Building Acoustics

Statistical energy analysis (SEA) 269

where η is the loss factor. It is assumed that the RMS-value is taken as a mean value over
the area of the plate and also over a frequency band Δω to include a number of modes
around the frequency ω. The model energy Emodal is expressed as

(^) modal ,

E

E

n ω

=

⋅Δ

(7.3)

and assumption no. 2 tells us that the net power W 12 transmitted between the two
subsystems may be written

12 12 21 12 12

12 12

.

EE EE

WWW b b

nnωωnn

⎛⎞⎛

=−=⋅ − =⋅−′′⎜⎟⎜′

⎝⎠⎝⋅Δ ⋅Δ

⎞

⎟

⎠

(7.4)

The quantities b and b ́ = b/Δω are factors of proportionality. The equation may be cast
into a more suitable form by introducing the aforementioned coupling loss factors ηij. We
then have

12 1 12
and 21 2 21.

WE

WE

ωη

ωη

′ =

′ =

(7.5)

Hence, 12
1

b

n

ωη

′

= and 21

2

b

n

ωη

′

= , which gives

nn112 2 21⋅η =⋅η , (7.6)

which is called the reciprocity or the consistency relation. Equation (7.4) may then be

written as

(^1212112)
2

.

n

WE

n

ωη

⎛

=−⎜

⎝⎠

E

⎞

⎟ (7.7)

The last assumption given was that the forces driving the various subsystems were
statistically independent. This implies that we may calculate the total modal energy of
each subsystem by adding the responses resulting from each force input. For system
containing k subsystems we get the following set of equations:

11 21 1

1 in

1 1

12 2 2 in

(^22)
2
in
1

.

ik

i

i

i

k k

kkik

ik

E W

E W

E W

ηη η η

ηηη

ω

ηηη

≠

≠

≠

⎡⎤+− ⋅−

⎢⎥⎡ ⎤

⎢⎥⎡⎤

⎢⎥−+⋅⋅⎢⎥⎢ ⎥

⎢⎥⎢⎥⎢ ⎥

⎢⎥⎢⎥⋅ ⎢ ⋅ ⎥

⋅⋅⎢⎥⋅⋅⋅⋅ ⎢⎢⎥

⋅

=⎥

⎢⎥⎢⎥⎢ ⋅ ⎥

⋅⋅⋅⋅ ⎢ ⎥

⎢⎥⎢⎥⋅ ⋅

⎢⎥⋅⋅⋅⋅⎢⎥⎢ ⎥

⎢⎥⎣⎦⎢ ⎥

−⋅⋅+⎣ ⎦

⎢⎥

⎣⎦

∑

∑

∑

(7.8)