Acoustic frequency fluctuations

Abstract Acoustic frequency fluctuations of sound of music in general... 

Fluctuations, which was organized by the present author in Tokyo in 1977, that acoustic frequency fluctuations of musical sounds have in general a power spectral density which is approximately in inverse pro-... 

Acoustic vibrations should be said to be musical sounds if frequency fluctuations have Ilf spectrum."... 

There are an infinitely large variety of acoustic frequency sequences or melodies which satisfy the definition of musical sound. The all possible melodies make up a statistical set or ensemble. Existing pieces of music are members of this statistical ensemble, but still there are many, many other possible acoustic waveforms (or melodies) which can be musical sounds. Some of them are waiting for human composers and others will never been composed by human composers because they cannot be played by musical instruments or they are not so attractive. We tried to pick up these members mathematically and played them with musical synthesizers after proper arrangements made by a musician. Some of them became very nice background music. When this music was played with a synthesizer in such a way that the tempo was slightly fluctuated as 1//, it sounded milder than when it was played in a constant tempo. The 1//-fluctuated tempo simulates the music played by human players as will be described in 2. 

Fig. 1.1 The regions for transient cavitation bubbles and stable cavitation bubbles when they are defined by the shape stability of bubbles in the parameter space of ambient bubble radius (R0) and the acoustic amplitude (p ). The ultrasonic frequency is 515 kHz. The thickest line is the border between the region for stable cavitation bubbles and that for transient ones. The type of bubble pulsation has been indicated by the frequency spectrum of acoustic cavitation noise such as nf0 (periodic pulsation with the acoustic period), nfo/2 (doubled acoustic period), nf0/4 (quadrupled acoustic period), and chaotic (non-periodic pulsation). Any transient cavitation bubbles result in the broad-band noise due to the temporal fluctuation in the number of bubbles. Reprinted from Ultrasonics Sonochemistry, vol. 17, K.Yasui, T.Tuziuti, J. Lee, T.Kozuka, A.Towata, and Y. Iida, Numerical simulations of acoustic cavitation noise with the temporal fluctuation in the number of bubbles, pp. 460-472, Copyright (2010), with permission from Elsevier...

Measurements of sound velocity at ultrasonic frequencies are usually made by an acoustic interferometer. An example of this apparatus11 is shown in Fig. 2. An optically flat piezo-quartz crystal is set into oscillation by an appropriate electrical circuit, which is coupled to an accurate means of measuring electrical power consumption. A reflector, consisting of a bronze piston with an optically flat head parallel to the oscillating face of the quartz, is moved slowly towards or away from the quartz by a micrometer screw. The electrical power consumption shows successive fluctuations as the distance between quartz and reflector varies between positions of resonance and non-resonance of the gas column. Measurement of the distance between resonance positions gives a value for A/2, and if /... 

It can be concluded from the figures that the pressure fluctuation due to the impingement between the opposing streams in the SCISR has the charactistics of wide spectra and multi-frequency, and the fluctuation energy is concentrated mainly in a range of frequency no greater than 1000 Hz, with weaker fluctuation in the acoustic wave range. 

The results of power spectrum analysis show that the major fluctuation is concentrated in the range of frequencies < 1000 Hz while at higher frequency range the fluctuation has lower energy and there are some lower peaks in the acoustic wave range. 

The frequency correlation time xm corresponds to the time it takes for a single vibrator to sample all different cavity sizes. The fluctuation-dissipation theorem (144) shows that this time can be found by calculating the time for a vertically excited v = 0 vibrator to reach the minimum in v = 1. This calculation is carried out by assuming that the solvent responds as a viscoelastic continuum to the outward push of the vibrator. At early times, the solvent behaves elastically with a modulus Goo. The push of the vibrator launches sound waves (acoustic phonons) into the solvent, allowing partial expansion of the cavity. This process corresponds to a rapid, inertial solvent motion. At later times, viscous flow of the solvent allows the remaining expansion to occur. The time for this diffusive motion is related to the viscosity rj by Geo and the net force constant at the cavity... 

Here a is a dimensionless constant, 5p(R) is the density fluctuation of the medium at the position R (the center of symmetry of the benzoic acid dimer), 0)D is the Debye frequency, and N is the number of acoustic modes, cot = 7 sound k, (bk) is the Bose operator of creation (annihilation of a acoustic phonon with the wave vector k). In the localized representation we have... 

The first term is connected with isobaric entropy fluctuation, which gives a diffusive component, and the second term is connected with an adiabatic pressure fluctuation, which gives rise to a high-frequency acoustic wave. The pressure wave is an acoustic standing wave oscillating with a period of Tac = A/v. This component decays by a mechanical acoustic damping or run out effect of the wave if the number of the fringes is limited. After the complete decay of this wave, the isobaric wave appears. This wave just stays where it is and decays by the thermal diffusion process as described in Section I1.B.2. This equation may be further expanded as... 

T ight scattering in dense media is caused by fluctuations in the local " dielectric tensor c (i). In 1922 Brillouin (2) predicted that thermal acoustic phonons would lead to such fluctuations and hence to light scattering. In addition, the scattered light should be shifted in frequency because the phonons are moving. The frequency shift is given by ... 

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