9. Statistical Energy Analysis (SEA)

9. Statistical Energy Analysis (SEA) 80 

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9. Statistical Energy Analysis (SEA) 

9.1 Introduction 

In this chapter an introduction to a framework for the analysis of the dynamics of complex, 

built-up structural and fluid systems will be outlined. This framework denoted Statistical 

Energy Analysis was developed in the 1960's, to a great extent to clarify and handle structural 

acoustic problems in conjunction with space-crafts. Since then the formalism used within this 

framework has been successfully applied in many other areas such as ships, buildings, aircrafts 

and cars. 

Perhaps the most prominent text on this topic is by Lyon [1] and gives a valuable 

description of the framework by explaining each word in its name. The explanations are cited 

here: 

"Statistical emphasizes that the systems being studied are presumed to be drawn 

from statistical populations having known distributions of their dynamical 

parameters. Energy denotes the primary variable of interest. Other dynamic 

variables such as displacement, pressure, etc., are found from the energy of 

vibrations. The term Analysis is used to emphasize that SEA is a framework of 

study rather than a particular technique." 

Hence, the "statistical philosophy" introduced in SEA must be seen as a compilation of methods 

or ways to study structural acoustic questions where basic knowledge must not be set aside for a 

routine scheme. 

Apart from the textbook mentioned above, a few other references can be given which either 

exemplify the use of SEA in engineering practice [2] or bring out the limitations for the 

framework [3]. A rather comprehensive and advanced discussion of SEA is given in [4] where 

the objects under study are ships. 

The natural area of operation for SEA is the analysis of the high frequency behaviour of 

complex structures. Usually the analysis involves coupling between two or more, simply 

identifiable but complex structural and/or fluid systems. In this context, high frequencies refer 

to the frequency region where the system under study possesses a large number of 

eigenfrequencies in a limited frequency band. This means that if the system under study is 

excited by a force which has components at all frequencies - a broad band, white noise force - a 

lot of resonances will occur within the frequency band.


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It is of course justified to ask why introduce a statistical point of view for the analysis of 

the dynamics of systems that are normally considered to be rather deterministic. The answer 

lies in the many problems relating to the application of numerical or semi-analytical methods 

(e.g. FEM) for complex systems having a large number of eigenfrequencies in the frequency 

range of interest. Among those problems we may list: 

- uncertainties in the input data (e.g. boundary conditions) 

- unknown excitation (e.g. excitation points are not specified) 

- variations in the fabrication (e.g. two cars of the same type will not be exactly the same) 

Since the designer often wants to make predictions early at the design stage when details of the 

structural configuration are missing or incomplete, a simple "macroscopic" analysis is 

motivated. One of the advantages with SEA is namely the representation of the structural and 

fluid systems. These are represented by geometrical and material properties from which 

characteristics in the dynamic state such as: 

average modal densities 

n(ω ) = lim 

Δω →0 

N( ω + Δω ) − N( ω) 

Δω 

(9.1) 

where N(ω ) is the number of eigenfrequencies up to the angular frequency ω , 

average modal damping, 

η = W d 

2πV 


where W d is the energy lost during one period of oscillation and V is the peak value of the 

potential energy during that period, these quantities being based on the average modal energies, 

and 

average coupling data 

can be derived. The coupling between structural and/or fluid members, subsystems, of the 

system leads to energy flows in between them in order to maintain an energy balance in the 

presence of dissipative losses. In turn, the dynamic field variables of interest to the engineer are 

obtained as spatial and temporal averages which are directly related to the total energy of the 

subsystem.


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The most obvious disadvantage with SEA is that the energy quantities obtained for the 

different subsystems are statistical estimates of the true energy quantities and accordingly 

involve some uncertainty. Usually the uncertainty will be less pronounced at high frequencies 

where the number of resonant eigenfrequencies included is high for all subsystems. This means 

that there will be a low frequency limit in practice but no principle limitation exists as long as 

the subsystems have resonant eigenfrequencies 

The subsystems in SEA are presumed to be finite, linear, elastic structures or fluid cavities 

which can be described by their uncoupled eigenfrequencies, eigenfunctions (mode shapes) and 

losses. It is assumed that each eigenfunction can be modelled by a simple oscillator (one may 

here think of a mass-spring system) and that the interaction between two multi-modal 

subsystems can be represented by the coupling between two sets of oscillators, see Figure 9.1. 

System 1 

System 2 

System 1 

System 2 

1 

2 

1 

2 

1 

2 

1 

2 

α 

σ 

α 

σ 

N 1 

N 2 

N 1 

N 2 

Π σα 

Total energy flow 

Π 12 

Figure 9.1 Illustration of interaction and coupling between multi-modal systems. After [1] 

No attempt has been made in this chapter to fully compile all the theoretical considerations 

underlying SEA but the most necessary fundamentals required for the understanding of the 

procedure in working with SEA will be given.


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9.2 Fundamentals of Coupled Subsystems 

In SEA the complex system under study may be subdivided into a number of subsystems. 

Some can be directly excited by sources whereas others are indirectly excited through different 

couplings (junctions, discontinuities, transfer from one medium to another) to the former ones. 

Accordingly, a basic question in SEA is how to subdivide the system into subsystems. This 

division can intuitively be made at the boundaries of the different structural and fluid members. 

However, it is very important to bear in mind that for complex systems there will be different 

wave types present which realize different groups of modes. This implies that in addition to the 

previously mentioned subsystems we must subdivide the system with respect to these groups of 

modes. Hence, of a physical structural element we may have to form two or more sub-systems 

where one involves only flexural motion and the others different in-plane motions. 

The interaction between two simple oscillators which are assumed to be linear and coupled 

by linear, non-dissipative elements (no energy lost at the connecting boundaries) constitute the 

fundamental model from which some important theorems in SEA can be deduced. To 

summarize it is appropriate to start from the basic equation 

Π ′ 21 = B(E 1 − E 2 ) . (9.3) 

Consider the coupled system in Figure 9.1 if the energies in the two subsystems are E 1 and 

E 2 then the net energy flow between them is found from eqn. (9.3) above. In this equation B is 

a function of the coupling strength. This coupling is solely governed by the properties of the 

subsystems and the coupling elements. 

From this equation we may state: 

• the energy flow is proportional to the actual vibrational energies of the two subsystems 

• the coupling function or proportionality is positive, definite and symmetric in the system 

parameters therefore the system is reciprocal and the energy flows from the subsystem with 

the higher energy to the one with the lower energy 

• if only one subsystem is directly excited the highest possible energy for the indirectly driven 

subsystem is that of the first. 

The analogy with thermodynamics is not very far away. Thus, the picture with an energy flow 

between a hot body and a cold one in contact may be helpful although it should not be taken too 

literally.


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9.3 Energy Flow Between Multi-modal Subsystems 

Consider again the model in Figure 9.1. The basic assumptions when SEA is used in 

conjunction with this coupling problem can be cited from [1]: 

"1. Each mode is assumed to have a natural frequency ω iα that is uniformly 

probable over the frequency interval Δω . This means that each subsystem is a 

member of a population of systems that are generally physically similar, but 

different enough to have randomly distributed parameters. The assumption is 

based on the fact that nominally identical structures or acoustical spaces will 

have uncertainties in modal parameter, particularly at higher frequencies. 

2. We assume that every mode in a subsystem is equally energetic and that its 

amplitudes 

Y iα (t ) = ∫ ρ i y i ψ iα dx i / M 

are incoherent, that is, 

< Y iα Y iβ > t = δ αβ < Y iα 2 > . 

This assumption requires that we select mode groups for which this should be 

approximately correct, at least, and is an important guide to proper SEA 

modeling. It also implies that the excitation functions L i are drawn from 

random populations of functions that have certain similarities (such as equal 

frequency and wave number spectra) but are individually incoherent. " 

(In the expressions cited the following notation is used: 

ρ i is the density distribution of subsystem i 

y i is the physical response quantity for subsystem i 

ψ iα is the eigenfunction associated with the α th eigenfrequency of subsystem i 

x i is the local spatial co-ordinate for subsystem i 

M i is the total mass of subsystem i 

Y iα is the modal response quantity (generalised response) for mode α and 

subsystem i 

< > t denotes temporal average 

δ Kroneckar delta )


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Now, based upon the conditions cited the energy flow from system 1 to system 2 in Figure 

9.1 can be derived in analogy with eqn. (9.3) if an average value for the many, generally 

different coupling parameters B σα replaces B and E 1 and E 2 are replaced by the respective 

average modal energies E 1 and E 2 

m m 

Π ′ 21 = N 1 N 2 ( E 1 − E2 ) . (9.4) 

In the above equation denotes the average modal coupling and N 1 , N 2 the number of 

modes (eigenfunctions) in the two subsystems respectively. 

The average modal energies are defined by 

E i m = 

E i 

tot 

n( f ) Δf 


where n(f) is given by eqn. (9.1) and E i tot is the total vibrational energy of subsystem i. 

Since the energy flow between the subsystems, from one specific subsystem seen, can be 

interpreted as a loss of energy, the formalism can be developed by defining a coupling loss 

factor as 

η 21 = N 2 / ω (9.6a) 

which says that the loss from system 1 is proportional to the number of modes in system 2, i.e. 

how many energy reservoirs there are available in system 2. Since the complete system is 

reciprocal we could equally well have defined 

η 12 = N 1 / ω (9.6b) 

and from the two relations in (9.6) the reciprocity relation 

η 12 N 2 = η 21 N 1 (9.7) 

can be deduced. 

Introducing eqn. (9.6a) in eqn. (9.4) yields 

Π ′ 21 = ω η 21 N 1 E m m 

( 1 − E 2 ) (9.8)


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which in turn shows that the energy flow will be from the subsystem with the higher average 

modal energy tothe one with the lower. 

Equation (9.8) constitutes the fundamental block in any SEA calculation. The diagram in 

Figure 9.2 symbolizes the general situation in which eqn. (9.8) is to be applied. 

Π 

Π 

1, in 2, in 

E 1, tot 

Π 

21 

E 2, tot 

N 1 

N 2 

Π 

Π 

1, diss 2, diss 

Figure 9.2 

Illustration of energy flows to, form and in between two coupled subsystems 

From the diagram in Figure 9.2 one may see that not only does a flow between the two 

subsystems exist but also there may be both independent in-flow and out-flow, to and from each 

subsystem. Hence the average energy quantities sought for the prediction of the average 

response of a subsystem must be solved from a system of linear equations. 

7.3 SEA Modelling 

A number of questions are raised before and during the subdivision of a physical system 

into different, coupled subsystems. A basic feature of a subsystem must be that its response is 

determined by resonant modes. It is consequently not meaningful nor appropriate to apply an 

SEA approach to a system (or subsystem) below its fundamental eigenfrequency. However, 

coupling via non-resonant elements may be admitted. 

The sources of input power and the groups of eigenfunctions to which they are coupled 

must be identified. Moreover, the different groups of eigenfunctions with similar properties 

must be selected and represented by separate subsystems. As mentioned previously, these 

considerations usually mean that one physical part of the system must be represented by several 

SEA subsystems.


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The junctions between the subsystems must be assigned correct coupling loss factors. Very 

often the junctions are chosen to coincide with the geometrical boundaries of the physical parts 

but this is not necessary. A comprehensive discussion of the considerations involved in SEA 

modelling can be found in [4] where also some valuable hints on how to practically proceed 

from a physical object to an SEA model are outlined. 

9.4 The Average Modal Density 

The average modal density was introduced in section 9.1. This quantity will usually be 

estimated with sufficient accuracy from closed form expressions for elementary fluid and 

structural elements (cf. ch. 6) such as beams, plates, cylinders, cavities and so forth. 

Compilations of such expressions are found in many text books, e.g. [1] and [5]. It must be 

emphasized however that the closed form expressions normally are equivalent to the asymptote 

found at high frequencies. Therefore, the actual modal density, in the low and mid-frequency 

regions, may be different from the one predicted. At low frequencies where the number of 

modes in the frequency interval (band) of interest is small the estimates can be improved by 

measurements on the system if a sample of it exists and methods for such investigations are 

sketched in [1]. 

If in eqn. (9.7) the average modal density is introduced the reciprocity relation becomes 

η 21 n 1 = η 12 n 2 

(9.9a) 

the basic energy flow can be rewritten as 

⎛ 

m 

Π ′ 21 = ω η 21 Δω n 1 E η 1 − 

12 

⎜ n2 

⎝ 

η 21 

E 2 

m 

⎞ 

⎟ (9.9) 

⎠ 

9.5 The Dissipation 

In addition to the energy flow due to the coupling between subsystems also the energy flow 

out from the system under consideration - the dissipation - must be taken into account. This 

energy loss can be due to e.g. internal losses (transformation into heat) or to radiation or 

coupling to connected systems not included in the SEA-model. The dissipation is represented 

by an internal loss factor, η ii , for each subsystem i. 

Unfortunately, reliable prediction methods for the dissipation are not available to-day and 

the internal loss factors therefore either must be measured or rely upon experience. Theories 

exist for different kinds of mechanisms leading to losses but so far they are only applicable


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under rather specific and restrictive conditions. Moreover, experience may give the right order 

of magnitude in an overall sense but frequency trends are hard to foresee. This means that 

measurements rather often is what remains and a few experimental techniques are described in 

[4] and [5]. One may therefore conclude that the assignment of dissipative, internal loss factors 

to subsystems is a demanding task and further research in this area is desirable. 

9.6 The Coupling 

The description of the coupling is as shown above formalised by using coupling loss 

factors. Often these loss factors can be estimated theoretically whereby the duality of junction 

properties for infinite subsystems and statistical ensembles of finite subsystems is utilized [1]. 

In principle we may distinguish between three different types of junctions: 

• fluid cavity to fluid cavity (e.g. coupling through a partition wall) 

• structural element to fluid cavity (e.g. panel to cavity) 

• structural element to structural element (e.g. wall to floor). 

In the following these three types will be discussed in greater detail. 

9.6.1 Coupling Between Fluid Spaces 

It is hereby assumed that a transmission efficiency, τ, can be found for the interface 

between the two spaces. In the case that the two fluid spaces are separated by a partitioning 

wall the transmission efficiency is simply found from the sound reduction index 

R = 10 log (1 τ) 

commonly used in building acoustics. 

Using the transmission efficiency the coupling loss factor can be derived to be 

η 21 = τ c 1S 

8πfV 1 


where c 1 is the longitudinal wave speed of the fluid, S is the area of the partition and V 1 the 

volume of the sending space.


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It is thus found from eqn. (9.10) that the coupling loss factor increases with an increased 

transmission efficiency of the interface and with its area whereas an increase in the volume of 

the sending space reduces the coupling loss factor. 

9.6.2 Coupling Between a Structural Element and a Fluid Space 

For such a junction between two subsystems the coupling loss factor is related to the real 

part of the impedance which the structure sees looking into the fluid space. This leads to the 

ratio 

η fs = Re[Z] 

ω M s 


which hence implies a comparison of the input fluid impedance and the mass impedance of the 

structure. 

For the specific case of a surface coupling eqn. (9.11) can be explicitely written as 

η fs = 2 ρ c f σ 

ω m 


where the radiation efficiency, σ, of the structure has been introduced. ρ denotes the density of 

the fluid and m is the structural mass per unit area. 

From eqns. (9.11) or (9.12) it is seen that the coupling loss factor for the energy flow from 

the structure to the fluid increases with an increasing input impedance or equivalently radiation 

efficiency and decreases with an increased structural mass. 

Theoretical expressions for the input impedance (often designated radiation impedance) 

and the radiation efficiency have been derived for a variety of structures in contact with air or 

water and is readily found in the acoustic literature. 

9.6.3 Coupling Between Structural Elements 

This type of coupling can be derived in a number of ways. The most appropriate manner 

depends on the specific junction but often use is made of the infinite systems, associated with 

the actual structural elements. The most frequent classes of structural coupling are point-like 

(contact areas of dimensions smaller than the wave-length of the governing wave), line- or striplike 

(one-dimensional coupling) and surface coupling.


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For point-like coupling the ordinary or effective mobilities of the structural elements can be 

used and compilations of ordinary point mobilities are given and referenced in chapters 6 and 8. 

With respect to line or strip connections the coupling loss factor can be derived from the 

transmission efficiencies related to the associated infinite systems [5]. Thereby the coupling 

loss factor is obtained from 

η 21 = 

c g1 L 

τ 

2 ω S 21 (9.13) 

1 

where τ 21 denotes the transmission efficiency from element 1 to element 2, c g1 is the group 

velocity for waves in element 1 (the group velocity introduced here is the velocity with which 

the power is transferred which generally is different from the phase speed, ordinary wave speed, 

for dispersive waves), S 1 the area of subsystem 1 and L the length of the line interface. 

One may thus note that the coupling increases with an increased line length, with an 

increased transmission efficiency but decreases with an increased area of the sending surface. 

Moreover, the reciprocity relation obtained for normal incidence, τ 12 = τ 21 , is not valid in the 

case of "random" incidence but is replaced by 

k 1 τ 21 = k 2 τ 12 (9.14) 

where k 1 and k 2 are the wave numbers of the two structural elements respectively. 

The coupling of, for instance, plates over surfaces generally is more cumbersome to handle. 

Theoretical studies of waves in layered media describe the features of this class of coupling but 

it is beyond the scope here to further expand on the pertinent results. 

9.7 External Source Input 

The origin of wave propagation or, in the SEA philosophy, energy flow is of course input 

from one or more external sources. This input is according to the structural acoustic process 

penetrated in chapter 8, transmission. From an SEA point of view one may therefore consider 

the input power as given although all the question marks relating to transmission are far from 

straightened out. 

One particular case of power input to a system on which SEA is to be applied, however, 

will be discussed here since it does not belong to transmission in a structural acoustic sense. 

This case is featured by a source within a fluid space. If the fluid under study is air, typical air-


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borne sound sources are nowadays specified with respect to their air-borne sound power output. 

Commonly, the manufacturer already in product sheets state the acoustic power output which of 

course greatly facilitates the analyst's situation. In the work with SEA the external source input 

must not be considered as a problem. 

9.8 Energy Balance 

The energy balance for large systems has been extensively discussed in [4]. This section 

essentially recalls and summarizes the discussion therein although the notation used is different. 

The basic energy balance equation (9.8) is valid for two coupled, multi-modal systems 

when the underlying assumptions are fulfilled. The normal situation however, is that a lot more 

than two subsystems are involved. The block diagram in Figure 9.3 illustrates a general 

situation. 

As is seen from the figure, the general case includes many external source inputs as well as 

dissipation from all subsystems. 

Intuitively, it seems to be a straight forward task to generalize the two-subsystem case to N 

subsystems and obtain a system of N linear equations and indeed there are no formal 

difficulties. There is however a physical implication that must be taken into account before 

doing so. This implication is rather clearly demonstrated by considering the system in Figure 

9.4. 

In the two-subsystem case previously discussed, e.g. the two horizontally connected plates 

in Figure 9.4 with the vertical plate excluded, there is no ambiguity as to what is meant by the 

coupling loss factor. With respect to the three-plate system the quesiton arises: Should the rest 

of the system be connected or disconnected when one specific junctionis considered 

Obviously, it should make a difference when we are seeking the coupling loss factor at the L- 

junction if the horizontal plate consists of one or two elements (compare eqn. (9.11) with S 1 

being either the area of one or two plate elements). We may therefore conclude that in order to 

proceed straightforwardly from the two-subsystem case to that with N subsystems, the coupling 

losses at all junctions (connections between subsystems) must be small. This is equivalent to 

the prescription of weak coupling between the different subsystems. Consequently, the 

existence of adjoining subsystems must not markedly affect the properties of the eigenfunctions 

of a subsystem (in a statistical sense). 

At a first sight, the above-mentioned condition may seem rather drastic but comparisons 

made between SEA predictions and experimental results show that the condition of weak


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coupling does not impose serious estimation errors if the complexity of the complete system is 

large. Statistically the latter finding is acceptable since for large complex systems a 

"randomization" of the interaction between subsystems will always take place. 

Π 

1, in 

Π 

2, in 

E 1, tot 

Π21 

E 2, tot 

N 1 

N 2 

Π 

Π 

1, diss 

Π 

2, diss 

Π 

13 

Π 23 

Π 3, in 

E 3, tot 

N 3 

Π N1 

N, in 

Π 

3, diss 

Π N3 

E N, tot 

N N 

Π 

N, diss 

Figure 9.3 Block diagram of an SEA model with several subsystems. After [1]


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Plate 3 

Plate 2 

Plate 1 

Figure 9.4 A three plate-element system. From [4] 

For the specific case of two equal plates with no discontinuity in between, there is also no 

reason not to treat them as a single subsystem. Nevertheless, the condition of weak coupling 

discussed proves the statement that SEA is rather a compilation of methods where basic 

knowledge in structural acoustics is required for judging the applicability of the different 

assumptions made in an analysis. 

On the basis of the weak coupling condition being sufficiently well fulfilled, the 

generalization of the energy balance equation becomes 

in 

tot 

m 

⎧Π 1 

⎫ ⎡n 1 

η 1 

−n 2 

η 12 

L −n N 

η iN 

⎤ ⎧E 1 

⎫ 

in 

⎢ 

tot 

⎥ 

m 

⎪Π 2 ⎪ −n 

⎨ ⎬ = ω ⎢ 1 

η 21 

n 2 

η 2 

M 

⎥ ⎪E 2 ⎪ 

⎪ M⎪ 

⎢ M O M ⎥ 

⎨ ⎬ 

in 

⎢ 

⎩ 

⎪Π N ⎭ 

⎪ 

tot 

⎥ 

⎪ M⎪ 

m 

⎣−n 1 

η N1 

L L −n N 

η NN ⎦ ⎩ 

⎪E N ⎭ 

⎪ 


where η i 

tot 

includes all the types of losses from a subsystem, i.e. 

η i 

tot 

N 

∑ , 

j 

= η ii + η ji 

the out-flow from the i th subsystem.


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The loss factor matrix is symmetric and positive definite ({x} t [A]{x} > 0; every {x}) 

which means that the solution of eqn. (9.15) can be established by means of standard computer 

routines. If in addition some systematics is introduced in the numbering of the subsystems the 

loss factor matrix can be made banded (the elements are clustered around the main diagonal) 

which greatly facilitate the numerical handling. 

In this section it is appropriate with a warning concerning the fact that some confusion 

exists in the literature with respect to the concept of weak coupling. Sometimes, the concept 

weak coupling is used in connection with the assumption 

η ji = E i / M i (9.16) 

where M i is the total mass of the i th subsystem. 

Similarly for a fluid space or subsystem the response estimates are given by the spatial 

average, mean square pressure as 



2 Eiρ i c 

2 

i 

> = 

i 

Vi 


From eqns. (9.16) and (9.17) it is evident that no information concerning the response field 

within a subsystem is obtained. Rather, the distribution of the response among the subsystems 

is revealed. It is also very important to bear in mind the explanations of the name SEA, given in


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section 9.1, when interpreting the results. Thus, the predictions finally obtained through eqns. 

(9.16) and (9.17) are merely statistical estimates with an inevitable variance and not exact 

results from a calculation regarding a specified physical system. With these underlinings in 

mind the use of SEA has proved very valuable in most branches of noise and vibration control. 

9.10 References 

[1] Lyon, R.H. 1975. MIT Press. Statistical energy analysis of dynamical systems. 

[2] Maidanik, G. 1977. Journal of Sound and Vibration, 52, 171- . Some elements in 

statistical energy analysis. 

[3] Fahy, F. 1974. Shock and Vibration Digest. Statistical energy analysis: A critical 

review. 

[4] Plunt, J. 1980. Chalmers University of Technology, Dept. of Engineering Acoustics, 

Report 80-07. Methods for predicting noise levels in ships. 

[5] Cremer, L., Heckl, M. and Petersson, B. 2005. Springer Verlag, Berlin. Structure-borne 

sound.

9. Statistical Energy Analysis (SEA)

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