6.1.1 Wave-Energy Relationships

In order to reconstruct the averaged wave amplitude from the total energy we must know how energy is related to the diffuse wave field amplitude. We keep the derivation here as general as possible. Thus, an arbitrary wave quantity $\mathrm{\Psi }$ was chosen. For the expression of energy density in terms of complex wave amplitudes we introduce the following relationship

  (6.14)

For pressure waves ($\mathrm{\Psi }=p$) this constant would be $B=1/({\rho }_0{c}_0^2)$. In addition a link to the intensity is needed. According to Equation (3.238) this reads as

  (6.15)

because the group velocity defines the velocity of energy flow in contrast to the phase speed that defines the propagation of constant phase.

6.1.2 Diffuse Field Parameter of One-Dimensional Systems

In the context of one-dimensional waves the model of a diffuse reverberant field may seem artificial. Phase and direction are randomly but equally distributed over the phase angle $0$ to $2\pi$ and the available room angle. The latter is just the positive and negative propagation direction for one-dimensional systems so $\mathit{k}\mathrm{\Delta }\mathbf{r}=\pm k\mathrm{\Delta }x$. Thus the diffusivity of the one-dimensional field is given by the random phase and amplitude. The integration over the probability function for the expected value is replaced by a sum.

When we chose one point in the center of the one-dimensional system, the intensity is zero, because the flow of energy is equal in both directions. This changes when we look at the boundaries, for example at $x=0$. In this case we have only one irradiating direction with probability $1/2$:

  (6.16)

Thus, the intensity is half the intensity of a free field wave with similar rms-amplitude. For the calculation of the cross correlation we also use the sum

  (6.17)

6.1.3 Diffuse Field Parameter of Two-Dimensional Systems

The additional dimension creates more freedom in terms of available directions. For the integration we consider only the contributions that are normal to the edge. Choosing the edge parallel to the $x$-axis, the normal is $\mathbf{n}={\mathbf{e}}_y$. The according integration angle is $\phi \in [0,\pi ]$ and the intensity normal to the edge is $I_y=I\sin (\phi )$ (Figure 6.3)

  (6.18)

Similar considerations are performed for the cross correlation and we choose without loss of generality $\mathrm{\Delta }\mathbf{r}=\mathrm{\Delta }r{\mathbf{e}}_y$. With ${\mathit{k}}_i\mathrm{\Delta }\mathbf{r}=\mathit{k}\mathrm{\Delta }r\sin ({\phi }_i)$

  (6.19)

This follows from the antisymmetry of the sine function and the integral representation of the Bessel function. The shape of this cross correlation is the same for location in the center of the diffuse field area because integration over the half circle leads to the same functional shape.

6.1.4 Diffuse Field Parameter of Three-Dimensional Systems

For this case, the normal vector $\mathbf{n}={\mathbf{e}}_z$ of the $xy$-plane is chosen, and the intensity the integration runs from $0$ to $\pi /2$ for $\vartheta$. The normal intensity is $I_z=I\cos (\vartheta )$ and the integrand does not depend on $\phi$

  (6.20)

We select $\mathrm{\Delta }\mathbf{r}=\mathrm{\Delta }r{\mathbf{e}}_z$ and the wave number distance vector product is given by $\mathit{k}\mathrm{\Delta }\mathbf{r}=k\mathrm{\Delta }r\sin (\vartheta )$.

  (6.21)

The global form of the cross correlation is also kept for half-space integration. Two locations at the walls of a diffuse field have the above correlation function. This expression is used in experiments to check the quality of the diffuse field, see e.g. Jacobsen (1979) for details.

6.1.5 Topology Conclusions

With the different dimensions comes different behavior regarding correlation and surface intensity. The relationship between energy density and the normal surface intensity can be described by a proportionality factor ${\beta }_D$

  (6.22)

This factor scales from $\frac{1}{2}$ for one-dimensional systems to $\frac{1}{\pi }$ and $\frac{1}{4}$ for two-dimensional and three-dimensional systems, respectively, as summarized in Table 6.1.

In Figure 6.5 the different cross correlation functions are compared to each other.

6.1.5.1 Dissipation in the Reverberant Field

In the reverberant field there are two processes for sources of damping: first, the damping during wave propagation, for example described by the complex wave number $\mathit{k}=k(1-j\frac{\eta }{2})$, and second, the absorption due to the reflection at an absorbing surface at end sections, edges, or surfaces. In principle, all systems have a volume, a one-dimensional beam or tube has a cross section, and plates have thickness.

According to (1.67) the loss due to wave propagation is given by

  (6.23)

The dissipation due to surface absorption is given by the intensity irradiating all surfaces. The dissipated power due to boundary reflection in the diffuse field is

  (6.24)

$A_s$ is the total and given by

  (6.25)

$A_P$ is the perimeter area or boundary. In order to get an expression that fits to the global damping loss factor, the dissipation at the perimeter is expressed by the total energy of the random field.

  (6.26)

The sum of both dissipations provides the effective damping loss factor

  (6.27)

A random system with certain energy has a time decay $\eta \omega$ so that without any power flow into the system the energy will decay as

  (6.28)

6.1.6 Auto Correlation and Boundary Effects

The autocorrelation of the above given function follows from $\mathrm{\Delta }r=0$. Please note that this is

  (6.29)

At the boundaries of such fields we have two effects. First, for the total sum of energy only half of the space is available. Thus, the mean square value of the amplitude is half of the amplitude if the full angle is available. Second the reflected wave amplitude is $\mathit{R}\hat{\mathrm{\Psi }}$. So, the boundary amplitude of every ensemble wave is

  (6.30)

In case of hard walls and fixed edges this means a doubling of the pressure or stress, leading to four times higher mean square value of the amplitude. This is partly compensated by the fact that only waves from half space can reach the boundary

  (6.31)

The amplitude of each noncorrelated wave doubles for full reflection and ($R=1$), but for the summation of all waves only half of the room, area, or line is available. In other words, the double amplitude is partly compensated by the fact that only half the space is available

  (6.32)

Note, that one coherent wave would give twice the amplitude. The factor of $\sqrt{2}$ instead of two in the reverberant field sometimes leads to confusion.

6.1.7 Sources in the Diffuse Acoustic Field – the Direct Field

Every system requires sources to get power input into the system. The waves that radiate from the sources or boundaries are obviously deterministic, because no reflection that may generate uncertainty has taken place. In the vicinity of the source around a radius of a few wavelengths there is a deterministic wave field with similar shape and behavior as in free field conditions as long as the wave amplitude reaches the diffuse field level. In room acoustics this distance is called the source radius. For correct measurement of the diffuse sound field level the distance between source and microphone must be larger than this radius. In the following examples, the direct field is included, except for the one-dimensional systems. The reason for this is that there is no geometrical attenuation for one-dimensional systems which is the case for two-dimensional ($\mathrm{\Psi }(r)\sim r^{-1/2}$) and three-dimensional systems ($\mathrm{\Psi }(r)\sim r^{-1}$).

6.1.8 Some Comments on the Diffuse Field Approach

The considerations in Sec. 6.1.7 regarding the diffuse field are very stringent. The system must be constructed and uncertain in such a way that the reverberant field assumptions are met. Thus, the properties of the system must allow a large number of reflections with uncertain phase so that there are enough waves with equally distributed angles and random phase.

This is hard to achieve when the system is excited, for example, only at one point. Consequently, the diffuse field approach (and also the statistical energy method that is based on this concept) is of limited precision when local and single excitation occurs. So, when this is the case we should consider using deterministic and numerical methods for systems that are locally excited.

However, when many excitations exist the diffuse wave field can be applied. When the frequency is high enough for geometrical acoustics, this is one further option to be applied to random wave fields but allowing energy distributions along the systems described by random wave fields.

As mentioned at the start of this chapter, one approach is to average in addition over frequency bands. However, with the ensemble concept the properties of the reverberant field can be kept, so the more critical question is how large the variation from the mean value is and under which conditions the mean value fulfills the reverberant field assumptions. Please note, that also frequency averaging does not solve the issues regarding the strict diffuse field assumptions. In the remainder of Chapter 6, we will deal with some examples of random systems in order to explain these phenomena.

6.3.1 Fluid Tubes

Let us imagine that we have an ensemble of tubes with different temperatures and consequently variations in the speed of sound or small imperfections in length. This would lead to propagating waves with random phase for both directions if the imperfections and variations in sound speed lead to differences in the propagation paths exceeding one wavelength.

Those imperfections will cause a diffuse sound field in an ensemble of one-dimensional systems. The wave quantity $\mathrm{\Psi }$ is the acoustic pressure $p$ and the pressure field inside the tube is given by (4.24). Damping is introduced to ensure dissipation.

6.3.1.1 Energy

The reverberant pressure field is characterized be the mean square amplitude that is related to energy density of the wave and the total energy of the system. As the wave quantity is pressure, the constant in Equation (6.16) is $B=\frac{1}{{\rho }_0{c}_0^2}$.

Thus, the random state is defined by the energy density $e$, the energy per length $E^{\prime }$, or the total energy $E$ with a given length $L$

  (6.37)

In the diffuse field model the energy is equally distributed over the length. The irradiated intensity at both tube end cross-sections follows from (6.16)

  (6.38)

6.3.1.2 Power Input to the Reverberant Field

The power input into the field is defined by Equation (4.20). For large values $k\eta L\gg 1$ the radiation impedance of the system becomes a smooth function that approaches the impedance of a half infinite tube $z_0$, and the resonance peaks disappear.

We would now like to present an instructive explanation for this effect using the modal response.

We consider the half bandwidth of one modal peak that is given by $\mathrm{\Delta }{\omega }_{3dB}=\omega \eta$. One can expect a smooth frequency response when the distance between the modal frequencies $\mathrm{\Delta }{\omega }_n$ is equal or smaller than the half power bandwidth because the modes are overlapping in this case. This relationship is quantified by the factor

  (6.39)

So, we have two parameters to control the smoothness of the response (independent from random effects): The damping loss $\eta$ and the modal density $n(\omega )$. When the model overlap factor is larger than 1 we can estimate the power input into the system by Equation (4)

This effect is depicted in Figure 6.6. When the modal overlap factor is less than one the gap between the modes is large enough so that single peaks can be identified; when $M$ is approaching 1 there is still some waviness. For $M>1$ the slope of the power input is nearly constant. Note that there is no ensemble averaging involved – the only condition is that all modes are equally distributed over the frequency band.

6.3.1.3 Dissipation

Surface dissipation occurs at the interaction with the tube end sections (neglecting the tube wall) with the total absorption area

  (6.40)

With the tube volume $V=AL$ and ${\beta }_D=\frac{1}{2}$ the combined dissipated power is

  (6.41)

providing the effective damping loss factor.

6.3.1.4 Power Balance

In a stable configuration the input power equals the dissipated power. This is the birth of our very first and simple SEA model

  (6.42)

Looking at the above equation we see that the input power depends on the free field impedance of the pipe on one side; on the other side, the loss of energy is proportional to energy and frequency given by the factor $\eta$. With Equation (6.37), the rms pressure is then given by:

  (6.43)

6.3.1.5 Energy Ray Tracing

For increasing frequencies, the wave attenuation (2.58) $\alpha =\eta k/2$ is getting larger, so the wave propagation is highly damped. Consequently, there are not enough reflections available to guarantee reverberance and an equal distribution of wave energy in the field. A description of random systems by diffuse wave fields is no longer valid, and methods that account for the damping along the propagation and system dimension are required.

The frequency limit of this effect can be estimated when we say that the energy damping along the system extension $L$ must not exceed $\frac{1}{e}$, hence

  (6.44)

Though SEA is often mentioned as a high frequency method, this important criteria for the application of SEA is neglected. If this criteria is not fulfilled, the error in the propagation of energy is not caused by an erroneous coupling of systems but the nonuniform energy distribution in the system. This issue can be overcome by geometrical acoustics. For very small wavelengths the wave propagation and scattering can be described by rays (Pierce, 1991; Kuttruff, 2014). Each wave is traced similar to the ray model in optics, and the wave field is considered as the sum of these rays. For a one-dimensional system, this can be done analytically.

We consider each propagation in the positive and the negative direction as one ray. We still assume a vibrating piston with velocity ${{\mathit{v}}_x}_1$ on port 1. A ray in the positive direction will radiate into the tube the mean squared pressure as follows

  (6.45)

At the end of the tube, the wave is reflected with the reflection coefficient ${\mathit{R}}_1$, so the reflected mean square pressure is

  (6.46)

with $R_1^2={\mathit{R}}_1{\mathit{R}}_1^{*}$; this wave is reflected at port 1 with the coefficient ${\mathit{R}}_0$ and so on and on:

  (6.47)

The last term in parenthesis is an infinite geometrical series and we get for the slope of the root mean square pressure

  (6.48)

6.3.1.6 Load at the Boundaries due to the Reverberant Field

The reverberant field will have an impact on the boundaries of the system. For deterministic systems the answer is given by Equation (4.24) or the pressure distribution itself. The field pressure at the boundaries creates a reaction force or pressure. The question now is what is the reaction force of a diffuse field that is only described by its energy or rms pressure average in the system?

The pressure in the diffuse field with energy $E$ is:

  (6.49)

The ratio $L/c_0=\pi n(\omega )$ can be expressed by the modal density providing

  (6.50)

According to Equation (6.31) the mean square pressure at rigid surface doubles

  (6.51)

Finally, the auto spectrum of the blocked force that is generated by the diffuse field is given by

  (6.52)

Here $\mathit{Z}$ is the mechanical free field impedance of the one-dimensional tube.

We have compiled all the information required to describe a random system by a diffuse field. This includes the interaction of diffuse field at the boundaries with external systems:

1. The input impedance of the reverberant field determining the power input.
2. The load generated by the reverberant field at the boundary.

6.3.1.7 Monte Carlo Experiment

The applies uniform random variation to parameters of a system that is solved numerically. By this method a large ensemble of systems and its response can be simulated. The pressure field for a deterministic system is known from Equation (4.24) and therefore an excellent case for fast calculation of a large set of results that can be averaged. In order to work out the limits of the diffuse reverberant field we deal with two realizations of damping.

1. Damping is dominated by dissipation in the wave propagation given by the complex wave number $\mathit{k}=k(1-j\frac{\eta }{2})$ with $\eta =0.01$ and rigid ends.
2. Damping is dominated by absorption at the boundary by applying an impedance $\mathit{z}=19.1z_0$ leading to a reflection factor of $R=0.9$. Damping loss for propagation is $\eta =0.001$.

On the LHS of Figure 6.8 it can be seen that the system damping loss for case 2 starts with high values but is decreasing. The damping loss is defined per cycle, and for higher frequencies the wave passes many cycles before the reflection dissipates energy from the system. Thus, this case reaches a modal overlap of $M=1$ at $kL>300$ and case 2 is not in the investigated frequency range.

Those values are selected to investigate the statistical aspect and to reduce the effect of high modal overlap that would smooth even the deterministic response.

For ensemble realization, we randomize the speed of sound by $\mathrm{\Delta }{c}_0=\pm 2\%c_0$ and $\mathrm{\Delta }L=\pm 1\%L$. We generate $N=1000$ representations of this ensemble, with pressure $p_n(x)$ and calculate the following ensemble averages:

  (6.53)

  (6.54)

The first average shall show that the mean value vanishes in an ensemble average; the rms value should verify the reverberant field assumption for high frequencies. The spatial average over length will also be presented in the following numerical experiments to check if a single value can be representative for the total system.

  (6.55)

6.3.1.8 Input Power and Impedance

From section 4.1.1 and the modal overlap discussions, it was shown that the radiation impedance approaches the free field impedance. This effect should occur even earlier in frequency due to incoherent reflection according to the diffuse field concept. In Figures 6.9 and 6.10 this effect is clearly visible for both cases. The light grey background represents the ensemble of different realizations.

For case 1 with dominating wave absorption the ensemble mean reaches very early ($kL\approx 50$) the free field limit with higher peaks in the ensemble variations. In case 2 the ensemble peaks are still quite sharp, because of the low modal overlap even for higher frequencies.

The same effect can be seen in Figure 6.10. The second case’s mean value oscillates more, but the ensemble variation is smaller.

Doing the same investigation with the input power, this is further confirmed. In Figure 6.11, the free field limit is reached much earlier than for the deterministic case shown.

6.3.1.9 Random Fields

We investigate the rms and mean pressure along the tube. The results are shown in Figures 6.12 and 6.13 for cases 1 and 2, respectively. Several modal frequencies and overlap factors are shown to investigate the validity and limits of the random field assumptions in both cases. For the first frequency, the main impact of the uncertainty is a large variation in amplitude, but each wave form remains stable in phase. So, mean and rms averages result in the similar pressure distribution along the pipe. This is still true to frequencies around the 10th mode as shown in the upper right axes. When we compare the rms pressure averaged over length to the result of the SEA model the agreement is quite good.

Reaching higher frequencies as shown in the lower left axis, both rms values don’t coincide, because damping along the tube prevents a constant pressure along the pipe. The mean pressure becomes zero for $x=L$ but not for $x=0$ near the vibrating piston side. Here, the coherence of the near field keeps the constructive interference near the source; the wave field has not yet gathered enough uncertainty to establish a diffuse sound field. This is the direct field component which can be added to the reverberant background to improve the precision of the sound field description.

For the high frequency results in the lower right axis it can be seen that rms pressure shows a strong decay along $x$. The wave propagation is strongly affected by damping, and the diffuse field assumption is not met any more. It is the assumption of equivalent amplitude for both directions that is explicitly violated. The system is dynamically complex, but the damping is too high to establish a reverberant field. In this case ray-tracing or geometrical acoustics shall be applied, which agrees very well with the average result.

However, there are many options to achieve a reverberant field. Our case is very stringent, because there is only one source at one end of a one-dimensional field. If there are more sources, a reverberant field can be realized even for damped waves.

So, even with such a simple system the effects of deterministic and random regimes are visible. At low (deterministic) frequencies, there is the typical wave dynamics with maxima and minima regarding the related wave shape. The wave motion, the local minima and maxima, can be clearly determined. In the random regime the wave behavior is smeared out. In the statistical energy description, neglecting the phase is sufficient to describe the dynamics of a complex system. For higher frequencies when damping along the system dimensions becomes important, the damping must be considered, for example, by energy ray tracing methods. This high frequency limit is often neglected in the literature. Lyon and DeJong (1995) and Le Bot (2015) derived the same equation as (6.44) but expressing an upper limit of system size

  (6.56)

For the power input the local dynamics of the area near the excitation must be known precisely, meaning that the free field impedance is required. Thus, even when we later deal with fully random models the dynamic properties of the system in the vicinity of junctions must be precisely known and is thus considered as deterministic.

6.4.1 Plates

We apply a point force excitation somewhere at the plate and focus on bending waves, because the wave speed for the other propagation modes is too high to create large modal overlap at audible frequencies even for large plates. Similar to the acoustic tube, the high modal overlap causes the mechanical impedance to reach the free field radiation value (3.226) of the infinite plate. The modal approximation is used to calculate the system response for the Monte Carlo analysis. Plates are two-dimensional systems; so, all equations for the two-dimensional wave field apply, and the field quantity is the displacement $\mathit{w}$ in the $z$-direction.

6.4.1.1 Energy

In section 3.7.6.1, the energy per area is given by

  (3.227)

providing energy density and energy by

  (6.57)

Thus, the constant $B$ is

  (6.58)

and for the two-dimensional expression of diffuse field intensity at the edges (6.18) we get

  (6.59)

Please note that the intensity expression is per edge area – for intensity per length this expression must be multiplied with the plate thickness $h$. In addition, the group velocity is different to the phase velocity and depends on the frequency

  (6.60)

6.4.1.2 Power Input to the Reverberant Field

For systems and frequencies of high modal overlap the mechanical impedance converges to the free field impedance of an infinite plate. Thus, the input power can be calculated using the assumption (3.226)

  (6.61)

However, the free field assumption is not fulfilled for low frequencies so we will use Equation (5.42) in addition to correct the power input.

6.4.1.3 Dissipation and Power Balance in the Reverberant Field

The same dissipation assumptions as for one-dimensional systems apply and the inert damping due to wave propagation is the same

For two-dimensional systems, the edge intensity is determined from our considerations in section 6.1.1, the volume is $V=Ah$, and the dimensional constant is ${\beta }_D=1/\pi$. The power balance provides the energy of the subsystem with given input power

  (6.62)

with $E=\frac{m^{″}A{\omega }^2{\hat{w}}^2}{2}$ the average displacement result is

  (6.63)

6.4.1.4 Direct Field Correction

For the direct field part in the vicinity of the point force, we add the results from Equation (3.231)

to the reverberant field.

6.4.2 Monte Carlo Simulation

The dimensions of the rectangular plate test case are $L_x=1.0$ m, $L_y=1.8$ m and $h=2$ mm. The ensemble is realized by varying the thickness $h$ by $\pm 2\%$ and the area mass $m^{″}$ by $\pm =2\%$, and the system average response is presented by the rms displacement averaged over the plate area

  (6.64)

6.4.2.1 Input Power and Impedance

In Figure 6.14 the mean value and the ensemble impedances are shown together with the modal overlap. Due to the properties of the plate, the modal density is constant over frequency (5.35); the consequence for the modal overlap is, that it increases linearly with frequency

  (6.65)

The modal overlap is above 1 around a frequency of $\omega >1000$ s⁻¹. Even though there is an ensemble average the input impedance of the free field is hardly approached. There is a remaining variability even at high frequencies. The reason for this is that the point excitation does not excite all modes equivalently. This is different to the tube case where we excited at the $x=0$ and thus made sure all the modes are excited. This is not the case for the plate. A mode that has zero amplitude $\hat{w}=0$ at the point of excitation will not be excited and will not contribute to the statistics of the assumed random field. This is the reason why we also use the real input power calculated from (5.42) to correct the power input for the final random field comparison. It is worth mentioning that commercial software codes implement the point force in the same way leading to similar error. Thus, if the response of plate structures to point forces is targeted, the input power should be derived by alternative methods, e.g. experiments or modal methods as suggested above.

6.4.2.2 Random Displacement Field

When we do the random response analysis based on the power balance according to Equation (6.63) and the Monte Carlo analysis with the various random quantities, we get the result in Figures 6.15 and 6.17 for $\omega =2000$ s⁻¹ . The rms wave field shows still some variation even in the ensemble average. However, when we compare the displacement averaged over the surface, the order of magnitude is met, and the corrected power input leads to an improvement in the prediction quality.

The assumptions of the diffuse field are better fulfilled at frequencies around $\omega =10000$ s⁻¹ as shown in Figures 6.16 and 6.18. The oscillation of the displacement rms still exists. But the SEA result and the surface average coincide very well. Here, the SEA random result plus the direct field component shows a much better agreement with the Monte Carlo result.

6.5.1 Energy and Intensity

The link between pressure field and system energy is given by

  (6.66)

The intensity at the boundary is

  (6.67)

The above expression is known as the irradiated intensity in a twin chamber arrangement where the average pressure is measured in the room. The power irradiating a specimen of surface $A$ at the room walls is

  (6.68)

6.5.2 Power Input to the Reverberant Field

From section 4.2 and Equations (2.95) and (2.96) the radiated power is determined by:

The power input based on modal room theory is given by Equation (4.89)

  (6.69)

6.5.3 Dissipation

The dissipation in room cavities depends on propagation and boundary absorption (6.41). For the surface absorption we must consider the angular dependency of the absorption according to Equation (2.103) related to the normal component of the intensity.

  (6.70)

with

  (6.71)

Usually, the angular integration is performed numerically. Assuming that the room is surrounded by surfaces $A_i$ and with absorption ${\alpha }_{s,i}$ the total $A_s$ follows from Equation (6.25), and the final dissipation formula (6.27) reads in case of cavities as

  (6.72)

6.5.4 Power Balance

The pressure in the room follows from the balance of energy flow ${\mathrm{\Pi }}_{\text{in}}={\mathrm{\Pi }}_{\text{diss}}$

  (6.73)

Similar to the considerations for the plate, the direct pressure field amplitude of the point source (2.91) is added to the random field component.

  (6.74)

6.5.5 Monte Carlo Simulation

The test case applies the modal room acoustics from section 4.2 considering two different cases.

room 1 Damping is dominated by dissipation in the wave propagation with ${\eta }_{\text{field}}=0.01$ and rigid wall surfaces.

room 2 Damping is dominated by absorption at the boundary by applying an impedance $\mathit{z}=10z_0$ at floor $z=0$ and ceiling $z=L_z$ leading to a diffuse field absorption coefficient ${\alpha }_s=0.165$ and an absorption area $A_s=1.5$ m². Wave damping is ${\eta }_{\text{field}}=0.001$.

The ensemble variation of the room lengths $L_{x,y,z}$ is $\pm 2\%$ and for the speed of sound $\pm 1\%$.

6.5.5.1 Room Absorption

In Equation (4.79), a complex wavenumber was introduced to determine the impact of absorbing boundaries on the modal coefficients. For nearly rigid walls, the off diagonal terms can be neglected and we get

  (6.74)

Thus, we select $\mathit{Z}=20z_0$ and get specifically for the floor area the following expression

  (6.75)

When we evaluate expression (6.70) for the floor area $A=L_x{L}_y$ we get for the average absorption coefficient for the floor ${\langle {\alpha }_s}_E=15.6\%$ leading to a total absorption area of $A_s=1.5$ m².

6.5.5.2 Input Power and Impedance

Studying the ensemble average radiation impedance, the results from the sections before are further verified. The modal overlap becomes larger than one (Figure 6.19) above $\omega =2000$ s⁻¹.

Both real ensemble impedances (Figures 6.20 and 6.21) show resonance variation at low frequencies but approach the free field value. The modal peaks of room 2 are lower at low frequencies because of the larger modal overlap from the beginning, whereas the variations between each ensemble representation are higher for room 2 because of the lower damping contributions from the fluid.

6.5.5.3 Random Pressure Field – Room 1

The pressure field at lines passing through the source is shown in the Figures 6.22 and 6.23 for $\omega =5000$ s⁻¹. The need for near field consideration is further confirmed, but with this simple direct field consideration the random field pressure corresponds surprisingly well to the Monte Carlo result. At the boundaries the pressure is roughly $\sqrt{2}$ higher than the field pressure as predicted by Equation (6.31). Due to the high expense of the calculation time, only 100 averages were performed leading to limited statistics.

6.5.5.4 Random Pressure Field – Room2

At 5000 s⁻¹, the second room with damping via absorption shows a similar agreement between random and the Monte Carlo results (Figures 6.24 and 6.25). The suitability of three-dimensional systems for reverberant field modelling is further verified.

6.7.1 Diffuse Field Energy

In the diffuse field model with equal power from all directions, the homogeneous energy distribution is evident. For systems described by modes, it is not that obvious that modes (with a specific uncertainty) lead to an equally distributed mean square amplitude in the wave field.

6.7.2 Free Field Power Input

In the discussions of the modal overlap factor of section 6.3.1.2 we have established the criteria that allows treating the point impedances of resonant and deterministic systems as free field impedances. In that case the damping is so high that no reflection is received at the point of excitation (independent from the fact of whether the reflection is coherent or not).

We will now show that the free field assumption, which is easy to explain in the diffuse wave field model, can also be proven when we assume a Gaussian orthogonal ensemble of modes (see details in Langley (2016)). In this approach, the uncertainty in an ensemble of systems leads to small variations in the modal frequencies as shown in Figure 6.26 for one ensemble representation. When dealing with structural systems, the input power into the system excited by a harmonic source at a specific location and frequency $\omega$ is according to (1.49)

  (6.85)

The real part is called, the imaginary part. For the derivation of the conductance $G$ of modal systems we take Equation (5.42) as a typical example.

  (6.86)

In principle, every closed dynamic system can be described by the above equation. Independent from the fact of whether a discrete or homogeneous aproach is selected. In order to assess the real and imaginary parts of the mobility we rationalize the denominator.

  (6.87)

and the conductance reads as

  (6.88)

with $a_n=\frac{{\mathrm{\Phi }}_n(\mathbf{x}){\mathrm{\Phi }}_n^{*}({\mathbf{x}}_0)}{m^{″}\eta \omega }$, and $g_n(\omega )$ that can be further simplified assuming that the major contribution is for frequencies near the resonances $\omega \approx {\omega }_n$.

  (6.89)

We stay first with one mode $n$ and investigate the ensemble average of $g_n$ of modal frequency ${\omega }_n$. When the modal frequencies are equally distributed over the frequency interval $\mathrm{\Delta }\omega$ the probability density of ${\omega }_n$ in this interval is $p({\omega }_n)=1/\mathrm{\Delta }\omega$ as shown in Figure 6.27. Thus, the expected value of $g_n(\omega )$ is

  (6.90)

This interval $\mathrm{\Delta }\omega$ must be larger than the modal bandwidth $\eta \omega$ for getting a smooth average, or in other words, the variation in modal frequency must be so large that the peak in frequency is shifted such that every value of $g_n(\omega )$ is possible.

The sum over all $g_n(\omega )$ is approximated by the number of modes $N$ times ${\langle g_n(\omega )}_E$. $N$ is simply the product of the frequency interval $\mathrm{\Delta }\omega$ and the modal density. Thus, the expected value of (6.88) is

  (6.91)

6.7.2.1 Point Conductance

For the point conductance, excitation and response points are the same $\mathbf{x}={\mathbf{x}}_0$, and the ensemble average over the sum of all modes is given as

  (6.92)

The mode shape function from (5.33) is

 

Similar to the diffuse wave field, we assume that the mode shapes are not correlated. So, arguments in the sine functions in an ensemble of mode shapes have uncorrelated phase, and we take the expected value of each squared sine function $E[{\sin }^2(\phi )]=1/2$. Thus,

  (6.93)

and finally

  (6.94)

When we use the modal density for a flat plate (5.35), the result is

  (6.95)

which corresponds exactly to the point impedance of infinite plates (3.226). We can conclude that for an ensemble of modal systems with modal frequency uncertainty with stable modal density and frequency variation $\mathrm{\Delta }\omega \gg \eta \omega$, the point conductance and also the impedance are equal to the free field expression.

6.8.1 Power Balance in Diffuse Fields

In the introduction to Section 6.8, the model of the diffuse acoustic field was applied to describe the state of a dynamically complex system, as shown in Figure 6.28. For the single system, the equation is simple, and the energy of the system that determines the amplitude of pressure, displacement, or other wave amplitudes is given by

  (6.96)

Engineering systems consist of several components and systems. Therefore there are usually many diffuse acoustic fields (encapsulated in specific systems) that are exchanging energy. This transmitted energy is related to the system surface intensity as given by Equations (6.16), (6.18), and (6.20) as far as the connecting area and the transmission of radiated waves into the other connected subsystems. The details of this coupling will be discussed in Chapter 7. In the following we use the upper index $(n)$ for the $n$th subsystem for quantities that are already using a descriptive index as for example ${\mathrm{\Pi }}_{\text{in}}$, ${\mathrm{\Pi }}_{\text{diss}}$.

In Figure 6.29, two connected random systems are depicted. In order to cope with the damping loss ${\eta }_m$, we use a similar expression for the loss due to transmission into the connected subsystem. Thus, the power transmitted by subsystem $(1)$ into system $(2)$ reads as

  (6.97)

The expression for the other direction is

  (6.98)

The coefficient ${\eta }_{mn}$ is called in accordance with the definition of the loss factor. The total power balance for subsystem (1) is

  (6.99)

and for subsystem (2)

  (6.100)

When we consider a number $N$ of subsystems the power balance of the $m$th subsystem would be given by

  (6.101)

and this can be written in full matrix form as

  (6.102)

or in short form

  (6.103)

$\left[\begin{array}{c}L\end{array}\right]$ is called the power-flow matrix or the SEA matrix.

6.8.2 Reciprocity Relationships

Reciprocity is a general principle that relates the conjugate input variables of a two port system. Applied to the mechanical systems of Chapter 5, that means that the velocity at position 1 generated by a force at position 2 is similar to a velocity at position 2 generated by the same force at position 1. This general principle applies to all systems that are composed of linear, passive, and bilateral elements.

The same principle must be fulfilled for random systems. Thus, if we apply a force at system 1, the velocity response at system 2 at one point and therefore also at the average must be similar to the reciprocal configuration – that is, the force excitation at system 2 and the velocity response at 1.

  (6.104)

Now imagine a two system SEA configuration as given by Equation (6.99). When the force is exciting exclusively system 1 (${\mathrm{\Pi }}_2=0$), the solution of this system with regards to the energy of system 2 is

  (6.105)

and a similar solution is found for the excitation at subsystem 2

  (6.106)

The reciprocity relationship must now be converted into energy and power expressions. The energy of the subsystem is given by the total mass $m_i$ and the average velocity of the subsystem

  (6.107)

and the power input is determined by the conductance

  (6.108)

When expressing power and energy by the above expressions and squaring the reciprocity relationship (6.104) we get

  (6.109)

This leads to an expression that relates the SEA matrix coefficients using conductance and mass quantities.

  (6.110)

In section 6.7.2 a relationship for the input conductance that links the input power created by force excitation and the modal density was found

  (6.111)

leading to

  (6.112)

This is the reciprocity relationship of SEA and it leads to a symmetric form of the equation when dividing each energy by the modal density of the system. When we define the double index ${\eta }_{nn}={\eta }_n$ for the damping loss, we may write

  (6.113)

This matrix is symmetric and better suited for computationally efficient solvers. In addition, only half of the coupling coefficients must be determined. We denote this symmetric matrix with $\left[\begin{array}{c}{L}^{\prime }\end{array}\right]$

  (6.114)

6.8.3 Fluid Analogy

The principle of energy per modal density motivates the fluid analogy of SEA shown in Figure 6.30. The amount of water corresponds to the total energy $E_m$ in each subsystem, the width to the modal density (linearly dependent on the number of modes in band), and the height of the fluid in the tank is the energy per modal density. The power input and losses are the amount of fluid per time that are entering or leaving the systems. The exchange of fluid depends on the energy difference in the systems.

6.8.4 Power Input

The input power is determined from the source properties and the convenient property describing the reaction to this excitation. When we studied the different random systems we showed that in the ensemble average the dynamics of the system is equal to the free field radiation dynamics of the involved wave physics. In other words: sources connected to a random system get the dynamic reaction of the free field. This fact is quite obvious in the model construct of the diffuse wave field model, because it is one main assumption of this model that there is no coherent reflection from the boundaries of the diffuse wave field system. But, even in modal space this assumption can be proven given the right statistics and uncertainty for the involved modes. When damping is high enough, this fact can even by shown without any ensemble averaging. In this case, the reflecting waves are simply damped out.

Thus, the important message is that random systems have the same dynamic characteristics as the according free field. Assuming a surface impedance ${\mathit{Z}}_{\text{rnd}}$ of a random system, the ensemble average of the free field radiation impedance is supposed to be

  (6.115)

6.8.5 Engineering Units

The random description of systems involves the energy of systems but not directly the required quantity. In general we must separate between the potential and the kinetic energy of the waves, but as the model of the diffuse field is linked to the propagation of plane waves, we can use the statements from the plane wave energy descriptions: the potential and kinetic energy is equally distributed, and we chose the typical wave quantity usually taken to describe the system.

For fluid systems such as pipes and cavities, this quantity is the acoustic pressure (6.37):

  (6.116)

and for plates it is the displacement (6.57):

  (6.117)

For practical reasons and better scaling over the frequency range, the results are often expressed as velocity or acceleration

  (6.118)

The same can be done for one-dimensional system such as beams and pipes. For pipes the first equation applies; but, concerning beams and rods, we will not enter deeper into the details of random modelling, because the high bending stiffness in engineering systems (this is what their task is in typical constructions – to be stiff and to carry loads) doesn’t make random modelling of such systems realistic. In most practical cases, those systems must be considered as deterministic.

In the introduction to Chapter 6, we mentioned the ratio between wavelength and the dimensions of the subsystems as possible criteria to seperate between random and deterministic systems. Either the modal overlap $M=\eta \omega n(\omega )$ must be greater than one, or the ensemble variation in modal frequency must be large enough to make the modal responses overlap to allow for a random description of the system. In all cases the major criteria is the modal density in combination with the damping as a factor in the first case or the frequency variation in the second case. To establish the link between the Helmholtz number $He=\omega L/c_0$, it is helpful to present the modal density in terms of the. Usually $\lambda$ is linked to the phase velocity, but we set $c_{\text{gr}}=c_0$ to allow for simple expressions that are used here as fuzzy criteria to separate between random or deterministic systems.

  (6.119)

  (6.120)

  (6.121)

For the different exponents of the Helmholtz number, we assumed that $A=L^2$ and $V=L^3$. With this view it becomes clear why two- and especially three-dimensional systems become random much earlier in frequencies than one-dimensional systems.

6.8.6 Multiple Wave Fields

In plates, there occur different wave types: the longitudinal, the shear, and the bending wave. In beams, there are even more types of wave propagation that are not treated in detail here. Those waves have different wavelengths, so there can be simultaneous waves that fulfill the requirements for a reverberant field and that do not. From the energy perspective, we may use a combination of wave types to describe a random system. It is good practice to consider the one wave type field as one SEA subsystem and not all wave fields in one system. We will see later that the calculation of the coupling loss factors will motivate a particular wave field selection. For fluids, the situation is simple, because there is only one mode of wave propagation.