A method to calculate acoustic radiation modes based on spheroidal wave functions

As early as 1990, Borgiotti (1990) developed a method that used singular value decomposition (SVD) to represent the sound power radiated by a vibrating surface. That procedure was later referred to as the acoustic radiation mode (ARM) method, which has since been widely used in noise control applications. Naghshineh and Koopmann (1993) introduced a strategy based on the ARM method to actively decrease the radiated power of a vibrating beam by minimizing the coupling between the structural surface velocity and the acoustic basis functions that radiate sound efficiently. The design of quiet structures by employing the ARM method is also discussed extensively in the book by Koopmann and Fahnline (1997); for example, more recently, Liu et al. (2017) used ARMs to guide the design of an engine oil pan by finding the surface vibration pattern that corresponded to significant sound power radiation; then the appropriate structural mode was selected and suppressed through structural modifications.

ARMs are usually obtained by applying SVD to the radiation resistance matrix of the radiating surface. The resistance matrix for a surface with an irregular geometry is usually calculated by using the boundary element method (BEM) (Wu et al., 2013; Wu et al., 2014). However, for radiating surfaces with simple geometries, such as spheres, the ARMs can be found analytically as solutions of the Helmholtz equation (Cunefare et al., 2001). Additionally, the ARMs for baffled plates can be obtained from the Rayleigh integral by representing sound power radiation with the method of elementary radiators (Fahy and Gardonio, 2007; Liu et al., 2018). Alternatively, for baffled plates, the ARMs can be obtained through mapping of structural modes to radiation modes (Fahy and Gardonio, 2007). It is generally true that obtaining ARMs analytically is much more efficient than using BEM, due to the large number of surface elements required to ensure a converged solution, especially for high frequencies.

To predict the sound power generated by vibration of an arbitrarily shaped convex surface, a mapped ARM method was proposed by Wu et al. (2014). That method is based on mapping the ARMs of spherical harmonic functions onto an arbitrarily shaped convex surface, so that the “contribution” of each of the spherical harmonics can be obtained numerically. Then the power radiated by the enclosing surface can be solved for on the surface of the sphere by using the contribution coefficients of each of the spherical harmonics. However, note that the mapped ARMs on the enclosing surface are not the actual ARMs of that surface. Recently, a new method was developed to calculate the ARMs for an arbitrarily shaped convex surface by using these mapped spherical harmonics (Liu et al., 2019). In that method, the resistance matrix of the enclosing surface can be obtained analytically with the combination of spherical harmonics and a mapping matrix that relates the sound pressure on a sphere to that on the enclosing surface. Then the ARMs of the enclosing surface can be calculated by using generalized singular value decomposition (GSVD). Since that method relies on a mapping from a sphere to the targeted surface, it is most suitable for a structure with an aspect ratio close to 1, i.e., for a surface that is similar to a sphere.

In this paper, a similar GSVD-based method is proposed to obtain the ARMs on an arbitrarily shaped surface by using the mapped spheroidal harmonic functions instead of mapped spherical harmonics. In principle, by choosing a spheroid with an appropriate aspect ratio as the reference surface for sound pressure mapping, the proposed method can be efficient for surfaces with a wide range of aspect ratios. Spheroidal wave functions are solutions of Helmholtz equations in a spheroidal coordinate system (Flammer, 2014), and they are widely used in scattering-related problems (Boisvert and Van Buren, 2002; Chertock, 1961). Recently, the ARMs and spheroidal wave functions have also been used in the area of active noise control to solve sound radiation problems (Maury and Bravo, 2015; Maury and Elliott, 2005). One important difference between spherical harmonic functions and spheroidal harmonic functions is that the acoustic potential modes of the latter are not orthogonal to each other on the spheroid surface; instead, they are orthogonal to corresponding velocity modes on the surface of a spheroid, which means that the spheroidal harmonics are not orthogonal, but they nevertheless radiate acoustic power independently (Skudrzyk, 2012). The spheroidal harmonic functions are actually orthogonal on a spheroid surface with a different weighting function, which enables one to calculate the contribution of each mode with a similar approach. These properties of spheroidal harmonics make the method proposed here more complicated mathematically than the previous work of Liu et al. (2019).

In Sec. 2.1, derivations of the spheroidal wave functions in the prolate spheroidal coordinate system are given. The orthogonal properties of spheroidal wave functions are presented in Sec. 2.2. In Sec. 2.3, a method to find acoustic radiation modes with GSVD is described. Conclusions are drawn in Sec. 3.

An arbitrary point in a prolate spheroidal coordinate system, which is an orthogonal coordinate system, can be described using three variables (⁠$ξ,η,φ$⁠). The mapping to Cartesian coordinates is expressed as

where d is the distance between the two foci (a constant for a particular coordinate system). The two distances r₁ and r₂ between an arbitrary point in the spheroidal system and the two foci can be expressed as

On a spheroid, the variable ξ is a constant. The variable η, which is equivalent to the polar angle, θ, in the spherical coordinate system, is defined as

The scale factors between the two coordinate systems are

These factors are used to convert measures between the Cartesian coordinate system and the spheroidal coordinate system. For example, an area on the surface of a spheroid can be calculated by using the relation

The directional derivative normal to the surface of a spheroid is

The Helmholtz equation in prolate spheroidal coordinates can be expressed as

where $h=ωd/2c$⁠, ω is the angular frequency, c is the sound speed in the medium, and $Ψ$ is the acoustic potential. The Helmholtz equation in prolate spheroidal coordinates can be solved by separation of variables, and the eigenfunctions that satisfy the radiation boundary condition can be expressed as follows (Flammer, 2014):

where

is the spheroidal radial function of the fourth kind representing outgoing waves. The function $Sml(h,η)$ is the spheroidal angle function of the first kind. And in the $φ$ direction, the eigenfunctions are simply $sin mφ$ and $cos mφ$⁠. The acoustic potential on the surface of a spheroid can then be expressed as

where

and

where A_ml and B_ml are the coefficients of the different spheroidal wave components. When assuming an $eiωt$ time convention, where t is the time variable, the acoustic pressure can be expressed as

and the normal acoustic velocity on the surface of a spheroid is

Equation (10) can be substituted into Eq. (14) to obtain the following:

where $R′ml(4)$ is the derivative of $Rml(4)$ with respect to ξ.

According to the Sturm–Liouville theory, the angle functions, $Sml(h,η)$⁠, form an orthogonal set on the interval $η∈[−1,1]$⁠, so that

where N_ml is a normalization factor for the prolate angular functions. The normalization constant, N_ml, can be calculated with the Meixner–Schäfke scheme (Meixner and Schäfke, 2013), which results in

The orthogonality expressed in Eq. (16) on the interval $[−1,1]$ is similar to that of associated Legendre polynomials, since the spheroidal angular functions can be represented as a series of associated Legendre functions.

However, it is important to note that spheroidal wave functions do not have the same orthogonal properties on a spheroid as those of spherical wave functions on a sphere. It is well-known that spherical harmonics are orthogonal on the surface of a sphere (Abramowitz and Stegun, 1948). For example, for a unit sphere

where $Ylm(θ,φ)$ are the spherical harmonics of degree l and order m, θ and $φ$ are the polar angle and azimuthal angle, respectively, and ^* denotes the complex conjugate. Note that $sin θ$ is the weighting (or the Jacobian) for the surface area measure of a differential element $dθdφ$ on the unit sphere. A similar rule cannot be applied to spheroidal functions. The functions

are not orthogonal over the surface of a spheroid with a surface area Jacobian (or weighting function) of

Instead, Eq. (19) is orthogonal over the $η−φ$ spheroid surface with a weighting function of 1. For example, the $cos φ$ modes in Eq. (11) satisfy the relation

where $ϵm=1$ if m = 0 and $ϵm=2$ if m > 0. The same property applies to the $sin φ$ modes in Eq. (12).

For convenience in the following derivations, a new set of functions is defined in a way similar to the treatment of spherical harmonics (Flammer, 2014): i.e.,

A set of normalized complex spheroidal wave functions can then be defined as

Because of Eq. (22), the normalized complex spheroidal wave functions satisfy the relation

The spheroidal functions $Glm$ are then orthonormal on a spheroid with respect to a unit weighting function: i.e.,

Note that these spheroidal functions are not real surface “harmonics,” since they are not orthogonal on the spheroid surface with respect to a standard surface area integral. However, they help to simplify the expressions of a number of related equations in a way similar to the treatment of spherical harmonics.

By using $Glm$⁠, the acoustic potential can now be expressed as

and the corresponding velocity can then be expressed as

Now, consider a single acoustic potential mode

It is, as mentioned earlier, not orthogonal to other modes on the surface of a spheroid with respect to standard surface area integral but, instead, with respect to the integral with a unit weighting function. However, the acoustic pressure,

due to this single spheroidal potential mode is orthogonal to the corresponding velocities of other potential modes on the surface of a spheroid with respect to the standard surface area integral. The corresponding velocity of one potential mode, $Ψ¯ml$⁠, is

where the leading term, $[(ξ2−1)/(ξ2−η2)]1/2$⁠, turns out to be the reciprocal of the Jacobian of a spheroidal surface [Eq. (20)]. This is the reason why the sound pressure of a mode is orthogonal to the velocities of the other modes with the standard surface area integral on a spheroid. The radiated power on the surface on a spheroid with constant $ξ=ξ0$ can then be expressed as

Because of the orthogonality property described above, the coupling of different modes makes no contribution to the power, and thus,

The radiated power expression shows that although the spheroidal wave functions, i.e., the acoustic potential modes, are not orthogonal on the spheroid surface in the sense of a surface area integral, each mode radiates power independently. This, again, results from the fact that the $(ξ2−η2)1/2$ term produced in the directional derivative [Eq. (27)] cancels out the non-constant scale factor $1/(ξ2−η2)1/2$ on the surface of a spheroid. It should be noted that Eq. (30) is a complex equivalent of the “Chertock weighted mode,” which was first used by Chertock to evaluate the radiation of spheroidal structures (Chertock, 1961).

Based on Eq. (32), the total sound power radiated by any velocity distribution can be calculated as

If the velocity distribution is given on the surface of a spheroid, the mode participation coefficient can be calculated analytically with

A generalized GSVD method was used by Liu et al. (2019) to obtain the acoustic radiation modes on an arbitrarily shaped convex surface by using spherical harmonic functions. A similar approach can be followed for spheroidal wave functions. However, due to the special orthogonality of spheroidal wave functions, the derivation is more complicated than in the spherical case. To derive the mapped ARMs, it is convenient to use vector forms in the following steps. Consider a spheroid surface defined by a constant ξ₀.

Given a velocity distribution, v₁, the mode participation coefficient defined in Eq. (34) can be represented in vector form on the spheroid surface as

where the matrix $[H1]$ is the area weighting matrix of the spheroid, which depends only on the geometry of the surface mesh, $[G¯]$ is a vector described by the function $(2/d)[1/R′ml(4)(h,ξ0)](1/ξ02−1)Glm(h,η,φ)*$⁠, and ${v1}$ is the normal velocity vector on the surface nodes of the spheroid. After substituting Eq. (35) into Eq. (33), the acoustic power radiation of the spheroid can then be represented in vector form as

where $[Σ1]$ is a diagonal matrix that can be calculated with (d/4)$ωρ(1−ξ02)Im[Rml(4)R′ml(4)*]$⁠, and $[ϕ]$ is a matrix defined with the equation

It should be noted that although $[ϕ]{v1}$ gives the participation coefficient of the velocity mode associated with the potential mode, $[ϕ]$ is not the well-known acoustic radiation mode, since it is not orthogonal with respect to the surface weighting (surface Jacobian) on the spheroidal surface.

The mapped velocity distribution on an arbitrarily shaped surface can be calculated with a propagation matrix, [T]; i.e.,

where the matrix [T] is

and where

and

Here $[z1]$ and $[z2]$ are acoustic impedance matrices and can be solved analytically on the spheroidal surface and on the arbitrarily shaped surface, respectively, with the relation

where $n→$ is a unit vector normal to the corresponding surface. Also, $[Tp]$ is the propagation matrix that can be calculated with the pressure modes in Eq. (29) associated with the spheroidal potential modes. A direct mapping of the velocity [e.g., in the previous work of Liu et al. (2019)] is not possible, since the velocity modes associated with acoustic potential modes are not real spheroidal wave functions.

Since both surfaces radiate the same power, we obtain the following:

where $[Z1]$ and $[Z2]$ are resistance matrices on the spheroid surface and on the arbitrarily shaped enclosing surface, respectively. And since that relation holds for any arbitrary $v1$ distribution, we can also obtain the following (Liu et al., 2019):

After using the velocity modes associated with the acoustic potential modes on the spheroid, the resistance matrix can be expressed according to Eq. (36) as

After substituting Eq. (45) into Eq. (44), we can obtain

where

Since Eqs. (44) and (45) hold [see Eqs. (12) and (13) in Liu et al. (2019)], it is then possible to apply the same GSVD approach to $[E]$ and $[T]$ to obtain

and

where $[M], [N]$⁠, and $[X]$ are unitary matrices, and $[σ1]$ and $[σ2]$ are nonnegative diagonal matrices. By using Eqs. (48) and (49), we can obtain

After substituting Eq. (50) into Eq. (46), we can further obtain

and, after denoting $[Δ]=([σ1][σ2]−1)H([σ1][σ2]−1)H$⁠, which is nonnegative and diagonal, the resistance matrix $[Z2]$ on the arbitrarily shaped convex surface can then be expressed as

where the columns in $[N]$ are the ARMs on the enclosing surface if the surface elements of the arbitrarily shaped surface have uniform area. Otherwise, a weighted version of Eq. (52) should be used, which can be written as

where the rows of $[Ñ]$ are acoustic radiation modes that are orthogonal with respect to the surface weighting matrix $[H2]$ associated with a non-uniform mesh grid used on the arbitrarily shaped surface for surface integration calculation. The detailed derivation of the latter result can be found in the  Appendix.

In this article, a new method of obtaining the radiated power and the ARMs for an arbitrarily shaped convex surface has been introduced. The method maps the spheroidal wave functions on a spheroid surface onto an enclosing convex surface. The sound power can then be calculated with the mode participation coefficients of the velocity modes associated with the acoustic potential modes, and the ARMs can be obtained by using GSVD. This approach is potentially more efficient for radiation problems involving spheroid-like surfaces than the previous spherical harmonic approach.

For weighted version, the definitions of [T] and [E] should be changed to [H₂][T] and [E][H₁]. We use $[H]$ to represent $[H2]$ in the following derivation to improve the readability. First, define

It should be noted that the propagation matrix $[T]$ is square in this article, which requires an equal number of nodes on the spheroid surface and on the enclosing surface. After applying GSVD to $[E]$ and $[T̃]$⁠, we obtain

and

It should be noted that $[PE]$⁠, $[PT]$⁠, and $[X̃]$ are unitary matrices. We can then obtain the expression of $[E][T]−1$ with Eqs. (A2) and (A3) as

where

Then the resistance matrix can be represented as

where

is a nonnegative diagonal matrix. Since the element shape matrix $[H]$ is symmetric and positive definite, we can use Eq. (A5) to arrive at

which suggests that the rows of $[Ñ]$ are orthogonal with respect to the weighting matrix $[H]$⁠. The rows of matrix $[Ñ]$ are then the weighted acoustic radiation mode according to Eqs. (A6) and (A8).