Calculation of acoustic radiation modes by using spherical waves and generalized singular value decomposition

Borgiotti (1990) was the first to apply singular value decomposition (SVD) to identify a set of orthonormal velocity patterns on a source surface, each member of which contributes to the radiated sound power independently. He also pointed out that a relatively small number of the orthonormal velocity patterns could account for most of the radiated sound power due to the rapidly decreasing radiation efficiencies associated with the progressively higher order velocity patterns. He also noted that the orthonormal velocity patterns are only dependent on the shape of the radiating structure. Cunefare (1991) further demonstrated the converging nature of the radiation efficiencies of the acoustic radiation modes (ARMs) and showed that the most efficient radiation modes are not strongly affected by introducing more degrees of freedom into the formulation. Cunefare et al. (2001) then showed that even for shapes other than spheres, the radiation efficiencies of the ARMs display a grouping behavior, very similar to that of spherical harmonics, and that the radiation efficiency changes within a group are dependent on the aspect ratio of the geometry.

On the application side, ARMs are commonly used in active noise control, simply because of the rapid reduction of radiation efficiency as the ARM mode number goes up, and since by attenuating the velocity distributions that couple with the most efficient radiation modes, an overall reduction of the sound power is guaranteed. Baumann et al. (1991, 1992) were among the first to implement active noise control techniques by suppressing the modes that radiate the most sound power. In contrast, Naghshineh and Koopmann (1992) proposed that inherently quiet structures could be designed by first solving for the ARMs, and then, by tailoring the material properties, a vibration pattern that results in minimal sound power radiation could be obtained, so that active noise control was unnecessary. ARMs have also been used as the basis functions in acoustical holography, which not only allows the velocity of a radiating surface to be reconstructed, but also gives the sound power contribution of each of the superimposed vibration patterns, thus revealing the potential sound power reduction that can be achieved (Liu et al., 2018). In the context of structural optimization for sound radiation reduction, ARMs have also been employed to identify the vibration patterns which radiate the most sound power, so that precise countermeasures can be taken to reduce radiated sound power at a certain frequency (Liu et al., 2017).

However, the estimation of sound power is not always easy, and in fact, can be computationally intensive when a large number of elements are required to ensure accuracy and/or a wide frequency range is being considered (Marburg, 2002). Recently, Wu et al. (2014) proposed a method to compute the radiated sound power based on mapped ARMs, which involves using the spherical harmonics and Gram–Schmidt orthogonalization. This method successfully eliminated the requirement to first solve the boundary integral equation (BIE) by using the boundary element method (BEM). However, the mapped ARMs are not, strictly speaking, the ARMs, as explained next.

For a mode set to qualify as ARMs, it must satisfy two requirements: (1) the particle velocity vectors associated with the various modes should be orthogonal; and (2) the particle velocity vectors should be orthogonal to the sound pressure vectors. It is important to differentiate between several commonly seen definitions of “modes” as used in acoustical studies: first and foremost are structural modes. While the structural modes are orthogonal with respect to the mass and stiffness matrices, the particle velocity vectors associated with radiation from the structural modes are not orthogonal to the associated sound pressure vectors: therefore, one cannot simply add up the sound power radiated by each of the scaled structural modes to obtain the overall sound power. The latter is also the case for the mapped ARMs described by Wu et al. (2014). Because of the use of the Gram–Schmidt procedure in creating the mapped ARMs, the pressure vectors associated with the mapped ARMs are not orthogonal to the particle velocity vectors, and so, the total sound power cannot be calculated simply by summing up the sound powers of the individual mapped ARMs. Another interesting set of modes related to sound radiation are referred to as pellicular modes (Coyette et al., 2009), which are derived from the cavity modes of a thin cavity extruded from the radiating surface. These modes do not possess the properties of ARMs since a rigid boundary condition is applied when calculating the pellicular modes, and, as a result, the velocity vectors are always zero, and therefore, the overall sound power cannot be calculated as the sum of the contributions of each of the pellicular modes.

In this article, the classical ARM calculation method based on utilizing the BEM and SVD is first reviewed. Then, the mapped spherical harmonic method of Wu et al. for calculating the sound power radiated by a general-shaped source is presented; however, it is shown that that method does not yield true, independent ARMs. So, here, a new way of obtaining the ARMs is proposed that is based on using a projection of spherical harmonics onto an enclosing surface, followed by the application of generalized singular value decomposition (GSVD), thus avoiding a solution of the BIE, which reduces the computational effort and thus enables the application of the very useful properties of ARMs to a wider variety of engineering problems.

Historically, the most direct way of obtaining ARMs for arbitrarily-shaped radiators has been by using the BEM, since the latter directly reveals the relationship between the sound pressure and the velocity on the radiating surface, and little effort is then needed to calculate the sound power. By discretizing the BIE, one arrives at the influence matrices relating the sound pressure and particle velocity on the radiating surface,

where ${p̃}$ and ${ṽ}$ are the pressure and normal velocity on the surface, respectively. The radiated sound power can then be calculated by using the expression

where $[G]$ is the area weighting matrix (Liu et al., 2018). The radiation resistance matrix is

and $[R]$ can be expressed as

after performing a SVD. Here the row vectors of $[Q]$ are the radiation modes and $[λ]$ is a diagonal matrix with each diagonal entry being the radiation efficiency of the corresponding radiation mode.

To reduce the effort necessary to obtain the sound power by using the BEM, Wu et al. (2014) proposed a mapped radiation mode method. That method takes advantage of the spherical harmonic functions, which when weighted and superimposed, can represent the solution of the Helmholtz equation. The sound pressure associated with the spherical harmonics is expressed as

where r is the distance from a reference to point $x⇀$⁠, $θ$ and $ϕ$ are the polar angles of point $x⇀$⁠, $hl1$ is the spherical Hankel function of the first kind representing out-going waves, and $Ylt$ is the normalized spherical harmonic function of degree l and order t. The mapped ARM on an arbitrary convex shape is then

where n is the normal direction at point x. Application of the Gram–Schmidt process then yields

where ${vn}$ in Eq. (7) are the columns of $[V]$⁠, also the mapped ARMs, $[E]$ is an orthogonalized unitary matrix, and $[U]$ is an upper triangular matrix. Therefore, any velocity distribution ${v}$ can be expressed as a combination of the mapped ARMs, i.e.,

where the vector containing the coefficients, ${α}$⁠, can be calculated as $[U][E]H{v}$⁠. The sound power radiated by the velocity distribution {v} is then easily evaluated as

where $wn$ is the sound power of each mapped ARM, which equals the sound power radiated by each spherical harmonic on the unit sphere, since the spherical harmonics radiate sound power independently. This method was recently applied to a sound-power-based structural optimization of an air compressor (Zhang et al., 2017). Note, however, that the pressure and velocity vectors associated with the mapped ARMs are not orthogonal. Hence, the mapped ARMs are not true ARMs, by definition.

By applying Eqs. (5) and (6), a transfer matrix $[T]$ can be obtained that expresses the relation

where ${v1}$ is the velocity on the unit sphere, and ${v2}$ is the mapped velocity on an arbitrary, convex radiating surface. Similarly, the radiation resistance matrix $[Z1]$ of the unit sphere can be obtained. Since it is known that the unit sphere and the arbitrary surface radiate the same sound power, a relation between $[Z1]$ and the radiation resistance matrix $[Z2]$ of the arbitrary surface can be obtained:

Since the latter relation holds for any arbitrary ${v1}$⁠, it follows that

a proof of which can be found in the  Appendix. Further, $[Z1]$ can be expressed as

where

Here, $[φ]$ contains the ARMs of the unit sphere, which can be easily populated by using the spherical harmonics with appropriate scalings. Then rewrite Eq. (12) as

and after denoting $[K]=[E][T]−1$⁠, Eq. (15) can be expressed in a compact form as

Next, it is possible to apply GSVD (Golub and Van Loan, 1996) to $[E]$ and $[T]$ to obtain

and

Here, $[M]$ and $[N]$ are unitary matrices. The non-negative diagonal matrices $[Σ1]$ and $[Σ2]$ satisfy the relation

It is important here to distinguish GSVD from “normal” SVD. After substituting Eqs. (17) and (18) into $[K]$⁠, we obtain

Thus, after substituting Eq. (20) into Eq. (16), we obtain

After denoting $[Δ]=([Σ1][Σ2]−1)H([Σ1][Σ2]−1)$⁠, $[Z2]$ can now be expressed in a SVD form as

and here, $[Δ]$ is a non-negative diagonal matrix. Thus, the column vectors in $[N]$ are the ARMs, and the self-radiation efficiencies are proportional to the elements in $[Δ]$⁠. It is also important to note that it may be necessary to apply a weighted form of GSVD to Eq. (22) (Abdi, 2007) if the surface cannot be given a uniform element size or when the sizes of the elements are not sufficiently small. In that case, the weighted version of Eq. (22) is simply

where

and where $[S]$ is a diagonal matrix with the corresponding element sizes for the entries in $[N]$⁠.

In this article, a new procedure for obtaining the ARMs for arbitrarily-shaped, but convex sources, has been presented. The procedure entails the projection of spherical harmonics onto an enclosing surface, followed by the application of GSVD. This procedure avoids the need to solve the BIE, thus reducing the computational effort required to apply ARMs to radiation problems.

To justify the operation on Eq. (11) to obtain Eq. (12), the relation

with $[A]=[A]H$ and $[B]=[B]H$ must hold for any arbitrary vector ${x}$⁠, with the result $[A]=[B]$⁠. Denote $aij$ as the element at the ith row and jth column in $[A]$⁠, and $bij$ as the element at the ith row and jth column in $[B]$⁠. Let ${x}=[0,…,1,0,…]T$ with 1 as the ith element, so that we obtain

Then, let ${x}=[0,…,1,0,…,1,…]T$ with 1 at the ith and jth elements, so that we obtain

Then, substitute Eq. (A2) into Eq. (A3), and take advantage of the fact that $[A]=[A]H$ and $[B]=[B]H$⁠, to finally obtain

and hence, $[A]=[B]$⁠.