CC BY
48Electronic Journal «Technical Acoustics» http://www .ejta.org
2011, 11 Jalil Olia, Vahid Afshinmehr
Iran University of Science and Technology, Tehran, School of Architecture and Environmental Design, e-mail: [email protected], [email protected]
Newton's method for numerical analysis of the eigenvalue equation for acoustic problem in the rectangular room
Received 21.06.2011, published 29.09.2011
Numerical procedure for finding the acoustic eigenvalues in the rectangular room with arbitrary (uniform) wall impedances is developed. This numerical procedure is Newton’s method. The eigenvalues are found by increasing the impedances of each wall pair in small increments up to the terminal impedances. These eigenvalues then are used to find the reverberation times of the modes in the room with the floor lined with sound absorbing material of known acoustic impedance. Validity of the Sabine’s model for reverberation time is discussed. Keywords: acoustic eigenvalues, Newton’s method, room acoustics
INTRODUCTION
Since the earliest discoveries of Sabine [1] and until today, the reverberation time is considered to be one of the most important factors in determining the sound quality of a room. When the sound decay is linear, on a decibel scale, the slope is related to the reverberation time, which is then the only parameter necessary to describe the sound decay process. In practical rooms, such as concert halls, theatres, classrooms, meeting rooms and industrial rooms, considering the process of sound decay to be linear is quite often only a crude approximation to the actual conditions. In such rooms, the sound decay is usually non-linear and peculiarities such as double slopes, stepped decays and curvature are often reported.
Sabine [1] considered the room as a lumped system, where the sound energy was assumed spatially uniform at any instant of time during the decay process. The transient response during the decay determined by formulating an overall energy balance on the room, in which the sound power loss at the surfaces was related to the rate of change of the internal sound energy.
Morse and Bolt [2] established a complete formulation for sound waves in the rectangular room in terms of the normal modes. They established an eigenvalue equation and obtained the steady state and transient solutions for large wall impedances (hard walls). They used of these classical solutions to study low-modal-density acoustical phenomena in small rooms.
In cases where there is a non-uniform impedance distribution over each wall, the separation of variables for finding eigenvalues is not possible and the method cannot be applied. Thus, a procedure for finding the eigenvalues is a necessary complement for the application of the method of Morse and Bolt.
In this paper we wrote a velocity potential for the standing waves. We obtained an equation that gives the set of eigenvalues for each wall pair. Then, we use numerical method to finding the eigenvalues. The numerical method is Newton’s method. After finding the acoustic eigenvalues numerically in a rectangular room, we discuss the results in the light of Sabine’s theory.
PHYSICAL THEORY
Morse and Bolt [2] wrote the velocity potential for a standing wave in a rectangular room with dimensions {LX,LY ,LZ } :
y/(a;x,y,z) = D(x )E (y )F(z )eiat , (1)
where a is the driving angular frequency and the set of the functions are defined by the following equation:
{D(x), E(y), F(z)} = \ cosh
in.
X -6
x tx
cosh
Lv y
cosh
inz
~LZ
In the case of damped standing waves, the wave number vector k =
(2)
. The
sizes of the room are described by three real numbers{LX,LY ,LZ }. The ratio of the sound
pressure p to normal particle velocity vn (normal to the surface and ingoing to the wall
surface) equals the impedance of the surface. For the walls normal to the X direction this ratio is given by
P
dy
dt
dv dx
Using the wave function given by (1) and it’s eigenfunctions (2) we obtain
(3)
Zx =-^ coth X.
inx
L
X -6
X tx
'X
(4)
The specific acoustic impedance for the wall at x = 0 which substituted into Eq. (4) gives dx =-coth-1 Xx . (5)
_ Ax _
The specific acoustic impedance for the wall at x is
X
+ coth 1 —xx + coth 1 Zx 2 xx
_ Ax X _ _ A. x _
(6)
As discussed by Morse and Bolt [2], Eq. (6) has an infinite number of roots. There is at least one root (and not more than two roots) locates between zero and 1, another root with between 2 and 3, and so on. The different roots are distinguished by assigning to them different values of the subscript n, with n > 0 for the value of Xx. When both parallel walls
have arbitrary, however, equal impedances Zx 1 =Zx 2 and dropping the subscripts x; Eq. (6) reduces to
______________. (7)
x Ax '
Morse [3] and Morse and Bolt [2] gave plots of the real and imaginary parts of the roots of Eq. (7) as functions of the magnitude and phase angle of the specific acoustic impedance Zx. These plots are known as Morse charts. The transformation is multi-valued, with an infinite number of sheets, corresponding to the infinite number of roots. The Morse charts were given for only a few cycles of the transformation [2].
A numerical procedure may in principle be developed for finding the roots of Eq. (7). However, there are difficulties in solving this transcendental equation in the complex plane. The infinity of roots gives rise to numerical problems when solving Eq. (7), because the numerical method may jump without control from one solution to the other.
The basic rule for the positions of the branch cuts is that they must go through the branch points.
An alternative form of writing Eq. (7) is (Ax = A )
x (nx ^ . A
itanlTr l2Z (8)
Mechel [4] presents a procedure to find the modal solutions in rectangular ducts based on the direct determination of the branch points of an equation with the same basic form of Eq. (8) (See also Lippold [5]).
There is no simple numerical method for the case with one wall which has different and arbitrary impedance.
Now, we attack this problem by re-writing the defining equation, avoiding working with inverse functions. Making use of the identity
f—j . cothZ, cothZ2 +1 coth(Z, +Z 2) = Z 2Z (9)
coth Z1 + coth Z 2
with the aid of Eqs. (5) and (4), when dropping the subscripts x, Eq. (9) can be written as
x^1 ^( xZ2
Applying to Eq. (10) the identity
1 - e 2
we obtain
1 + e ~2z coth z =----------—
(11)
1 _ ^2
A
0.
(12)
Eq. (12) is an entire function. It has no branch points or branch cuts whatsoever. This form of the acoustic eigenvalue equation greatly simplifies the development of numerical solutions.
NUMERICAL PROCEDURE
In Eq. (12), the n-th root is found by increasing the impedance of the both walls, in small increments, from 21 to Z2 for the first wall of the pair. Here 21 and 22 will be called the specific terminal impedances.
We define s =C2/ 21 as a fraction of the terminal impedances. After each small increment
in the impedances of both walls, the root is found by Newton’s method, having as initial approximation the root found in the previous step.
Table 1 shows the ten eigenvalues with the smallest real parts in the x direction with LX = 1 (m) for different wave numbers k
It was observed, however, that the final solution for s depends on the step size adopted to increment the impedances.
Since, on the one hand, no solution should be attributed more than once to a root; on the other hand, no important solution should be missed by the numerical procedure. The latter difficulty was also experienced by this numerical procedure. As discussed earlier, there is the possibility of more than one root in some strips of unity width running parallel to the imaginary axis.
CONCLUSIONS
In the present we used the numerical procedures for finding the solutions of the acoustic eigenvalue equation in the rectangular room with arbitrary (uniform) wall impedances. These numerical procedures were made possible by a transformation of the original eigenvalue equation of Morse and Bolt [2] into an entire function. Our numerical procedure finds the eigenvalues using Newton’s method.
Once the eigenvalues were found, the natural frequencies and damping constants of the room modes could be obtained. Our results can be used for estimating the reverberation times of the modes. We can show that a single reverberation time, for all modes, is only supported in the rectangular room with hard walls and at the higher frequency bands, consistent with Sabine’s theory, which assumes a diffuse sound field. In the rectangular room with hard walls and at the lower frequency bands, and in the room with the floor lined with sound absorbing
material and for all frequency bands, modes with rather distinctive reverberation times may produce sound decays not always consistent with Sabine’s prediction.
Table 1. The ten eigenvalues with the smallest real parts in the x direction with LX = 1 (m) for different wave numbers k
k=2 k=10 k=20
0.0370+0.4524 i 0.7631+1.9528 i 2.9873+0.6154 i
3.0768+0.0106 i 2.6597+2.0652 i 4.8000+6.4000 i
6.2513+0.0052 i 5.6177+0.5628 i 6.0268+0.8930 i
9.4036+0.0034 i 9.0063+0.3390 i 9.1681+0.8216 i
12.5505+0.0026 i 12.2582+0.2467 i 12.3426+0.6812 i
15.6952+0.0020 i 15.4635+0.1947 i 15.5169+0.5668 i
18.8390+0.0017 i 18.6467+0.1611 i 18.6849+0.4814i
21.9821+0.0015 i 21.8177+0.1375 i 21.8437+0.4171 i
25.1248+0.0013 i 24.9813+0.1200 i 25.0053+0.3674 i
28.2672+0.0011 i 28.1398+0.1065 i 28.1602+0.3280 i
REFERENCES
1. W. C. Sabine. Collected Papers on Acoustics. Peninsula Publishing, Los Altos, CA, 1922 (republished).
2. P. M. Morse, R. H. Bolt. Sound waves in rooms. Reviews of Modern Physics, 16 (1944) 69-150.
3. P. M. Morse. The transmission of sound inside pipes. The Journal of the Acoustical Society of America, 11 (1939) 205-210.
4. F. P. Mechel. Modal solutions in rectangular ducts lined with locally reacting absorbers. Acustica, 73 (1991) 223-239.
5. R. Lippold. Numerical solution of the acoustic eigenvalue equation in the rectangular room with respect to boundaries between modes. Acustica, 83 (1997) 530-534 (in German).
