Introduction. Figure 1 W8LC Line Array, box and horn element. Highlighted section modelled.

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1 imuation of the acoustic fied produced by cavities using the Boundary Eement Rayeigh Integra Method () and its appication to a horn oudspeaer. tephen Kirup East Lancashire Institute, Due treet, Bacburn, Lancs, UK. s.irup@bacburn.ac.u Ambrose Thompson Martin Audio, Century Point, High Wycombe, Bucs, UK. ambrose@martin-audio.com Introduction In this paper a method based on couping the interior boundary eement method (BEM) and the Rayeigh Integra Method (RIM) for simuating the acoustic fied of a cavity with one opening is proposed. uch a method has a number of appications in acoustics. In this paper we wi be appying the method to the probem of determining the acoustic response of a typica horn oudspeaer. Figure W8LC Line Array, box and horn eement. Highighted section modeed. In order to coupe the BEM and RIM methods, a fictitious boundary is paced across the opening of the cavity. The interior boundary integra equation formuation of the

2 cavity is couped to the Rayeigh integra which governs the exterior acoustic fied, by enforcing continuity in the veocity potentia (sound pressure) and its derivative on the opening. The method is termed the Boundary Eement Rayeigh Integra Method () and it ony requires a mesh of the interior cavity and the opening, and hence it is generay much more efficient than the straightforward appication of the exterior BEM to this ind of probem. The couping of the integra equations gives a inear system of equations, the soution of which returns the veocity potentia (sound pressure) and its norma derivative (veocity) on the cavity surface and on the opening. The acoustic properties can then be found either within the cavity using the interior boundary integra equation formuation or in the exterior region by using the Rayeigh integra. The method is deveoped in 3D through a simpe trianguation of the cavity and the opening and the appication of coocation to the integra equations to give the Fortran subroutine 3. In this paper the 3 method is verified by appying it to the horn oudspeaer iustrated in Figure and comparing the resuts with resuts from the exterior BEM. A horn oudspeaer is a type of acoustic transducer which presents to the vibrating piston a higher acoustic resistance than experienced by a piston in free air. The shape of the horn contros the degree of oading and directiona characteristics. Practica horns do not generay conform to the cassica fares for exampe exponentia or hyperboic but are formed from geometry which prevents simpe anaysis. In professiona sound reinforcement horns have been an essentia feature for many years, Martin Audio (4). Amongst its virtues are higher efficiency and a contro over directiona characteristics. The atter has become very important in recent years, due to the advent of high power ampifiers and compression drivers buit to withstand them. It is for this reason that we concentrate on the PL and poar response in this paper; resuts from 3 appied to the horn oudspeaer of Figure are presented for a wide range of sampe frequencies. Boundary Eement- Rayeigh Integra Method () Mode There are at east three approaches to soving the open cavity probem using integra equation techniques. One method is to treat it as an exterior probem and appy the BEM by wrapping eements both around the exterior and the interior cavity was, for exampe by using the AEBEM* methods Kirup (4a). A second method is to cose the cavity and coupe boundary integra equation reformuations of the interior and exterior regions across the openings (eg couping the AIBEM* and AEBEM* programs of Kirup (4a)). An aternative method is to cose the (one) opening of the cavity and coupe the interior boundary integra equation with the Rayeigh integra (ie couping the AIBEM* and ARIM* methods of Kirup (4a,b). It is this the third idea, boundary eement- Rayeigh integra method () that we wi be deveoping in this paper.

3 Infinite refecting baffe Interior cavity Mouth of cavity, dividing the two acoustic fieds into interior and exterior Figure. Preparation of mode for appication of. The physica probem is now iustrated by Figure. The acoustic domain is the cavity and the haf-space beyond the mouth. The baffe is rigid and perfecty refecting. This mode can be appied to a range of acoustic cavity probems. In any practica probem the baffe must be finite but, even if there is no baffe, at east the continuity in the acoustic fied is maintained across the mouth and the mode can sti be appied with due care. In the case of a horn oudspeaer, such as the one iustrated in Figure, there is a substantia baffe and the mode is considered very appropriate. Let be the surface of the cavity and be the opening. The boundary condition is appied on the surface of the cavity and the condition is presumed to be in the foowing form: a ( ϕ ( + b( = f ( (p ), though for the horn oudspeaer appication in this wor ony the Neumann condition is considered: a(=, b(= (p ). Method The method is derived through couping the interior boundary integra equation formuation for points on interior cavity surface: { M + I} + ϕ ( = { L} + (p +) and the Rayeigh integra for points on, v ( { L } ϕ( (p ). = In equations () and (3), represents the veocity potentia and v its derivative with respect to the norma that is outward to the cavity. The operators are defined as foows: { L µ } Γ ( p ) = G( p, q) µ ( q) and G( p, ) { M µ } Γ ( p ) = q µ ( q) d n q d q Γ Exterior fied Γ q () () (3)

4 where G is the free-space Green s function for the Hemhotz equation and is used here to represent any surface or part of the surface (incuding ), I is the identity operator. If we consider equation (3) for points on and foowing equations: separatey then we obtain the { M + I} ϕ ( + { M } ϕ ( = { L } + { L } v ( (p ) { M } + ϕ ( + { M + I} ϕ( = { L} + { L} (p ) The computationa method is appied by a trianguation of the interior surface of the cavity and the opening aone. By approximating and v by constants on each triange and through coocation the integra equations (3),(4), and (5) can be written as inear systems of equations: v = [ L ϕ, [ M + I ϕ + [ M ϕ = [ L v + L [ v, [ M ϕ + [ M + I ϕ = [ L v + [ L v, respectivey. The correspondence between (3-5) and (6-8) shoud be cear. The operators L, M and I are repaced by the matrices [L,[M and [I and the boundary functions and v are repaced by vectors and v. For more detais on this see Kirup (4a). In the coocation method the centres of the trianges, the coocation points, are the representative points on the cavity surface and opening at which the surface functions and v are observed. If the cavity surface is divided into n eements and opening is divided into m eements then is an n-vector, is an m-vector, [L is an nxm matrix etc. With equations (6-8) we then have n+m equations with potentiay n+m unnowns. The system is competed with the n equations that are provided by the discrete form of the boundary condition (): [D a +[D b v = f (9) where [D a and [D b are diagona nxn matrices with the diagona made up of the vaues of a( and b( at the coocation points on. Using equations (6-9) we can form a (n+m)x(n+m) system of equations that returns approximations to the vaues of and v at the coocation points. For purey Neumann or Dirichet boundary conditions we can simpify (9) and in these cases we can write the couped system as an (n+m)x(n+m) system, this simpification is (4) (5) (6) (7) (8)

5 made in 3. Once the surface functions are determined, resuts on the cavity D can be found using the integra ϕ( { L } { M } ϕ( (p D), = + s+ and the Rayeigh integra can be used again for points in the exterior E v ( { L } ϕ( (p E). = Appication of 3 to the Horn Loudspeaer In order to appy 3 to the horn oudspeaer shown in figure, first the 3D soid mode is generated automaticay from a set of around parameters. This is then introduced into the popuar GID pre/post processor where a trianguation of the interior surface and mouth is made and subsequenty soved. A typica GID post process mesh is shown in figure 3. A veocity of m/s was set at the throat (assumed to be fat) and zero everywhere ese. In order to mitigate the numerica effects of the sudden change in boundary conditions where the cavity surface meets the mouth, a sma fange was added. A description of each cacuation can be found in Tabe, where number of eements and approximate running time on a AMD PC patform are given. () () Figure 3 Typica 3 mesh showing surface PL at 3Hz The sound pressure is observed on poar paths of m radius. The resuts from 3 are compared with measured resuts in Figure 4, showing poar pots of the sound pressure eve (sp) in the vertica and horizonta poar pane and an iustration of the mouth veocity ampitude for 3,6,9,,and 5Hz. The popuar GID pre/post processor was used to mesh and dispay the resuts.

6 Fig 4 Resuts Cac No. 3Hz Mouth Veocity Magnitude Cac No. 4 9Hz Cac No. 3 6Hz Cac No. 7 5Hz Cac No. 5 Hz

7 Comparison of 3 with BEM By way of comparison and further vaidation, the appication of 3 is compared with the appication of the boundary eement method (AEBEM3) to the same probem, but at 3Hz ony. In order to appy the BEM, the mesh in Figure 5 is used. The horizonta and vertica poar pots of the PL at m is shown in figure 6. Fig 5 AEBEM Mesh PL 8 5 Fig 6 AEBEM Resuts EBEM EBEM 7 3 Cacuation over Freq Eement ize Num Eements Time AEBEM 3Hz mm 89 3min 3Hz mm 994 min 3 6Hz mm 994 min 4 9Hz mm 994 min 5 Hz 8mm 35 5min 6 Hz 7mm 6 56min 7 5Hz 7mm 6 56min Tabe Computation Timings Concusion For a structure such as a horn oudspeaer, which consists of a cavity (the horn) opening out on to a pane, the Boundary Eement Rayeigh Integra Method () seems most appicabe. In Figure 3 it is shown that requires a mesh of the interior surface and opening pane aone whereas the appication of the boundary eement method (BEM) to the same probem requires consideraby more eements. Hence when it can be appied 3 reduces the meshing required and typicay uses an order of magnitude ess computer time than the straightforward BEM. The resuts in Figure 6, compared with Figure 4 at 3Hz show good agreement between computed and measured resuts, there are a number of other points. 3 seems to give better agreement with measured than the BEM in the forward fied, however, near the baffe the BEM has more agreement. The proposed reason for this

8 is that the BEM accuratey meshes the baffe whereas assumes and infinite baffe; 3 gives more support to the wider fied than the true finite baffe. Taing into account the comment in the previous paragraph on the modeing of the wider fied, the resuts generay show good agreement between measured and computed in Figure 4. In generay the obes in the sound fied are captured. There is ony significant drift in the horizonta poar at 5Hz: this woud probaby benefit from a further refinement in the mesh. The present method represents a significant improvement over our initia acoustic modes Webb, Baird (3). In genera 3 is a powerfu too for the simuation of the sound fied of a horn oudspeaer; returning resuts for a given probem and given frequency within a few minutes at ow and medium frequencies on a typica modern PC. Martin Audio (4) Website Historica and Product information B Webb, J Baird (3) Advances in Line array technoogy for Live ound, Institute of Acoustics Conference tratford UK 3.M. Kirup (994). Computationa oution of the Acoustic Fied urrounding a Baffed Pane by the Rayeigh Integra Method, Appied Mathematica Modeing, 8, M. Kirup (998). Fortran Codes for Computing the Discrete Hemhotz Integra Operators, Advances in Computationa Mathematics, 9, M. Kirup (4a). The Boundary Eement Method in Acoustics, econd edition. M. Kirup (4b). RIM3 Manua : Computing the Acoustic Fied of a Radiating Pate with Fortran code on CD.. M. Kirup (4c). 3 Manua: Computing the Acoustic Fied of a Radiating Pate with Fortran code on CD. GID Pre/Post processor