Sounding mechanism is numerically analyzed to elucidate physical processes in air-reed instruments. As an example, compressible large-eddy simulations (LES) are performed on both two and three dimensional ocarina. Since, among various acoustic instruments, ocarina is known as a combined system consisting of an edge-tone and a Helmholtz resonator, our analysis is mainly devoted to the resonant dynamics in the cavity. We focused on oscillation frequencies when we blow the instruments with various velocities.
Elucidation of acoustical mechanism of air-reed instruments is a long standing problem in the field of musical acoustics [1]. The major difficulty of numerical calculations of an air-reed instrument is in strong and complex interactions between sound field and air flow dynamics [2], which is hardly reproduced by hybrid methods [3] normally used for analysis of aero-acoustic noises [4,5].
There are two types of air-reed instruments, which are different in acoustics mechanism: in the group of flute, recorder, organ pipe, the pitch of an excited sound is determined by the length of an air column, i.e., resonance of the air column; on the other hand, it is considered that sound of an ocarina is produced by Helmholtz resonance where the pitch is determined by resonance of the entire cavity and the placement of each hole on an ocarina is almost irrelevant. It should be noted that the Helmholtz resonance is based on an elastic property of the air, not the sound propagation. This is another reason that the usual hybrid model consisting of fluid mechanics and sound propagation cannot reproduce the oscillating dynamics in an ocarina. Thus, when we study the acoustic mechanism of the ocarina, the whole calculation of a compressible fluid mechanics for the air-reed and the resonator is essentially required. Taking an ocarina as a model system, we investigate how the Helmholtz resonator is excited by the edge tone [6] created by a jet flow collided to an edge of aperture of the cavity. The ocarina is a relevant model for our purpose, since it creates a clear tone and is enough small in size to calculate with present computational resources.
In addition to the sound propagation, the oscillation in an ocarina is described by an elastic dynamics of air. In the present calculation, the LES solver coodles in OpenFOAM 1.5 is used to directly solve the compressible Navier Stokes equation in order that both the radiated sound and flow dynamics are simultaneously reproduced. This paper is organized as follows. In Section 2, by the use of two dimensional ocarina model, we explore suitable playing conditions with changing the velocity such that a well sustained sound vibration is excited in the cavity. In Section 3, a reproduced sound by three dimensional ocarina model is analyzed. Based on these analysis, a conclusion is given in Section 4.
At first, a two-dimensional air-reed instrument is configured. The geometry studied in this section is shown in Fig. 1. The aperture of this instrument is 0.5 cm, and the area of the cavity is 1.635 cm 2 . The maximum length in the cavity is 21.5 mm.
Numerical calculations are performed by OpenFOAM-1.5. In order to solve in two-dimension, front and back planes in z-direction are set to empty type. The following boundary conditions are introduced: the fluid velocity and the pressure gradient are set to zero on walls (fixedValue and zeroGradient are used); inletOutlet walls with U = (0, 0, 0) m/s and p = 10 5 Pa are introduced at boundaries of the open part.
We are interested in relation between the pitch of sound vibration and the jet velocity. The excitation of over-tone will be prohibited if the sound vibration in ocarina mainly originates from the Helmholtz resonance. Then, the pitch of fundamental only depends on the volume of the cavity but irrespective to its shape.
The resonance frequency of our two-dimensional model is estimated. We approximate that the effective length L of open end correction is in proportion to the length of aperture l and the proportional constant is α. Then the effective mass m and the spring constant K for Helmholtz resonator are given by
where ρ, c, and s are the density of the air, the sound velocity, and the size of the cavity, respectively.
The resonant frequency f 0 can be written as follows:
Thus, the frequency of the two-dimensional Helmholtz resonance is
Next, let’s estimate the f 2D for our 2D model. Parameters of the model are
By the use of the above parameters, the value of Eq. ( 3) is estimated to
(5)
Numerically obtained pressure values and their normalized spectrum are shown in Fig. 2. While the oscillation is very small and unstable for the blown velocity 20 m/s, the oscillations are sufficiently grown for velocities 30 m/s and 40 m/s. In the figures of the normalized spectrum, almost no higher harmonics are observed in (b’) and (c’). The single oscillation without harmonics is one of the properties of the Helmholtz resonance.
The frequency for the most prominent peak in each Fourier transformed data of the pressure are shown in Fig. 3. As the velocity becomes larger, the frequency slightly increases. This shows a broad resonance which is often observed in ocarina, and the frequency as the Helmholtz resonance is about 2.5 × 10 3 Hz. It is very difficult to obtain only from theoretical considerations with analogies from those simple Helmholtz instruments found in textbooks [1]. If we use this resonant frequency 2.5 × 10 3 Hz for the for
This content is AI-processed based on open access ArXiv data.