JPH0887281A - Diffuser for two-dimensional primitive root shape - Google Patents

Abstract

PURPOSE: To laterally and uniformly scatter sounds, to suppress specified mirror- shaped reflection and to increase indirect sound fields for a listener by providing the well of two-dimensional well and determining the depth of the well by operating a primitive root sequence theory. CONSTITUTION: The matrix of 12×13 showing the positions of wells 1-156 on a matrix according to an instruction is calculated. Concerning such a matrix, numbers are obliquely continued at -45 deg. until reaching a final enable spot, a sequence continues from the apex of the next column, and the next row of the 1st column is started when the final column is completed. The numbering sequence is continued until all 156 wells are properly positioned. After the primitive root is multiplied corresponding to the number of wells, the number generated as a result is further divided by a prime number such as '157' in this case, the prime number '157' is multiplied to a numeral less than a decimal point, and the result is defined as a sequence value corresponding to the number of wells. The respective sequence values are multiplied with a designed wavelength and when it is divided by the double prime number, the real value of well can be provided.

Description

Detailed Description of the Invention

[0001]

FIELD OF THE INVENTION This invention relates to acoustic diffusers, and more particularly to two-dimensional primitive root diffusers.

[0002]

2. Description of the Related Art Acoustic analysis of a diffraction grating, which has played an important role in the field of spectroscope for more than 100 years, is until the invention and development of a reflection phase grating diffuser within the past 10 years. , Not used in architectural acoustics. The one-dimensional reflective phase grating described in US Pat. No. D219,501 and shown in FIG. 1 linearly and periodically groups an array of wells of equal width and different depth separated by a thin divider. It is made up of The depth of the well is determined by calculation using the square residue theory. In a one-dimensional reflective phase grating, a number of theoretical phase changes occur in one direction on the plane of the unit and
It doesn't change 0 degrees. The reflective phase grating can also be designed two-dimensionally, in which case a number of theoretical phase changes occur not only in one direction but in two orthogonal directions. As with the one-dimensional diffuser, a well depth sequence of wells with a quadratic residue is used. A two-dimensional diffuser consists of a two-dimensional array of square, rectangular or circular wells of different depth separated by a thin divider, FIG.
The registered trademark "Ominiffuso" described in No. 306,764
r "is a commercially available two-dimensional quadratic residue diffuser. It can be seen that the Ominiffusor diffuser has two symmetrical vertical mirror planes and four rotationally symmetric planes, but the primitive root diffuser does not include symmetrical elements, as will be described later.

FIGS. 3 and 4 respectively show schematic comparisons of the semi-disc-shaped cover pattern of the one-dimensional quadratic residue diffuser and the hemispherical cover pattern of the two-dimensional quadratic residue diffuser, respectively. . In FIG. 3, the incident plane wave is represented by the arrow arriving at 45 degrees to the vertical surface. Radial arrows contacting the half-disc envelope indicate the diffraction direction. In FIG. 4, the incident plane wave is represented by the arrow arriving at 45 degrees to the vertical surface. The arrows emanating from the hemisphere envelope indicate some of the many diffraction directions.

The planar coset sequence provides uniform diffusion over all of the diffraction orders, while the coset root sequence suppresses the zero order and Zecklogarithm suppresses the zero and first order diffractions at the design frequency and its integral multiples. Applicants have discovered that the scattered intensity pattern of the primitive root sequence excludes certain lobes that occur in the scattered intensity pattern of the planar residue number theory sequence.

[0005]

[Equation 1]

[0006]

[Equation 2]

[0007]

(Equation 3)

[0008]

[Equation 4]

The diffraction direction for each wavelength λ of the incident sound scattered from the reflection phase grating (FIG. 5) is determined by the size of the repeating unit NW of Equation 1. Where N is the number of wells per space, W is the width of the wells,
αi is the incident angle, αd is the diffraction angle, and n is the diffraction order. The intensity in any direction (FIG. 6) is determined by the Fourier transform of the reflectance rh as a function of the in-space depth (dh) or phase of (Equation 2). Equation 1 shows that as the repeating unit NW increases, larger diffraction lobes occur and the diffusion increases. Further, as the number of spaces increases, the energy is concentrated in the diffraction direction (Fig. 6).

FIG. 6 (a) shows the theoretical scattering intensity pattern for a quadratic diffuser. The diffraction direction is shown by the dotted line, and scattering from a finite diffuser occurs over a wide lobe. The maximum intensity is normalized to 50 dB. In FIG. 6B, the number of spaces is increased to 2 to 25, and the energy is concentrated in the diffraction direction. Figure 6
In (c), the number of wells per space is increased to 7-89, and the number of lobes is tripled. The arrows indicate the incident direction and the specific reflection direction. The reflection phase grating improves the time distribution of the backscattered sound due to the unevenness of the surface, and makes the cover of a wide angle uniform over a wide design frequency bandwidth regardless of the incident angle. The diffusion properties are practically invariant with respect to incident frequency, incident angle and viewing angle.

The well depths of the one-dimensional quadratic residue diffuser (Equation 3) and the two-dimensional quadratic residue diffuser (Equation 4) are based on a mathematical number theory sequence, which is The Fourier transform of exponentiation sequence values has the unique feature of having a constant value in the diffraction direction. The symbol h indicates the number of wells in the one-dimensional quadratic residue diffuser, and the symbols h and k indicate the number of wells in the two-dimensional quadratic residue diffuser. For square sequence elements S _h = h ² _modN and S _h , k = {h ²
+ K ² } _modN , where N is an odd prime number.
For example, if N = 7, the one-dimensional sequence elements for h = 0 to 6 are 0, 1, 4, 2, 2, 4, 1. Repeat this sequence for larger values of h. Table 2 shows that N = for a two-dimensional quadratic residue diffuser.
The values of Sh and k in the case of 7 are shown.

[0012]

[Table 2]

[0013]

(Equation 5)

Two-dimensional polar response or diffraction order (m, n)
Equation 5 can be conveniently represented in the plot of the iterative grating reflective phase grating shown in FIG. The diffraction order is determined by the interference condition. When the depth variation is determined by the quadratic residue sequence, the non-infinitesimal small scattering lobes have a dimensionless N
Displayed as equal energy shapes in a circle equal to W / λ. The effect of changing the frequency is easy to understand,
This is the preferred plotting method. Therefore, reducing λ ₂ to λ ₁ increases the number of accessible diffraction lobes contained within a circle of diameter NW / λ ₁ and increases diffraction. The one-dimensional reflective phase grating with horizontal wells has (n = 0, ±
Diffraction from a one-dimensional reflective phase grating with vertical wells scattered in the direction indicated by the vertical line in the repetitive grating reflective phase grating (1, ± 2 etc. and m = 0) is (m
= 0, ± 1, ± 2, etc. and n = 0). Coordinates on a repetitive grating reflection phase grating plot are unidirectional. These scattering directions are shown in the three-dimensional banana-shaped plot in FIG. 8, and in this FIG.
The nine diffraction orders occurring in the circle of / λ ₂ are plotted from the perspective view. A conventional pole pattern for a one-dimensional iterative phase grating with vertical wells at λ ₂ is obtained from a plane slice through lobes 0, 2 and 6 in FIG. Including.
The width of the scattering lobe is proportional to the number of spaces contained in the reflective phase grating.

In the primitive root sequence underlying the present invention, S _h = g ^h _modN, where g is the primitive root of N. For N = 11, the primitive root g = 2. This means that the number of remainders after dividing 2 ^h by 11 is a unique combination only once, and the value of (N-1) S _h , that is, 1,
It means that it becomes 2, ... 10. Repeat this series periodically for larger values of h. Since each digit occurs only once, the symmetry obtained with a quadratic diffuser does not exist with a primitive root diffuser. Primitive root diffusers have the property that the scattering at the design frequency and its integral multiples decreases in a particular direction due to the fact that the phase is evenly distributed between 0 and 2π. FIG. 9 shows a one-dimensional analysis pattern of the primitive root diffuser based on N = 53 at normal incidence. Note here the reduced specific lobes at integer multiples of the design frequency f0.

Applicants have found that in order to form a two-dimensional primitive root array, the prime number N must be chosen such that N-1 has two common prime factors that cannot be divided into each other. Storing the one-dimensional sequence elements in a "Chinese Remainer" fashion utilizing horizontal and vertical matrix transformations, these common prime factors form a two-dimensional matrix. When the applicant repeats this matrix periodically, consecutive numbers are simply -4.
It was found to follow a 5 degree diagonal, namely S ₁ , S ₂ , S ₃ , S ₄ etc. (these are shaded in Table 3). This can serve as a check when forming the appropriate matrix. It can be seen that in the two-dimensional array there is the preferred Fourier characteristic of the one-dimensional array, ie the flat power response.

In Table 3, the value Sh of the one-dimensional sequence is N =
It is shown that it is formed in two spaces of 11 primitive root sequences.

[0018]

[Table 3]

For a prime number, for example N = 17, N-1 is 2
Not all prime numbers can be two-dimensional because they do not contain two common prime factors. Two numbers h and k that do not have a common factor are called common prime numbers. In reality, a two-dimensional primitive root array cannot be square.

As will be described in more detail below, acoustic diffusers with wells determined according to a primitive root sequence in which the wells are arranged to follow a -45 degree diagonal line, have a scattered sound as compared to a quadratic diffuser. The lateral ratio pointing to is greater. As mentioned above, the spreading pattern for the primitive root diffuser is devoid of central specific reflection lobes at the design frequency and integer multiples thereof. There is no such specific reflection lobe that forms the sound field of the primitive root diffuser of the present invention.

In addition, a diffuser designed according to the sequence of squared residue theory has wells with depths that exhibit symmetry about a centerline in a diffuser manufactured according to the primitive root theory. , Each well has a unique depth that differs from the depth of other wells. Therefore, the depth of one well is not repeated in the entire sequence, so the diffuser made in accordance with the principles of the present invention is asymmetric.

[0022]

Therefore, the first aspect of the present invention
The purpose is to provide a two-dimensional primitive root diffuser. Another object of the invention is to provide such a primitive root diffuser with asymmetrically arranged wells.

Yet another object of the present invention is to provide a primitive root diffuser that scatters uniformly laterally and suppresses certain mirror-like reflections, thus increasing the indirect sound field to the listener. Especially.

Yet another object of the present invention is to provide such a diffuser in which there is no particular reflection lobe in the analysis pattern of the diffuser at the design frequency and its integral multiples.

A better understanding of the above and other objects, features and advantages of the present invention will be obtained from a reading of the following detailed description of the preferred embodiments, taken in conjunction with the accompanying drawings.

[0026]

Examples In developing the present invention, not only the development of a two-dimensional primitive root diffuser having advantageous acoustic characteristics but also aesthetically preferable, such a two-dimensional primitive root that can be incorporated into the present room shape. Attention was also paid to the development of the shape diffuser. As a first feature, it has been found that current suspended ceiling grid systems typically have square openings of dimensions 5.08 cm x 5.08 cm (2 inches x 2 inches). The preferred embodiment of the present invention uses these dimensional dimensions.

From an aesthetic point of view, the Applicant has found that when molding a two-dimensional primitive root diffuser, it can maintain its acoustic function while giving it an aesthetic appearance. In addition, manufacturing a diffuser with a large number of wells (recesses, each well having a unique depth) is extremely time consuming, so molding the diffuser saves costs.

For the primitive root diffuser of the present invention to be effective in the environment in which it is intended to be used, at least 500-5 per second
The sound must be spread over a bandwidth of 000 cycles. In addition, Applicants have ensured that each primitive root diffuser has a Class A ASTM E-84 rating, i.e., a frame spread of 762 cm (25 ft) and a smoke generation index of 450 compared to red oak.

According to the gist of the present invention, 5.08 cm × 5.
Each diffuser measuring 8 cm (2 inches x 2 inches)
It weighs less than 11.4 kg (25 lbs),
It has sufficient rigidity to minimize partition wall absorption.

Due to the design constraints required to make each diffuser approximately square in shape, each cell is rectangular with an aspect ratio to camouflage non-square cross sections. In considering the prime numbers used in calculating the depth of each well, several different prime numbers were considered. It was found that the larger the prime number used, the more sensitive the non-square cross section of the well. Furthermore, the Applicant has found that an effective primitive root diffuser can be manufactured from a calculation with a prime number of 157 such that N-1 = 156 with a prime factor of 12x13. The 156 rectangular blocks that make up the acoustic well make the surface shape extremely balanced and aesthetic,
Non-square aspect ratios are indistinguishable at reasonable viewing distances.

In addition, Applicants have come up with an algorithm that can be used to determine 157 primitive roots. This primitive root was calculated so that g = 5. This algorithm is also used to calculate sequence values because the primitive root g = 5 exponentiation operation exceeds the capabilities of most computers and cannot display the result of 5'156. Table 4 below is a copy of the algorithm used for this.

[0032]

[Table 4]

In the above example, which is the preferred embodiment of the present invention, the algorithm described in Table 4 is used to calculate the well depth values in the diffuser of the present invention. Prior to performing calculations using the algorithm shown in Table 4, a 12 × 13 matrix showing the positions of wells 1-156 on the matrix according to the above instructions was calculated. In this matrix, the number continues diagonally at -45 degrees until the last possible spot is reached, the sequence continues from the vertex in the next column, and when the last column is completed, it goes to the next row in the first column. Has become.

[0034]

[Table 5]

Thus, referring to Table 5, well 1 is at the top left corner of the matrix, wells 2-12 traversing the matrix diagonally until the bottom row is reached, with well 13 at the top of the last column. Well 13 is located at the top of the last row, so well 14 is located one position below the highest position in the first row, ie, just below well 1. Wells 15-24 proceed diagonally at an angle of -45 degrees,
After well 24, well 25 is naturally located at the top of the next row and well 26 is located below well 13. After well 26, well 27 is located third from the top in the first row, and the numbering sequence continues as shown until all 156 wells are in place.

In such a preferred embodiment, the total number of wells is 156, the primitive root g is 5, and the specific value for the well depth is calculated as follows.

The primitive root is raised to the power of the number of specific wells selected. For example, for well 3, the primitive root is set to 5, and this is cubed. The resulting number 125 is the prime number 1 selected
Dividing by 57 gives 0.7961783. Multiplying the number less than or equal to this decimal number by 157 is 125.

[0038]

[Table 6]

Therefore, in Table 6, the position corresponding to the numeral 3 in Table 5 contains the number 125 corresponding to the depth of the well corresponding to that position. In another example where the well number h is equal to 6, g ^h = 5 ⁶ or 15,625,
Dividing this by 157 gives 99.5222292. In this case, the number after the decimal point is 0.5222292, which is multiplied by 157 to obtain 82. As shown in Table 6,
The numeral 82 is placed at the same position as the numeral 6 in Table 5.

In this way, after the primitive root is raised to the power corresponding to the well number, the resulting number is further divided by a prime number, in this case 157, and the number after the decimal point is set to the prime number 1.
Multiply by 57 and take the result as the corresponding sequence value for the number of wells. Each sequence value is multiplied by the design wavelength λ and divided by twice the prime number (157 in Table 6) to give the actual well depth value. In the preferred design, 1
It should be understood that the powering of the primitive root g with a large number based on the use of 56 wells exceeds the capabilities of most computers, so it should be understood that the algorithm shown in Table 5 was created.

The primitive root is a primitive root sequence expression or Table 4.
Using the algorithm of, the trial-and-error known to form the matrix of Table 6 is a prime number less than N. Applicant has found these results only for such prime numbers.

Reference is now made to FIGS. In these figures, a particular diffuser with the values shown in Table 6 is shown.

Turning to FIGS. 10-27, representative examples of wells having the numbers shown in Table 4 and the well depth values shown in Table 6 are indicated by the reference numbers corresponding to the numbers in Table 5. Has been done.

FIG. 10 shows an isometric view of a 12 × 13 two-dimensional primitive root diffuser forming a preferred embodiment of the present invention. FIG. 11 shows a plan view of the diffuser of FIG. 10 when viewed from below, with FIGS.
Shows the respective cross sections shown in FIG.
Refer to Table 5 when mapping FIGS. 12-23 to FIG. The reference numbers in FIGS. 12 to 23 correspond to the number of well identifications in Table 5, and in order to facilitate understanding of FIGS. 12 to 23, identification of wells at each end of each cross section line is shown in FIGS. The numbers are shown.

24-27 show four side views from each side of the inventive diffuser best shown in FIG. To facilitate the understanding of the perspective view from which these side views were created, the identification numbers from Table 5 are displayed at each end of the first row in each side view.

FIG. 28 shows the theoretical diffraction pattern further than the field for a single diffuser as shown in FIGS. 10-27. There are no bright spots in the center of this pattern, indicating that there is no central, especially reflective lobe, as would be expected with a two-dimensional primitive root-based diffuser. .

FIG. 29 shows the far field diffraction pattern at the design frequency for an array of diffusers as shown in FIGS. 10-27 with an array containing three rows and three columns of diffusers. It should be pointed out that there is no central, especially reflective lobe, so that the specific response is not degraded at the center of the pattern.

In the preferred embodiment of the invention, each of the diffusers must be manufactured at low cost to be marketable, yet lightweight and flame retardant for installation in a building. Under these circumstances, in a preferred embodiment of the present invention, each diffuser of the present invention is manufactured by a molding process.
The applicant has found that the method using glass fiber reinforced gypsum or glass fiber reinforced plastic is suitable. The glass fiber reinforced gypsum molding method utilizes a hydraulic two-piece mold that uses a strength-enhancing and lightweight mixture of gypsum and glass fiber. This glass fiber reinforced plastic method uses a special composite two-piece mold to produce a diffuser with substantially no draft angle in the vertical rise of various rectangles. AS
These flame retardant compositions were used to meet the requirements of TM E-84.

As another feature, the Applicant has employed a primitive root diffuser manufactured in accordance with the teachings of the present invention for use with the variable acoustic modular actuation system described in Applicant's earlier US Pat. No. 5,168,129. And found it extremely effective.

Accordingly, the present invention has been described in terms of a preferred embodiment of the invention which provides a novel and useful two-dimensional primitive root diffuser which meets each of the above-identified objectives of the present invention.

Naturally, a person skilled in the art is able to contemplate various variations, changes and modifications within the spirit of the invention without departing from the spirit and scope of the invention.

Thus, it should be understood that the invention is limited only by the appended claims.

[Brief description of drawings]

1 shows a perspective view of a one-dimensional quadratic residue diffuser, corresponding to FIG. 1 of the applicant's earlier US Pat. No. D291,601.

FIG. 2 shows a perspective view of a two-dimensional quadratic residue diffuser, corresponding to FIG. 1 of the applicant's earlier US Pat. No. D306,764.

FIG. 3 shows a half-disc-like scattering pattern of plane acoustic waves incident at 45 degrees to a surface normal to a one-dimensional quadratic diffuser.

FIG. 4 shows a half-disc-like scattering pattern of plane acoustic waves incident at 45 degrees on a surface normal to a two-dimensional quadratic diffuser.

FIG. 5 is a graph of incident waves (A and E) and diffracted waves (D and H) from the surface of a periodic reflective phase grating with a repeating distance NW.

FIG. 6 shows a quadratic diffuser and its theoretical spreading intensity pattern.

FIG. 7 shows a two-dimensional iterative grating reflective phase grating exhibiting equal energies of diffraction orders m and n for a reflective phase grating based on the quadratic residue number theoretical sequence.

FIG. 8 shows a three-dimensional banana-like plot resulting from FIG.

FIG. 9: 3/4 of the design frequency of a primitive root diffuser, 1,
Diffraction patterns of 4, 8 and 12 times are shown.

FIG. 10 is an isometric view of a two-dimensional primitive root diffuser made in accordance with the teachings of the present invention.

11 is a plan view of the primitive root diffuser of FIG.

12 shows a cross section A-L shown in FIG.

13 shows a cross section A-L shown in FIG.

14 shows a cross section A-L shown in FIG.

15 shows a section A-L shown in FIG.

16 shows a section A-L shown in FIG.

FIG. 17 shows a cross section A-L shown in FIG. 1.

FIG. 18 shows a cross section A-L shown in FIG. 1.

FIG. 19 shows a cross section A-L shown in FIG. 1.

20 shows a section A-L shown in FIG.

21 shows a section A-L shown in FIG.

22 shows a section A-L shown in FIG.

23 shows a section A-L shown in FIG.

FIG. 24 is a side view of the primitive root diffuser of the present invention.

FIG. 25 is a side view of the primitive root diffuser of the present invention.

FIG. 26 is a side view of the primitive root diffuser of the present invention.

FIG. 27 is a side view of the primitive root diffuser of the present invention.

FIG. 28 shows the theoretical far field diffraction pattern from one space of a two-dimensional primitive root diffuser based on N = 157 and g = 5.

FIG. 29 shows a theoretical out-of-field diffraction pattern from a 3 × 3 array of two-dimensional primitive root diffusers based on N = 157 and g = 5.

Claims (12)

[Claims] 1. A prime number N is selected such that a number N-1 has two common prime factors X and Y which are indivisible from each other, b) a primitive root number g is determined based on the prime number N, and c) Forming a rectangular matrix with dimensions XxY, having N-1 spaces inside, d) putting the number 1 in the upper left corner of said matrix, then -45 to the horizontal row of said matrix
Putting integers running diagonally in the direction of degrees, and putting an integer in a particular column in the bottom row of the matrix, the next integer in the top row of the matrix in the adjacent column to the right of the particular column. Then enter integers one after another diagonally from the next integer in the -45 degree direction until the integers are in the rightmost column of the matrix, and after the integers are in the rightmost column, the 1-N-1 by placing the next integer below the number 1 and then continuing the integers until all spaces in the matrix are filled.
The integer "h" up to the space of the matrix, and e) calculate the sequence value for each said integer by calculating the expression g ^h / N, then subtract the total whole number part of the result and the remainder number N And then the sequence value S
And f) multiply each sequence value by the design wavelength λ, divide by 2N and convert each sequence value to a well depth value, g) calculate the well depth value calculated as above. A method of manufacturing a two-dimensional primitive root diffuser comprising the step of forming a two-dimensional primitive root diffuser having. 2. The method of claim 1, wherein N = 157. 3. The method according to claim 2, wherein g = 5. 4. The method according to claim 3, wherein X = 13 and Y = 12.
The described method. 5. The method according to claim 3, wherein X = 12 and Y = 13.
The described method. 6. A method according to claim 4, wherein steps d) and e) are carried out by computing the following algorithm. [Table 1] 7. A formula S ^_h = g _{h modN} (here, Sh is the particular sequence value, N is a prime number, h is an integer of 1 to N-1, g is a primitive root of N A two-dimensional primitive root diffuser containing a two-dimensional matrix of wells having respective depths calculated according to. 8. The diffuser of claim 7, wherein g = 5 and N = 157. 9. The diffuser of claim 8, wherein the matrix has dimensions X and Y. 10. The diffuser according to claim 9, wherein X = 13 and Y = 12. 11. The diffuser of claim 7 comprising glass fiber reinforced gypsum. 12. The diffuser according to claim 7, which is made of glass fiber reinforced plastic.