COMSOL 6.4 - Baffled Piston Radiation

An axisymmetric model of a rigid piston in an infinite baffle is used to exemplify the Acoustics Module’s Exterior Field Calculation feature. The radiation results obtained with COMSOL Multiphysics are compared with analytical results for the on-axis radiation pattern (as a function of distance), the spatial far-field response, and the total radiated power. The transition from near field to far field is smooth and continuous.

Model Definition

The model consists of a 2D axisymmetric piston recessed in a rigid baffle and vibrating in air. The piston radius is a = 0.1 m. It will be assumed to move at a constant velocity of u0 = 1 m/s

Figure 1: Geometry of the piston in an infinite baffle model. The colors represent the acoustic pressure in the air surrounding the piston at f = 10 kHz.

The piston will be excited by a range of frequencies going from 10 Hz to 20 kHz. The model parameters are given in Table 1.

Table 1: model Parameters.
Name	Expression	Description
u0	1[m/s]	Piston velocity amplitude
a	0.1[m]	Piston radius
Rmodel	0.2[m]	Domain radius
f0	20000[Hz]	Maximum frequency
c0	343[m/s]	Speed of sound
lam0	c0/f0	Wavelength at f0
ka0	2pia/lam0	k0*a
R0	ka0*a/2	Rayleigh distance at f0
Rfar	20*Rmodel	Far-field evaluation distance
Scale	a/lam0	Help variable

As described in Jacobsen and others (Ref. 2), theoretically, the piston can be modeled as a group of uncorrelated monopoles. Supposing linear superposition, the pressure radiated by the piston can be calculated at any point of interest as an integration of the power radiated by a monopole over the surface of the piston:

(1)

Here, ω is the angular frequency, ρ the density, k the wave number, S the surface of the piston, and x the distance between the point of interest and the running position on the piston. This is a special case of the Rayleigh integral.

The Exterior Field Calculation feature uses the full Kirchhoff–Helmholtz integral, which is valid at any point outside the computational domain. When applied to a flat surface, it reduces to the Rayleigh integral.

Far Field

As explained in Jacobsen and others (Ref. 2), the pressure in the far-field approximation can be written using Bessel functions J0(z) and J1(z), giving

(2)

where θ is the polar angle and r the distance from the center of the piston.

Near Field

In the near-field approximation, it is not possible to derive an analytical expression for the pressure. However, the force exerted on the piston (or the radiation impedance) can be calculated and is presented in Ref. 1. In the analytical force expression, there is a need to introduce Bessel and Struve functions. The Struve function can be approximated by (see Ref. 3)

(3)

(The Struve function is also often denoted H1.) The force on the piston can then be written as

(4)

which by division with the velocity and area gives the radiation impedance. The analytical expressions given above are implemented using variable definitions and integration coupling operators.

Results and Discussion

Figure 2 shows the computed COMSOL on-axis pressure as well as the analytical results, the asymptotes M and MM, and the far-field limit. First, it is noticeable that the COMSOL and analytical results are really similar.

Figure 2: On-axis pressure as a function of z/R0 (logarithmic axis).

From a theoretical aspect, the pressure zeros in Figure 2 can be explained with the so-called monopole model. All the monopoles will interfere destructively and cancel out, resulting in the pressure being zero. According to Blackstock (Ref. 1), the number of zeros, n, depends on the value of a/λ. If this ratio is an integer, then n = a/λ and the first zero is at the piston itself. If it is not an integer, then

(where

means “the integer part of”) and the first zero happens in front of the piston.

The asymptotes are also worth noticing. There is good agreement between the COMSOL model and the far-field limit. The asymptotes M and MM represent the following functions (see Ref. 1):

(5)

Figure 3: Far-field radiation pattern at f = 10 kHz.

In Figure 3, the radiation pattern of the actual far field, further away than the Rayleigh distance R0, is plotted at f = 10 kHz. As expected, it shows strong directivity at this frequency; there is a clear focus in the on-axial direction. However, there are still some sidelobes present in other directions. It should be noted that at low frequencies, the piston can behave as a strictly omnidirectional source, because, according to the representation by monopoles, they would all move in phase. Figure 3 also shows good agreement between the simulated and analytical solutions.

In Figure 4, the analytical total radiated power is compared with the result of the COMSOL simulation. They have really similar results from low to high frequencies. It goes from no radiated power to a maximum for the critical frequency before decreasing again to reach a steady state.

Finally, Figure 5 represents the acoustic field created by the piston, both in the computational domain and exterior to the domain. This is achieved using the Exterior Field Calculation feature combined with visualization in a Grid 2D dataset. The feature allows the calculation and the visualization of the pressure field outside the computational domain at any distance including amplitude and phase.

Figure 4: Total radiated power as a function of frequency.

Figure 5: Pressure field near the piston surface.

References

1. D. Blackstock, “Fundamentals of physical acoustics,” Wiley-interscience, 2000.

2. F. Jacobsen, T. Poulsen, J.H. Rindel, A.C. Gade, and M. Ohlrich, Fundamentals of Acoustics and Noise Control, Technical University of Denmark, Department of Electrical Engineering, 2008.

3. R.M. Aerts and A.J.E.M. Janssen, “Efficient approximation of the Struve functions Hn occurring in the calculation of sound radiation quantities,” J. Acoust. Soc. Am., vol. 140, pp. 4154–4160, 2016.

Acoustics_Module/Verification_Examples/baffled_piston_radiation

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