Schroeder Diffuser in 2D

Sound diffusers are objects or surfaces that are designed to reflect incident sound energy evenly across angles. They are widely used in room acoustics as a way to influence the spatial distribution of the sound field without necessarily attenuating it. One common type of diffuser is the Schroeder diffuser, also called well-based diffuser, which consists of a series of wells of different depths. The depths are determined from a mathematical sequence, such as quadratic residue or primitive root. The principle behind this type of diffuser is that each well will re-emit the incident wave with a different phase shift, causing interferences between the waves emitted by the different wells. The mathematical sequence used to determine the depths of the wells will then dictate the interference pattern and, hence, the polar response of the diffuser.

This model demonstrates how to calculate the scattering coefficient with the Pressure Acoustics, Frequency Domain interface. This coefficient is one of the main inputs for expressing boundary conditions in typical room acoustic simulations. Its measurement procedure can be complicated to set up. Therefore. it is more efficient to determine the data numerically. In addition, the effect of periodicity is investigated by studying the responses from different arrangements of the same diffuser.

Model Definition

The model represents a primitive root diffuser, a type of Schroeder diffuser which aims at eliminating the specular component in the reflected energy polar response (see Ref. 1). The model at hand is based on the primitive root sequence with 7 as odd prime, resulting in a diffuser with 6 wells. The study is carried out in 2D in order to considerably reduce the number of degrees of freedom in the simulation. Three cases are taken into account as shown in Figure 1:

•

Asingle diffuser flushed in an infinite baffle

•

An arrangement of 5 diffusers adjacent to each other flushed in an infinite baffle

•

An infinite arrangement of diffusers (a single unit cell is shown)

In the first two cases, a semicircle with radius r0 = 10 m is added to act as an integration line in the far field. The diffusers are surrounded by an air domain terminated by a , and the sound field outside of the calculation domain is obtained from . In the case of the infinite arrangement, is used to virtually extend the domain in both directions along the x-axis.

A plane wave with incidence angle θ0 from the y-axis is defined as incoming on the diffuser arrangements. When present, the specular reflection from the infinite baffle is also included in the ; this allows the solver to only compute the effect of the diffuser relative to the ideal sound field from a single infinite baffle. The incidence angle is varied in an from θ0 = −80° to θ0 = 80° with an increment of 10°, resulting in 17 incidence angles. Octave bands from 125 Hz to 4000 Hz are taken into account with 6 frequencies per band by setting up a on the band number parameter n.

Figure 1: Geometries of the diffuser arrangements. Top left: single diffuser; Top right: 5-unit arrangement; Bottom: infinite arrangement.

Scattering coefficient

The scattering coefficient s is defined as the power ratio (see Ref. 1)

(1)

with Pspec the power reflected in the specular direction and Ptot the total reflected power. In this model, power is calculated by evaluating sound intensity on a semicircle Ω with radius r0 located in the far field. After defining the reflected sound pressure pr, the total power is obtained from the line integral (see Ref. 2, 3)

(2)

This quantity can be evaluated directly with . For the specular power, sound intensity is integrated over a fraction of the semicircle centered around the specular direction θs. With ε an angular increment, the integration arc is defined as Γ:

. This leads to

(3)

(4)

When ε tends to zero, the integral becomes

(5)

With δ the delta function. From convolving the absolute squared pressure with the delta function, it follows that the power reflected in the specular direction can be evaluated as

(6)

Recalling Equation 1, the scattering coefficient can then be expressed as

(7)

For the infinite arrangement, it is not possible to define a semicircle that would encompass the whole geometry. The scattering coefficient is therefore based on its intensity definition

(8)

with Ispec the reflected intensity magnitude in the specular direction and Itot the total reflected intensity magnitude. The reflected intensity Ir can be obtained in the model by checking the box . The specular magnitude can then be calculated by taking the dot product with the specular direction unit vector es, and the total magnitude is a straightforward calculation. However, since the geometry is infinite, every point on the x-axis is a valid option as the origin for the unit vector. Therefore, the values are integrated over the top boundary L of the domain to give the scattering coefficient

(9)

The coefficient in Equation 7 or Equation 9 gives a value for each incidence angle and frequency included in the study. To obtain the random incidence scattering coefficient in octave bands, the quantity calculated is averaged over angles and frequencies per band in by selecting and .

Results and Discussion

The sound pressure scattered by the diffuser arrangements and their infinite baffles is shown in Figure 2 for θ0 = 0° and f = 1335 Hz. In this case, the plane waves propagating in the wells of the Schroeder diffuser are clearly visible. Moreover, it is seen that the planar wavefront that would result from a perfectly specular reflection is disrupted by the presence of the wells. This is especially true in front of the diffuser arrangements, where the wavefront tends to a cylindrical wave. It also appears that destructive interferences create directions with greatly reduced energy. Although this effect is present in the single diffuser, it is seen to be more prominent in the 5-unit arrangement.

Figure 2: Scattered pressure for θ0 = 0° and f = 1335 Hz. Top left: single diffuser; Top right: 5-unit arrangement; Bottom: infinite arrangement.

Further investigation of the scattered sound field can be made from the SPL polar responses of the single diffuser in Figure 3 and the 5-unit arrangement in Figure 4. They are evaluated in the far field at a distance r0 = 10 m, for θ0 = 30° and the six frequencies composing the 250 Hz octave band in this study. This type of plot could not be generated for the infinite arrangement as it requires to evaluate sound pressure on a surrounding semicircle. It can be observed that the primitive root diffuser does reduce the energy reflected in the specular direction as intended. The difference between the SPL in the specular direction and the maximum SPL lies between 1 dB and 4 dB for the single diffuser, with energy being distributed rather evenly across angles. For the 5-unit arrangement, the SPL difference goes from 4 dB to 9 dB between the specular and maximum directions. However, it is seen that the reflected energy is concentrated between 15° and 45°, as the lobes exhibit large amplitudes for both peaks and dips in that region of the polar response. This confirms the idea that a periodic arrangement reinforces certain reflection and attenuation directions for a given diffuser.

Figure 3: Polar response of the single diffuser for θ0 = 30° in the 250 Hz octave band.

Figure 4: Polar response of the 5-unit arrangement for θ0 = 30° in the 250 Hz octave band.

The scattering coefficients of the three diffuser arrangements is shown in Figure 5. Despite the differences observed in their polar responses, it is seen that the single diffuser and the 5-unit arrangement return similar values for all frequencies. This illustrates one of the shortcomings of the scattering coefficient: it only quantifies non-specularly reflected energy without any consideration of the energy distribution over angles. Two diffusers could therefore have the same scattering coefficient and yet behave very differently. Nevertheless, this coefficient still provides valuable information in a simple manner. The infinite arrangement also gives interesting results. In this case, the scattering coefficient takes much lower values especially at low frequencies. This is a sign that scattering is dominated by diffraction from the finite size effect at low frequencies, and the wells’ influence is only effective at mid and high frequencies.

Figure 5: Random incidence scattering coefficient of the diffuser arrangements.

Notes About the COMSOL Implementation

This model was studied with the Pressure Acoustics, Frequency Domain interface. Alternatively, the Pressure Acoustics, Boundary Elements interface could also be used and return accurate results, but with the drawback that solving time would be larger for this small 2D model.

References

1. T.J. Cox and P. D’Antonio, Acoustics Absorbers and Diffusers: Theory, Design and Application, 2nd edition, Taylor & Francis, 2009.

2. F.A. Everest and K. C. Pohlmann, Master Handbook of Acoustics, 5th edition, McGraw Hill, 2009.

3. A.D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, 2nd edition, Acoustical Society of America, 1991.

Acoustics_Module/Building_and_Room_Acoustics/diffuser_schroeder_2d

Modeling Instructions

This section contains the modeling instructions for the Schroeder Diffuser in 2D model. They are followed by the Geometry Modeling Instructions section.

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Start by importing the three geometry cases and the parameters needed for the study.

Single diffuser

In the window, click .

In the window for , type Single diffuser in the text field.

Geometry 1

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file diffuser_schroeder_2d_geom_sequence.mph.

In the dialog box, click .

Geometry 1

In the window, under click .

In the toolbar, click

Add Component

Right-click and choose .

5-unit arrangement

In the window for , type 5-unit arrangement in the text field.

Geometry 2

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file diffuser_schroeder_2d_geom_sequence.mph.

In the dialog box, select in the list.

Click .

Geometry 2

In the window, under click .

In the toolbar, click

Add Component

Right-click and choose .

Infinite arrangement

In the window for , type Infinite arrangement in the text field.

Geometry 3

In the toolbar, click and choose .

Browse to the model’s Application Libraries folder and double-click the file diffuser_schroeder_2d_geom_sequence.mph.

In the dialog box, select in the list.

Click .

Geometry 3

In the window, under click .

In the toolbar, click

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click the right end of the split button in the window toolbar.

From the menu, choose .

Click in the window toolbar.

In the toolbar, click

to close the window.

Global Definitions

Parameters 1 - Geometry

In the window for , type Parameters 1 - Geometry in the text field.

Parameters 2 - Physics

In the toolbar, click

and choose .

In the window for , type Parameters 2 - Physics in the text field.

Locate the section. Click

Browse to the model’s Application Libraries folder and double-click the file diffuser_schroeder_2d_parameters_physics.txt.

Proceed with setting up the physics and local variables. The incident plane wave is defined manually in order to include the specular reflection from the infinite baffle in the . This allows the solver to only compute the influence of the diffuser on the sound field.

Definitions (comp1)

In the window, under click .

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 29 only.

Variables 1

Right-click and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file diffuser_schroeder_2d_variables_single.txt.

Pressure Acoustics, Frequency Domain (acpr)

Background Pressure Field 1

In the window, expand the node.

Right-click and choose .

Select Domain 1 only.

In the window for , locate the section.

From the list, choose .

In the pb text field, type p_inc+p_inf.

Perfectly Matched Boundary 1

In the toolbar, click

and choose .

Select Boundaries 27 and 28 only.

Exterior Field Calculation 1

In the toolbar, click

and choose .

Select Boundaries 27 and 28 only.

In the window for , locate the section.

From the 0 list, choose .

Generate the mesh using the functionality. The frequency controlling the maximum element size is per default taken . Set the desired in the study step. In general, 5 to 6 second-order elements per wavelength are needed to resolve the waves. For more details, see Meshing (Resolving the Waves) in the Acoustics Module User’s Guide. In this model we use the default option, which gives 5 elements per wavelength.

Study 1

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
n (Band number)	range(0,1,5)

Step 1: Frequency Domain

In the window, click .

In the window for , locate the section.

In the text field, type 10^{range(log10(fL[1/Hz])+df_log/2,df_log,log10(fU[1/Hz])-df_log/2)}.

Click to expand the section. Select the check box.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
theta0 (Incidence polar angle from normal direction)	range(-80,10,80)	deg

Mesh 1

In the window, under right-click and choose .

Set up the physics and generate the mesh in a similar fashion for the second component with five diffuser units.

5-unit arrangement (comp2)

In the window, click .

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Definitions (comp2)

Integration 2 (intop2)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 125 only.

Variables 2

In the window, right-click and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file diffuser_schroeder_2d_variables_5unit.txt.

Pressure Acoustics, Frequency Domain 2 (acpr2)

In the window, under click .

Background Pressure Field 1

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

From the list, choose .

In the pb text field, type p_inc+p_inf.

Perfectly Matched Boundary 1

In the toolbar, click

and choose .

Select Boundaries 123 and 124 only.

Exterior Field Calculation 1

In the toolbar, click

and choose .

Select Boundaries 123 and 124 only.

In the window for , locate the section.

From the 0 list, choose .

Mesh 2

In the window, under right-click and choose .

Define the third component with a periodic condition to model an infinite arrangement of diffusers. As a consequence, the sound pressure in the exterior field cannot be evaluated on a circular arc in this case; the calculation of the energy reflected in the specular direction is thus based on sound intensity.

Infinite arrangement (comp3)

In the window, click .

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Definitions (comp3)

Integration 3 (intop3)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 3 only.

Variables 3

In the window, right-click and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file diffuser_schroeder_2d_variables_infinite.txt.

Pressure Acoustics, Frequency Domain 3 (acpr3)

In the window, under click .

Background Pressure Field 1

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

In the p0 text field, type 1.

In the c text field, type c0.

Specify the ek vector as


sin(theta0)	x
-cos(theta0)	y

Select the check box.

In the ρ text field, type rho0.

Perfectly Matched Boundary 1

In the toolbar, click

and choose .

Select Boundary 3 only.

In the window for , locate the section.

From the list, choose .

Periodic Condition 1

In the toolbar, click

and choose .

Select Boundaries 1 and 28 only.

In the window for , locate the section.

From the list, choose .

Specify the kF vector as


sin(theta0)*acpr3.k	x
-cos(theta0)*acpr3.k	y

Mesh 3

In the window, under right-click and choose .

Results

In the window, click .

In the window for , locate the section.

Select the check box.

Proceed with solving the study. Generate the default plots for Component 1 only so as to avoid duplicate plot groups.

Study 1

Step 1: Frequency Domain

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Physics interface	Solve for	Equation form
Pressure Acoustics, Frequency Domain 2 (acpr2)		Automatic (Frequency domain)
Pressure Acoustics, Frequency Domain 3 (acpr3)		Automatic (Frequency domain)

Click to expand the section. In the table, enter the following settings:


Component	Mesh
5-unit arrangement	No mesh
Infinite arrangement	No mesh

In the toolbar, click

In the window, click .

In the window for , locate the section.

Clear the check box.

In the window, click .

In the window for , locate the section.

In the table, enter the following settings:


Physics interface	Solve for	Equation form
Pressure Acoustics, Frequency Domain 2 (acpr2)	√	Automatic (Frequency domain)
Pressure Acoustics, Frequency Domain 3 (acpr3)	√	Automatic (Frequency domain)