (17)

0 ti g g   For , the integrand is the linear interpolation in the g domain given by the middle line of equation (14). The value of the integrand at g = 0 is z t 0 i and the value of the integrand at g = g ti is z tiA . This means that the exact value of the integral is given by the trapezoidal rule as

( ) ( ) 0 0 2 ti g

ti ti t i tiA ti z g dg g z z g = +      . (18)

Thus, the approximate formulae for the diffuse field excited vibration field is

( ) ( ) ( ) 2 0 2 1 1 ti ti t i tiA ti ti z g z z g g ks  = + + −     . (19)

At first glance it might be thought that z tli does not need to be calculated because it appears to have no influence on z ti or < z ti > because g e < g ti when g e < 1 implies that z tiA ( g ti ) = z thi ( g ti ) when g e < 1. This is not the case for two reasons. The first reason is that z tli does influence z t 0 i = z ti ( g = 0) for low values of k via Equation (13). The second reason is that z tli does influence z ti ( g = 1) = z tiA ( g = 1) when g e = g ti = 1 via Equation (7). The values of z t 0 i and z tiA ( g = g ti ) control the value of z ti in the interpolation

0 ti g g   zone given by and the value of the first term on the right hand side of Equation (19) for < z ti >.

The original approximate method [1], the new approximate method proposed in this paper and the numerical calculations for calculating z ti were compared for values of g corresponding to angles of incidence of θ equals 0, 15, 30, 45, 60, 70, 75, 80, 85 and 90 degrees and values of wave number k equal to values separated by half octave steps in the range from 0.25 to 1024 for a = b = 1. The only values of k for which the difference of the moduli of the values in dB of each approximate method minus the numerical calculation was greater than 0.02 dB were 0.71 and 1. The average of these differences of moduli was 0.001 dB and the maximum and minimum differences were 0.29 dB and - 0.20 dB where a positive value indicates that the original approximate method was in better agreement with the numerical results. The maximum of the difference in dB between the new approximate method and the original approximate method was 0.29 dB. This value is smaller than the maximum difference between the two approximate methods of 0.75 dB which was calculated above since the calculations were only made for certain specific values of g , k , a, and b . The new approximate method and the original approximate method were greater than the numerical calculations on average by - 0.01 dB and -0.02 dB respectively. The standard deviation and the maximum of these differences were 0.41 dB and 2.11 dB respectively for both approximate methods. The minima of the differences were -1.55 dB and -1.56 dB respectively for the new and the original approximate methods. These figures indicate that the new and approximate methods have a similar accuracy. The mean, the stand- ard deviation, the maxima, and the minima of the values in decibels of the new approximate formulae proposed in this paper minus the numerical calculations for calculating < z ti > were 0.00 dB, 0.18 dB, 0.36 and -0.55 dB.

Formulae for the real part of the normalized radiation impedance experienced on one side of a rectangular panel in an infinite rigid plane baffle by a forced travelling wave on the rectangle are given in [1].

worm 2022

3. THE DIFFUSE FIELD FORCED SOUND TRANSMISSION COEFFICIENT

 If an infinite plane sound wave, with an angular frequency , and with a root mean square sound ( ) , , i p  pressure is incident on a single leaf planar panel mounted in an infinite planar rigid baffle

  at an incidence angle of to the normal to the panel and at an azimuthal angle of , in combination

( ) 2 , , i p  with its reflection, it generates a blocked sound pressure at the surface of the panel of .

Note that the actual pressure at the surface of the panel will be different due to the sound pressure wave on the incident side of the panel generated by the transverse motion of the panel. The transverse velocity of the panel is

( ) ( ) ( ) ( ) 0

 

2 , , , ,

p v

   =   +  

i

, (20)

2 , , , ,

z z c

r p

( ) , , p z  where is the normalized impedance that the panel presents to the incident sound wave

( ) , , r z  described above and is the normalized radiation impedance of the panel when excited by

the incident sound wave described above. These two impedances have been normalized by dividing them by the characteristic impedance of the acoustic medium which is assumed in this case to be the same on both sides of the panel and baffle. This characteristic impedance is equal to the product of

c 0  the speed of sound in the acoustic medium and the ambient density of the acoustic medium .

t I The sound intensity transmitted by the panel on either side is

( ) ( ) ( ) ( )

2 2

    

    = =     +

4 , , Re , , , , Re , ,

p z I v c z

i r t r

(21)

0 2

( ) ( )

  

2 , , , ,

z z c

0

r p

( ) Re , , r z      Note that the real part of the normalized radiation impedance of the panel is the

radiation efficiency of the panel.

i I The incident sound intensity is

( ) ( ) 2

 

, , cos i i

p I c

 = (22)

0

( ) , ,  Thus, the sound transmission coefficient of the panel is

( ) ( )

 

    = =

( ) ( ) ( ) 2 Re , , , ,

z I I z z

r t

, (23)

  

+

, , , , 2 cos

i r p

This expression can also be obtained using variational techniques. For a thin limp isotropic panel

( ) ( ) , , 2 p pi z z jd   = = , (24)

where

m d c   = . (25)

0 2

worm 2022

m d is the mass per unit area of the panel. It should be noted that is the ratio of the magnitude of the low frequency impedance of the panel to the sum of the characteristic impedances of the acoustic media on each side of the panel.

To proceed further the approximation of replacing the forced normalized radiation impedance in the denominator by its average value over all possible solid angles of incidence on one side of the panel is made and the real part of the forced normalized radiation impedance in the numerator is replaced by its azimuthally averaged value.

z

( )

z z d 

   + +  

. (26)

tr

( ) ( ) 2 2 ,

cos

tr ti

It is reasonable to make these approximations because the forced normalized radiation impedance does not vary much when the azimuthal angle is varied, providing the shape of the panel is not too different from the shape of a square. At high frequencies, the square bracket term in the denominator

2 d of equation (26) is approximately equal to , unless the mass per unit area of the panel is very small or the area of the panel is very large. At low frequencies, the forced normalized radiation impedance is independent of both the angle of incidence and the azimuthal angle.

( ) d   ( ) ,  To predict the diffuse field sound transmission coefficient , is multiplied by the

( ) cos  incident sound intensity which is proportional to and integrated over all possible solid angles

of incidence to obtain the total transmitted intensity. This is then divided by the total incident sound intensity, which is obtained by integrating the incident sound intensity over all possible solid angles of incidence.

 

2 2

( ) ( ) ( )



     

, cos sin

d d

( )

=

0 0

 

d

2 2

( ) ( )



  

cos sin

d d

0 0

1 , cos sin

 

2 2

( ) ( ) ( )



    

=

d d

. (27)

0 0



2 1 2

( ) ( ) ( ) ( ) ( )

 

  = =  

     

2 , cos sin , cos

d d

0 0

Inserting Equation (26) into the first term on the last line of Equation (27) cancels the potentially

( ) cos  troublesome in the denominator of Equation (26).

     = + + + +  . (28)

2

( )

2 2 2 2 2 sin 2 tr tr d

z d z

( )

0

( ) ( )

z z d z z d

tr ti tr ti

d tr z ti z If is very much larger than and , then Equation (28) becomes

( ) 2 2 tr d

z d   . (29)

Equation (29) was first obtained by Sato [5] and Equation (28) is a less restrictive version of this

a b = ti z ti z equation. If the rectangular panel is square ( ), then the low frequency values of and are

given by Equation (4) as

worm 2022

k S z z k a k S = =  =  . (30)

2 0.473 0.473 2 ti ti

tr z tr z The low frequency values of both and are given by Equation (116) of [1], which is the

theoretically derived Equation (35) of [2], as

2 2 4 2 2 tlr

k ab k S z   = = , (31)

4 S ab = k S ti z tr z where is the area of the panel. For small values of , is larger than and the

tr z term in the denominator of Equation (28) can be ignored. Using Equations (30) and (31) in

k S Equation (28) gives for small values of

          + +

2 2 0 0 2 2 4 4

S S

( )

. (32)

( ) ( )

d

0.946

m S m S

o o

The final expression in equation (32) agrees with Equation (14) of Wang [6] when the incorrect value

 of in the denominator of the second fraction of Wang’s Equation (14) is removed.

tr z To obtain a rough simple high frequency approximation for from the equations in [1,2], the

q tr w tr  low frequency result will be ignored by setting equal to zero, will be set equal to 1, will be f set equal to 0 and it will be assumed that is very much greater than zero. This gives

( ) ( ) 1 1 ln 2 0.443 ln 2 2 tr z ks k S  +      (33)

and an approximation to Equation (29) is

( ) ( ) 2 1 ln d k S d   . (34)

Equation (34) agrees with equation (11) of Davy [7] and is an approximation of Equation (54) of Sewell [8] which was derived using a different method.

( ) , , r z  In the past, the finite size of the panel was considered by assuming that had the infinite

panel value of

( ) ( ) , , 1 cos r z   = , (35)

l  2  and limiting the upper range of integration to an angle of incidence which is less than rad or 90°. Inserting Equations (24) and (35) into Equation (23) gives

( ) ( ) 2 2 1 , 1 cos d   = + (36)

Inserting Equation (36) into the second term on the last line of Equation (27) and using the limiting

l  2  angle instead of rad, gives

  + = =   + +    . (37)

2 1

1 1 ln 1 1 cos l d

dx d d x d d    

( ) ( ) ( ) 2

2 2 2 2 cos

l

worm 2022

If 2 1 d

as is often the case, then Equation (37) can be approximated as

( ) ( ) ( ) 2 2 2 1 ln 1 cos d l d d      = − +   . (38)

If ( ) 2 2 1 cos l d 

( ) ( ) ( ) 2 2 1 ln cos d l d      = −   (39)

Equation (39) is the same as Equation (55) of Sewell [8]. Comparing Equations (34) and (39) gives

( ) 2 1 cos l k S  = . (40)

worm 2022

Equations (37) to (40) agree with Equations (8) to (10) and (12) of Davy [6]. 4. CONCLUSIONS

This paper has developed an approximate formula for the calculation of the imaginary part of the diffuse sound field forced normalized radiation impedance of the rectangular panel. It has used this formula to derive a formula for the sound transmission coefficient of a limp rectangular panel, mounted in an infinite rigid baffle between two infinite half spaces, which is correct at both low and high frequencies.

5. REFERENCES

1. Davy, J. L., Larner, D. J., Wareing, R. R. & Pearse, J. R. The acoustic radiation impedance of a rectangular panel. Building and Environment , 92 , 743-755 (2015). 2. Davy, J. L. The forced radiation efficiency of finite size flat panels that are excited by incident sound. Journal of the Acoustical Society of America , 126 ( 2 ), 694-702 (2009). 3. Thomasson, S.-I. Report TRITA-TAK-8201, Theory and experiments on the sound absorption as function of the area , Department of Technical Acoustics, Royal Institute of Technology, Stockholm, Sweden, 1982. 4. Leppington, F. G., Broadbent, E. G. & Heron, K. H. Acoustic radiation from rectangular panels with constrained edges. Proceedings of the Royal Society of London Series A , 393 , 67-84 (1984). 5. Sato, H. On the mechanism of outdoor noise transmission through walls and windows - A modification of infinite wall theory with respect to radiation of transmitted wave. Journal of the Acoustical Society of Japan , 29 ( 9 ), 509-516 (1973). 6. Wang, C. Formulae for the forced sound transmission coefficients. Acta Acustica united with Acustica , 103 , 227-231 (2017). 7. Davy, J. L. Predicting the sound transmission loss of cavity walls. Interior Noise Climates - Proceedings of the 1989/90 National Conference of the Australian Acoustical Society , pp. 1-16. Cottesloe Beach Resort, Perth, Australia, 19-20 April 1990. 8. Sewell, E. C. Transmission of reverberant sound through a single-leaf partition surrounded by an infinite rigid baffle. Journal of Sound and Vibration 12 ( 1 ), 21-32 (1970).