In room acoustics, the acoustic energy density is the solution of a diffusion equation to predict the sound pressure level in rooms. The assumption is that the variations of acoustic energy density and energy flow remain small along one mean-free path (λ = 4V/S), while the computational resource can easily run out with a large number of elements solving the Helmholtz wave equation.

To activate the acoustic energy density solver, specify the degree-of-freedom label ENKE for the acoustic element type by setting KEYOPT(2) = 4. The solution of a diffusion equation supports steady-state, harmonic, and transient analyses in room acoustics. In a harmonic analysis, frequency-dependent materials can be defined, and only the stationary problem is solved at the frequencies with specified materials. The element size must be less than the mean-free path of the room.

The mixed frequency-dependent boundary conditions are applied to the walls of rooms having defined absorption coefficients (SF,,ATTN,α). No special treatment is required to solve the acoustic energy density in two coupled rooms connected by an open aperture. When the two rooms are coupled through a partition wall, the acoustic energy transfers from one room to another. The energy exchanges are described by the energy balance with the absorption coefficients (α) of wall surfaces and the transmission loss (R) of the partition wall.

For a partition wall, one solid hexagon or prism element along the thickness of the wall must be used. The absorption coefficients on the coupled areas and the transmission loss of the wall should be defined on the two parallel faces of the coupled wall element (SF,,ATTN,α,R). Tetrahedral and pyramid elements are not supported in the coupled wall model.

During its propagation, the sound energy is scattered and absorbed by the objects (furniture) in the room, which can be modelled by their density n_f (that is, the number of scattering objects per unit volume), their average scattering cross-section Q_f, and their absorption coefficient α_f. The mean free path of the scattering objects is given by:

If there are N scattering objects with surface area S_i for each scatter in room V, the average scattering cross section and absorption coefficient may be expressed as:

Frequency-dependent material properties (empty room diffusion coefficient, coefficient of atmospheric attenuation, furniture diffusion coefficient, and furniture absorption coefficient) are defined by the TB,AFDM,,,,ROOM command (see Diffusion Properties for Room Acoustics in the Material Reference). The mean-free path in a room with scatters is characterized by the mean-free path λ_e of the empty room and the mean-free path λ_f of the scatters:

The omnidirectional radiated sound power source (BF,,MASS) is used to predict the sound pressure level in the rooms. A time- or frequency-dependent volumetric, surface, line, or point power source can be defined. In a transient analysis, the initial condition of the acoustic energy density is defined by the IC,,ENKE command. Note that a non-physical solution with negative acoustic energy density may be obtained in a transient analysis if the initial condition is applied and the gradient of the acoustic energy density varies rapidly. Use the power source as an alternative. The reverberant time of a room is evaluated by monitoring the decay of the sound pressure level at specified locations.

In a transient analysis the damping matrix is diagonalized by default to obtain a more stable solution, but some accuracy may be lost. To deactivate the diagonalization of the damping matrix, set KEYOPT(1) = 3 for the acoustic element type.

To view nodal energy density (ENKE), element energy density flux (PG), and sound pressure level (SPL), issue the following commands:

...
! Define the room dimensions
lw=5
vol=lw*lw*lw               ! volume of the room
surf=6*lw*lw               ! total wall area
wall=0.01*lw
ls=0.1*lw                  ! dimension of furniture
qf=6*ls/(4*vol)            ! average scattering cross section 
nf=1/vol                   ! the number of scattering objects per unit volume
! Material parameters
c0 = 343.99                ! speed of sound
rho = 1.204                ! mass density
TL=30                      ! transmission loss of coupled wall (dB)
alpha=0.1                  ! absorption coefficient of wall
mfp=4*vol/surf             ! mean-free path of empty room
mfpf=1/(nf*qf)             ! mean-free path of scatters
roomD=mfp*c0/3.            ! diffusion coefficient of empty room
attn=1.e-5                 ! attenuation coefficient of atmosphere
fitD=mfpf*c0/3.            ! diffusion coefficient of furniture
fitattn=0.01               ! absorption coefficient of furniture
roomDp=roomD/(1.+attn*mfp) ! modified diffusion coefficient of empty room
!
! Define materials
mp,dens,1,rho
mp,sonc,1,c0
tb,afdm,1,,,room           ! diffusion properties for room acoustics
tbdata,1,roomD,attn,fitD,fitattn
...
! Define wall absorption 
nsel,s,loc,x,a(1)
...
sf,all,attn,alpha          ! absorption coefficient
...
! Define coupled wall
cmsel,s,ewall              ! select coupled wall elements
nsle,s,all
nsel,s,loc,z,lw            ! nodes on one side of coupled wall
sf,all,attn,alpha,TL       ! define absorption coefficient and TL
nsel,s,loc,z,lw+wall       ! nodes on another side of coupled wall
sf,all,attn,0.1,30         ! define absorption coefficient and TL
...
! Define power source at a location
...
bf,all,mass,1.e-2          ! power source is 100 dB (10-2 W)
...

For a complete room acoustic example, see Example: Sound Transmission Between Coupled Rooms Through a Partition Wall.

For more information, see Room Acoustics in the Theory Reference.