This paper reports an acoustic phenomenon regarding a parametric sound source (also referred to as a parametric array): a secondary sound wave is generated from the nonlinear interaction of multiple primary sound waves with varied frequency components, particularly when two relatively moving sound sources face each other. It was found that the frequency of the secondary wave fluctuated according to the source movement and provided a theoretical explanation for this phenomenon. It is experimentally demonstrated that this frequency shift was approximately proportional to the velocity of the moving source toward the fixed source and to the driving frequency of the moving source. This phenomenon has much in common with the Doppler's effect, but its unique property is that the frequency shift depends on neither the observation position nor the source velocity toward the observer. These sound generation principles enable measurement of the velocity of slowly moving sound sources while maintaining a low-modulation frequency band and a short measurement time. This phenomenon can potentially be applied to an alternative approach for acoustic noncontact velocimetry of moving objects.

When two intense sound waves with different frequencies interact nonlinearly, secondary waves arise, whose frequencies are the sum and difference of the two waves. This phenomenon is called a parametric sound source, or parametric array, and is one of the representative nonlinear acoustic effects exhibited by intense sound waves (Bennett and Blackstock, 1975; Westervelt, 1960, 1963). Parametric arrays generate audible sound fields in a specific region by emitting a sound wave with two frequency components from a source. Here, the nonlinear mixing effect of intense ultrasound propagation causes the audible sound to be highly directional. The principle of parametric arrays has been applied to personalized audible sound sources (Gan , 2012; Yoneyama , 1983) and localized active noise control (Brooks , 2005; Tanaka and Tanaka, 2010). Similarly, some researchers investigated a parametric receiver array, an active and remote measurement method that utilizes locally engendered parametric sounds (Berktay and Shooter, 1973; Truchard, 1978, 1975). Other applications of parametric arrays, such as non-invasive medical examinations, have been reported (Averkiou , 1997; Cai , 1992; Ward , 1997; Zhang , 1996). They also utilize the nonlinear interaction of the ultrasound field at the fundamental frequency with itself, followed by the narrow directivity of its overtone field in places distant from the source. Therefore, various phenomena have been investigated for a wide range of applications. Here, we report a newly discovered and intriguing phenomenon regarding parametric arrays when two ultrasound emitters are in relative motion.

When two planar ultrasound sources face each other and emit sounds of distinctive frequencies, the sound fields interfere and produce parametric sounds of their difference frequencies (Hamilton and Blackstock, 1988). Let the difference between the sound source frequencies be α Hz. With this setting, we experimentally observed a drastic frequency shift in the difference sounds caused by one immobilized source and one moving source, depending on the moving source's velocity. Because the property of this phenomenon that magnitude of the frequency shift depended on the source velocity was similar to that with the Doppler effect, we speculated that this phenomenon was related to the Doppler effect. Nevertheless, the following two characteristics, confirmed by our qualitative observations, distinguished this phenomenon from those of the Doppler effect:

  • The shift was larger than that in the case with an audible source driven at α Hz moving at the same speed. The frequency shift was easily perceived by our very ears.

  • It was not the velocity toward an observer (microphone) but that toward the fixed source that contributed to the frequency shift. Thus, the magnitude of frequency change did not depend on the observation point.

As discussed in Sec. II, this frequency shift can be concisely explained as follows. First, the frequency of the primary wave emitted by a moving source is shifted by the source's motion (Doppler shift). Accordingly, a frequency shift is caused by the parametric sound arising from the interaction between the shifted primary wave and the wave emitted by the fixed source.

Here, the characteristic mentioned above, (A), seems to be attributed to the fact that the frequency shift of the parametric sound is proportional not to the frequency of the audible parametric sound itself but to that of the moving sound source because the Doppler shift becomes greater as the source frequency becomes higher for the same moving velocity of the source (Kinsler , 1999). Characteristic (B) is also understood by the fact that the frequency shift of the primary wave, which can be regarded as almost planar, is independent of the position of the measurement point and is determined only by the velocity component of the moving source parallel to its wavenumber vector: that is, the direction of propagation.

To the best of our knowledge, this phenomenon has not been previously reported, and we find it interesting from the viewpoint of remote velocimetry. The conventional Doppler effect has been widely utilized for this purpose, including a radar speed gun in air (García , 2013; Mao , 2012), laser Doppler velocimeter (Bilbro, 1980; Rajan , 2009; Wilson and Liu, 1980), and Doppler sonar in water (Kinsey and Whitcomb, 2007; Lhermitte and Serafin, 1984; Spindel , 1976). All of these are based on the principle that the frequency shift of a reflected wave emitted toward an object depends on the object's relative velocity to the observer. Note that a similar measuring principle works for a sound-emitting body in movement instead of a sound reflector.

The Doppler shift of the primary waves depends on the observation position because it is not an exact plane wave with an infinite width. Therefore, conventional Doppler-shift-based velocimetry assumes that the receivers capture the normal reflection of the incident wave, meaning that the receiver and emitter cannot be separated. Furthermore, it is necessary to increase the frequency of the radiated ultrasound to increase the observed frequency shift for accurate velocimetry of slowly moving objects. However, increasing the oscillation frequency can cause problems, such as increasing distance attenuation of the sound, a higher susceptibility to wave shielding, and less diffraction. From this perspective, the frequency shift of parametric sound from moving sources has beneficial features for velocity measurements. That is to say, characteristic (A) enables the larger frequency shift in received signals while keeping their frequency low, avoiding the aforementioned troubles caused by a shortened wavelength. In addition, characteristic (B) allows the configuration of a measuring system whose signal transmitter and receiver are independent and separate. Unlike conventional Doppler velocity measurements, which can only estimate the velocity component of a moving object in front of the receiver, a frequency shift of the parametric sound was observed independent of the microphone location. We consider this to be because parametric sound sources are generated in the intersection area of planar waves and do not affect the observed frequency shift, regardless of whether the microphone is inside or outside that area.

There have been several preceding studies that handled a velocimetry technique of a vibrating object where parametric sound was caused by the incident wave and the one reflected by the object (Casula and Royer, 1998; Mater , 1998). However, the parametric sound in those studies originates from a single emitter and the reflected sound is modulated by the vibration frequency of the target object. Therefore, the situation we handle, where two independent ultrasound emitters are used for estimating continuous movement of one emitter, is clearly different from that in those studies.

The remainder of this paper describes the construction of a physical model that quantitatively explains this phenomenon, along with an experimental verification of its validity. Here, we address a situation where two sound sources face each other. One was moved at a constant speed, and the other was immobilized. The generated parametric difference sound was measured utilizing a microphone placed outside the intersection of the sound fields. We performed a frequency analysis of the measured difference sound to quantitatively evaluate the frequency shift of the difference frequency compared to the case with two immobilized sources. We also observed the relationship between the frequency components of the measured sound and the velocity of the moving source to verify the hypothesis of our physical model. Thus, we demonstrated that the frequency shift of the parametric sound arising from direct radiation from the source reflected the moving velocity very accurately, regardless of the measurement position, and that the magnification of the frequency shift contributed to the remote velocimetry of a moving object at a low speed based on the measurement of a much lower frequency component than that based on the conventional Doppler effect.

The above discussions handled both the sum and difference frequency components included in the parametric sound. In our experiments, the sum frequency component becomes ultrasonic with even a higher frequency than the two primary ultrasound fields. The advantages of our method, being more likely to diffract and a high frequency-change to source-velocity ratio for a low observed frequency, are not available for the ultrasonic sum frequency component. Therefore, we focus on the difference frequency component in the following parts of the paper.

Generally, nonlinear acoustic phenomena are described utilizing nonlinear wave equations. To make it easier to deal with, a method of successive approximation is employed for cases with weak nonlinearity, where the sound field is represented as the sum of the solution of a linearized form of the wave equation (i.e., primary wave) and the components caused by nonlinear effects (i.e., secondary wave). In this method, the secondary wave is modeled as a solution to a linear wave equation whose source term is assumed to be generated by the primary wave field. (Note that the source term is not identical to the primary wave field itself.)

Parametric sound waves that occur when two planar waves progress in different directions have been discussed in previous studies. An outline of this discussion is provided below: Let p ( r , t ) be a first-order term of sound pressure, which is equivalent to what is called the “primary wave” in this and related preceding studies, where r and t denote the observation point and time, respectively. Suppose that we have two planar sound-emitting sources, and the primary wave field p ( r , t ) can be expressed as a superposition of two sinusoidal plane waves with angular frequency ωi, wavenumber vector k i and amplitude p 0 i, where i = 1, 2 denotes the source index:
p = p 01 sin ( ω 1 t k 1 · r ) + p 02 sin ( ω 2 t k 2 · r ) .
(1)
Next, let θ be the angle between the two wave directions. Subsequently, θ satisfies the following relationship with the wavenumber vectors:
cos θ = k 1 · k 2 k 1 k 2 .
(2)
These equations indicate that when the amplitude of the sound pressure of the primary wave, represented by Eq. (1) is high, the nonlinearity of the medium is not negligible, and a parametric wave is generated having various frequency components. Among these frequency components, we focus on secondary waves whose frequencies are the sum (referred to as the sum sound, indicated by p + with angular frequency ω +) and difference (referred to as the difference sound, indicated by p with angular frequency ω ) between the two. In this case, sound pressure p ± of those frequency components satisfies the wave equation ( ω ± : = ω 1 ± ω 2 ) (Hamilton and Blackstock, 1988),
( 2 2 t 2 ) p ± = ( ω ± 2 p 01 p 02 ρ 0 c 0 4 ) β ± ( θ ) cos { ω ± t ( k 1 ± k 2 ) · r } ,
(3)
where
β ± ( θ ) = cos θ ± 4 ω 1 ω 2 ω ± 2 sin 4 ( θ 2 ) + B 2 A .
(4)
Here, B/2A is called nonlinear parameter and is an inherent value that depends on the medium. The value of B/2A is 0.2 for air. As expressed in Eq. (3), the frequencies of the virtual source terms on the right-hand side of Eq. (3) are determined only by the angular frequencies of the primary waves ω 1 , ω 2. It is noteworthy that Eq. (4) reveals that these frequencies are not governed by the propagation directions of the primary waves, and only the amplitude of the parametric sounds is affected by the primary wave directions. Equation (4) indicates that the second term is dominant for difference sound for the case where ω 1 , ω 2 > | ω 1 ω 2 |, as in our experiments. In this case, the amplitude of the virtual source term is almost proportional to sin4(θ/2) and is the maximum at θ = 180 °.
In the above discussion, we consider the case of plane waves with infinite widths. In practice, the actual width of the sound beam is limited, as was the case in our experiments. However, we presume that the successive approximation method is valid, even for a wave with a finite width, and that the only difference between them is where the secondary sound sources are generated. In the finite planar wave model, the region where the sources arise is mostly confined to the intersection of the beams. For these cases, a more general modeling equation was proposed (Aanonsen , 1984). By applying the successive approximation, as in many related works (Eckart, 1948; Kamakura , 1992), the governing equations of secondary (parametric) pressure wave field ps can be described by
( 2 2 t 2 ) p s = β ρ 0 c 0 4 2 p p 2 t 2 ( 2 + 2 t 2 ) L ,
(5)
where
L : = ρ 0 2 u p · u p p p 2 2 ρ 0 c 0 2
(6)
denotes the Lagrangian of the sound field and u p and pp indicate the primary wave's particle velocity and pressure fields, β = 1 + ( B / 2 A ), respectively. These equations reveal that the source terms of the secondary waves comprise the squared waveforms of the primary waves, which corresponds to the fact that the intersection regions of the two primary sound beams yield the sources of the sum and difference in the frequency components.

Here, it should be noted that what the nonlinear interactions of primary waves yield here is not the secondary sound field itself, but the source terms on the right-hand side of Eq. (5), which also is a linear wave equation that governs the secondary sound field. Considering the existence of that source terms, we expect that the secondary wave from the localized virtual sources is also observed outside the crossing area of the beams and that it has the same frequency components as those derived from the nonlinear mixing effect of the primary waves. This is owing to the fact that the resulting secondary field, ps, is given as spatiotemporal convolution of the source term in Eq. (5) and the Green's function that is determined by the linear operator ( 2 ( 2 / t 2 ) ) and the boundary/initial condition of the ultrasound field. Thus, when two sound sources with limited width generate a parametric sound, it is expected that the frequencies are the sum and difference of those of the primary sound beams in their intersection region. This study deals only with the case in which parametric sound is caused by finite-width plane wave sources facing each other. Therefore, the experiments and discussions here are limited to cases where θ is fixed at 180 °, and cases where the angle between the sound beams varies are not treated.

In our research, we observed the frequency shift of parametric sound, where one of the face-to-face pairs of plane wave sources was fixed and the other was moved at a constant speed. The mechanism of the frequency shift of the parametric sound discussed here can be explained as follows:

  • The frequency of the ultrasound wave emitted by the moving source is shifted by the Doppler effect.

  • The Doppler-shifted ultrasound wave and the ultrasound emitted by the other fixed source interact in the area between the sources, resulting in local secondary sound sources, including the difference sound. The difference sound accordingly undergoes the frequency shift due to the effect described in note (i).

  • According to the method of successive approximation, it is expected that the secondary wave is scattered and propagates in various directions, and thus it can be observed by a microphone placed outside the sound beams.

Schematics of these processes are illustrated in Fig. 1. Nonlinear interactions of the two ultrasound waves distribute the virtual sound sources, whose frequency is the difference between them. The frequency of the sound field produced by the moving source at an arbitrary position, P, situated in front of the source, is shifted by the Doppler effect, although a specific observer is not considered here. Then, the frequency of the primary wave at P, f 1 , is given by
f 1 = c c v f 1 ,
(7)
where c is the sound velocity, f1 is the frequency of the moving sound source, and v is the velocity of the source in a direction normal to the emitting surface. Let v be positive when the source moves toward a fixed source. The sound radiation in a region in front of the moving source can be approximated as a plane wave with a finite width; thus, the Doppler shift of the ultrasound anywhere in the region can be attributed to the velocity component of the sound source in the direction of the progressing plane wave, that is, the normal direction of the emitting surface.
FIG. 1.

(Color online) Schematic of phenomenon. (a) Conventional position-dependent frequency shift of primary wave emitted from a moving source. (b) The reported position-independent frequency shift of secondary wave derived from emissions by sources in relative motion.

FIG. 1.

(Color online) Schematic of phenomenon. (a) Conventional position-dependent frequency shift of primary wave emitted from a moving source. (b) The reported position-independent frequency shift of secondary wave derived from emissions by sources in relative motion.

Close modal
Based on the above suppositions, this study utilized arrays of ultrasound loudspeakers with sharp directivity in the front direction as sound sources. The two sound source arrays faced each other, and one was moved while the two emitting surfaces were parallel. A microphone at a distant position measured the difference sound generated in that situation, which was derived from the interaction between the wave of frequency f 1 and that emitted by the fixed source, whose frequency is represented as f2. Then, the frequency of the difference sound is given by
f s = f 1 f 2 = c c v f 1 f 2 .
(8)
The frequency shift of this difference sound, Δ f diff, from frequency, f diff = f 1 f 2, and that of the parametric sound from a fixed source with an emission frequency of f1 is represented as
Δ f diff = ( f 1 f 2 ) ( f 1 f 2 ) = v c v f 1 v c f 1 .
(9)
This approximation holds under the assumption that | v | c, which is fully satisfied because the maximum speed of the moving source was 250 mm/s in our experiments. Here, a general Doppler shift, in which a source with an emitting frequency of ( f 1 f 2 ) moves at a constant velocity v is represented as
v c v ( f 1 f 2 ) .
(10)
Therefore, the frequency shift of the parametric sound, Δ f diff, is f1/(f1f2) times greater than that in Eq. (10) for the same velocity of the moving source. As mentioned previously, the frequency shift per unit moving velocity increases because the frequency of the primary wave is higher, which is directly observed in the frequency band of the parametric sound f 1 f 2.

Because the actual wave width is limited, parametric sound waves caused by interaction on the edges of the wave fields are not theoretically negligible, and it is presumed that these fields exhibit the different frequency shifts described by the model mentioned above. However, we considered these effects practically negligible because the area of these interactions was much smaller than that of the interactions between plane waves. Moreover, we considered that the sound pressure of the parametric sound generated on the edges is also much smaller than that derived from plane waves because the sound fields on the edges intersect with each other at a certain nonzero angle, which corresponds to the case in which θ 180 ° in Eq. (4), which yields a lower source intensity. Secondary sound fields from these interactions were not observed in the experimental results, as described later.

We conducted experiments to observe the frequency shift when two sound sources kept facing each other and one moved at a constant speed. Parametric speaker kits from Tristate Co., Ltd., Japan, were utilized as the sound sources. The sound sources comprised 50 ultrasound oscillators (UT1007-Z325R; SPL Co., Ltd., Hong Kong) with a resonance frequency of 40.0 ± 0.1 kHz. The oscillators were arranged on 55 mm × 98 mm boards. We amplified the sinusoidal voltage signal produced by a function generator (AWG1025F; AS ONE Co., Ltd., Osaka, Japan) employing an audio amplifier (M4; Douk Audio, Shenzhen, China) and input it into the sound sources. We measured parametric sound utilizing a microphone (type 4138-A-015; Brüel & Kjaer, Naerum, Denmark). The frequency response of the microphone differed within the range of ±2 dB for frequencies from 6.5 Hz to 140 Hz. The directivity of the microphone is expressed by 4 dB power reduction for incident angle of 90° than that of 0° (incident wave normal to the top plane of the microphone) for the ultrasound frequency between 40 and 45 kHz. Although the amplifier was a D-class amplifier, we experimentally confirmed that a sufficiently intense output around the driving frequencies of the transducers was obtained and that no significant overtones were observed in the driving signals. The measured time-series data were imported to a personal computer (PC) via an oscilloscope (PicoScope4262; Pico Technology, Cambridgeshire, United Kingdom). In all experiments, continuous sinusoidal waves were used as driving signals of the oscillators.

The sampling intervals of the recorded data were set to 1.0 μs in all the experiments to prevent aliasing. The amplifier's output voltage was manually adjusted such that the sound pressure was approximately 240 Pa (=141.6 dB sound pressure level [SPL]) at a point 350 mm away from the sound source. We operated a three-dimensional (3D)-cartesian coordinate robot (ICSB3; IAI Co., Ltd., Shizuoka, Japan) (Fig. 2) to move the sound sources at a constant velocity. Note that the robot was driven with a trapezoidal temporal profile of velocity (accelerated at 0.1 G = 0.98 m/s2) since the maximum acceleration of the robot was limited. The start and stop timings of the robot were recorded via the second channel of the measurement data by synchronizing the signal input into the oscilloscope.

FIG. 2.

(Color online) Measurement setup.

FIG. 2.

(Color online) Measurement setup.

Close modal

Herein, we outline the experiments performed. First, we measured the directivity of the ultrasound emitted by sound sources with the same power and frequencies as those utilized in the following experiments. Next, we measured the frequency shifts of the parametric sound generated by a pair of an immobilized ultrasound source and a moving one under three different conditions: the frequencies of the two sound sources were varied while one of them was moving, the velocity of the moving source was varied, and the direction of the moving source was varied. We compared these results with the frequency shifts obtained for both of the sound sources immobilized. Finally, we evaluated the parametric sound frequency shift measured employing a microphone placed at various positions and examined the validity of the hypothesis that identical frequency shifts should be observed independently of the measuring positions.

The ultrasound oscillators were driven at frequencies identical to those utilized in the following experiments. We measured spatial directivity utilizing a microphone mounted on the robot. Radiation frequencies were set to 40 and 41 kHz for the sound source that was immobilized in the following experiments and to 43, 44, and 45 kHz for the sound source that was moved in the following experiments. The measurement results are presented in Figs. 3(a)–3(e). We additionally measured the directivity for the driving frequencies of 40 and 44 kHz, with the direction between the microphone and the oscillators set to 1 m [Fig. 3(f)]. These results indicate that the oscillators generally have strong directivity to the front across all driving frequencies. The spatial patterns varied slightly owing to changes in the wavelength according to the driving frequencies. At the same time, secondary power peaks are observed around azimuthal angles of around 60–70°. These are so-called grating lobes caused by the ultrasound emitting elements arranged with intervals greater than half the wavelength. The effects of those grating lobes are discussed later in Sec. IV. The experiments described below present the results of the interactions between the ultrasound fields with these spatial distributions.

FIG. 3.

(Color online) (a)–(e) Measured two-dimensional power distribution of sound sources for individual driving frequencies (40, 41, 43, 44, 45 kHz). (f) Measured angular directivities of the sources for 40 and 44 kHz.

FIG. 3.

(Color online) (a)–(e) Measured two-dimensional power distribution of sound sources for individual driving frequencies (40, 41, 43, 44, 45 kHz). (f) Measured angular directivities of the sources for 40 and 44 kHz.

Close modal

Before the subsequent experiments under various conditions, we measured the parametric sounds of the sources with their driving frequencies set to f 1 = 44 kHz and f 2 = 40 kHz for the control condition. The experimental setup is demonstrated in Fig. 2 and the positional relationship between the sound sources and the microphone in Fig. 4. Let the point 350 mm away along the negative x axis from the center of the fixed sound source be O (with the y-coordinate identical) and the microphone be fixed at a point 40 mm away along the positive z-axis from O. First, the parametric sounds generated when both sources were immobilized were measured. Next, we measured the parametric sound with a source emitting ultrasound of frequency f1 moving at speed v = ± 150 mm/s on the line segment, whose length was 400 mm and center was O. Here, let v be positive when it is toward the source emitting ultrasound of frequency f2 in the positive direction along x axis. The y-coordinates of the sources were manually set to be identical. The measured signal was trimmed to 2.0 s and was fast Fourier transform (FFT)-processed utilizing the Hann window function. Frequency analysis of the measured waveform was performed in all experiments utilizing this procedure. The observed spectra under the condition where one of the sources was moved are demonstrated in Fig. 5.

FIG. 4.

(Color online) Experimental component positions in experiments 0-1, 0-2, 1, and 2.

FIG. 4.

(Color online) Experimental component positions in experiments 0-1, 0-2, 1, and 2.

Close modal
FIG. 5.

(Color online) Difference sound measured under the control condition. The orange arrows correspond to the frequency peaks predicted by Eq. (9). The other arrows correspond to the peaks whose origins are discussed in Sec. IV.

FIG. 5.

(Color online) Difference sound measured under the control condition. The orange arrows correspond to the frequency peaks predicted by Eq. (9). The other arrows correspond to the peaks whose origins are discussed in Sec. IV.

Close modal

The difference in the driving frequencies of the sound sources was 4 kHz. We observed that amplitude spectra corresponding to the opposite moving direction of the source exhibited a symmetric power distribution to one another with respect to 4 kHz. The frequency component of 4 kHz was not observed in either spectrum; however, multiple peaks were observed. Among the peaks, the frequencies corresponding to the maximum amplitude (indicated by orange arrows) were 4019.2 Hz for Fig. 5(b) and 3980.8 Hz for Fig. 5(c), respectively. These results are consistent with the frequency shifts theoretically given by Eq. (9), Δ f diff = 44 kHz × [ ( ± 0.15 m / s ) / ( 340 m / s ) ] = ± 19.41 Hz, which reveals that our models are appropriate. In Fig. 6, the time signals for the measurement shown in Fig. 5(b) are depicted. An envelope of signal corresponding to the difference frequency (4 kHz) is prominent for unfiltered signals, whereas the difference frequency component is observed after bandpass filtering.

FIG. 6.

(Color online) Time signals corresponding to the measurement shown in Fig. 5(b). (a) Signal for a duration of 2 s. (b) Temporally magnified signal. (c) Bandpass filtered version of signal to extract the difference frequency components.

FIG. 6.

(Color online) Time signals corresponding to the measurement shown in Fig. 5(b). (a) Signal for a duration of 2 s. (b) Temporally magnified signal. (c) Bandpass filtered version of signal to extract the difference frequency components.

Close modal

Regarding the conventional Doppler shift, the frequency shift caused when an audible sound source is moving at an equivalent speed, u = ± 150 mm/s, toward a microphone is approximately ± 1.8 Hz ( = 4 kHz × [ ( ± 0.15 m / s ) / ( 340 m / s ) ]); thus, the result indicates that the frequency shift was indeed magnified when compared with a case with conventional Doppler shift. In the spectrum diagrams, other remarkable frequency peaks were observed (indicated by arrows with different colors). We presume that these subordinate components were derived from reflections on the emitting surface of the sources and that the frequency shift of the main component can be predicted from the velocity of the moving source, as discussed later in Sec. IV. These subsidiary components are not too small to ignore; however, the frequency corresponding to the formula Eq. (9) was observed to have the largest amplitude under almost all conditions. Because this tendency holds in later experiments, we focused on and analyzed the main peaks in the experiments.

According to Eq. (9), the frequency shifts are expected to depend on the frequency of the moving source (f1) but not on that of the fixed source (f2). To verify this, we measured the parametric sound when we changed the frequencies of the sound sources from those under the control condition. We performed experiments utilizing three different combinations of f1 and f2: ( f 1 , f 2 ) = (43, 40 kHz), (45, 40 kHz), and (45, 41 kHz). Other experimental conditions, such as the velocity (v = 150 mm/s), moving distance (400 mm), the distance between the two sound sources, and the microphone's position and the method of signal processing, were the same as those in Sec. III B.

The frequency components that provide the main peaks for each frequency pair (f1, f2) are listed in Table I. When f1 was changed to 43, 44, and 45 kHz under the condition that f 2 = 40 kHz, the center of the frequency shift correspondingly varied to 3, 4, and 5 kHz. Moreover, the shift from the center became slightly larger in accordance with an increase in f1. The measured frequency shifts are consistent with the theoretical values obtained utilizing Eq. (9). This indicates that our model accurately explains this phenomenon.

TABLE I.

Parametric frequency shifts when (f1, f2) were changed.

Frequency shift (Hz)
(f1, f2) (kHz) at v Measured Theoretical
v = 150 mm/s     
(43, 40)  +18.8  +19.0 
(44, 40)  +19.2  +19.4 
(45, 40)  +19.7  +19.9 
(45, 41)  +19.6  +19.9 
v = −150 mm/s     
(43, 40)  −18.8  −19.0 
(44, 40)  −19.2  −19.4 
(45, 40)  −19.6  −19.9 
(45, 41)  −19.7  −19.9 
Frequency shift (Hz)
(f1, f2) (kHz) at v Measured Theoretical
v = 150 mm/s     
(43, 40)  +18.8  +19.0 
(44, 40)  +19.2  +19.4 
(45, 40)  +19.7  +19.9 
(45, 41)  +19.6  +19.9 
v = −150 mm/s     
(43, 40)  −18.8  −19.0 
(44, 40)  −19.2  −19.4 
(45, 40)  −19.6  −19.9 
(45, 41)  −19.7  −19.9 

In addition, the results of ( f 1 , f 2 ) = (44, 40 kHz) and ( f 1 , f 2 ) = (45, 41 kHz) indicate that centers of the frequency shifts were 4 kHz in both cases, but the shift for f 1 = 45 kHz was larger than that for the others, which is also consistent with our models. Thus far, the tendency was true for cases where the moving source also departed from the fixed source.

According to the derived model, the frequency shifts are expected to be nearly proportional to the source velocity v. We measured the parametric sound when we moved one of the sound sources along a line segment of length 400 mm at v = ± 100 , 150 , 200 mm/s, along a line segment of length 50 mm at v = ± 20 , 30 , 40 , 50 mm/s, and along a line segment of length 25 mm at v = ± 10 mm/s. The other experimental conditions were the same as those described in Sec. III B. The temporal length of the measured data varied depending on the velocity and moving distance.

Figure 7 presents plots of the relationship between the frequency at the principal peak and the moving speed of the source. We plotted the blue points representing the measured frequency shifts and the red dotted line representing the theoretical values of the shifts, Δ f diff = ( v / c ) × 44 kHz. The measured frequency shifts were almost proportional to the moving speed of the sound source v and revealed good agreement with the theoretical values in our models (Eq. (9)). We also drew a gray dotted line to compare the conventional Doppler shifts that should be observed when an audible sound source of 4 kHz proceeds toward a microphone, ( v / c ) × 4 kHz. The frequency shift of the parametric sound was magnified compared with the conventional Doppler effect of the same audible frequency component in the emitted sound. Hence, a frequency shift around 4 kHz was observed even for a moving object with a slow velocity of 10 mm/s. The frequency shift accurately reflected the velocity of the moving source.

FIG. 7.

(Color online) Relationship between the sound source velocity and the parametric frequency shift. The red dotted line shows theoretical frequency shift for our experiment, and the gray dotted line shows the frequency shift caused by conventional Doppler effect with the same source frequency.

FIG. 7.

(Color online) Relationship between the sound source velocity and the parametric frequency shift. The red dotted line shows theoretical frequency shift for our experiment, and the gray dotted line shows the frequency shift caused by conventional Doppler effect with the same source frequency.

Close modal

In the derived model, we consider that the moving sound source is approximately a plane wave source and that its normal direction determines the frequency shift. Regarding this argument, we measured the frequency shifts when one of the sound sources was moved toward a certain angle, ϕ, from the normal direction while maintaining a constant posture. We changed ϕ to various values in the experiment to confirm whether similar frequency shifts were measured, even when a source moved in other directions other than toward the right in front of the other fixed sources, as in experiment 2.

The experimental setup is illustrated in Fig. 8. Let the x axis be the line on which the two sources are located at their initial positions, as in the previous experiments. The traveling direction of the moving source is defined as the clockwise angle between its movement and the negative direction of the x axis. The sound source was moved in the direction of ϕ = 0 ° (parallel to the x axis in its negative direction), 15 , 30 , 45 , 60 , 75 , 90 , 180 , 195 , 210 , 225 , 240 , 255 , and 270 °, along a 400 mm line segment whose center was O. The other experimental conditions were the same as those described in Sec. III B.

FIG. 8.

(Color online) Experimental component positions in experiment 3.

FIG. 8.

(Color online) Experimental component positions in experiment 3.

Close modal

Figure 9 presents a plot of the relationship between the frequency at the main peak and the direction in which the sound source moved. The blue points represent the measured frequency shifts, and the red dotted line represents the theoretical values, Δ f diff = ( v cos ϕ / c ) × 44 kHz. As presented in Fig. 9, the observed frequency shifts plot almost exactly on a cosine curve with respect to the moving angle ϕ. Thus, the frequency shifts were determined only by the velocity component in the normal direction, even when the sound source moved obliquely.

FIG. 9.

(Color online) Relationship between the moving direction of the source and the frequency shift. Red dashed lines show theoretical values.

FIG. 9.

(Color online) Relationship between the moving direction of the source and the frequency shift. Red dashed lines show theoretical values.

Close modal

Finally, we experimented to examine the relationship between the observation positions and observed frequency shifts. We set the sound source frequencies to f 1 = 44 kHz , f 2 = 40 kHz and the moving velocity to v = ± 150 mm/s, and generated difference sounds by moving the f1 source in the same manner as in experiment 1. The five microphone positions (Pos.1 to -5) are presented in Fig. 10. The data length of the observed signals was set to 2.0 s, and the temporal waveforms were short-time Fourier transform (STFT)-processed and visualized as spectrograms. For comparison, we observed their portions around the frequencies of the difference sound (4 kHz) and moving sound source (44 kHz). This signal-processing procedure examines whether the spectrum's peak frequencies vary over time along with the movement of the sound source. The window function for STFT is a Hamming window with a temporal width of 0.2 s and an overlap ratio of 50%. We conducted zero padding during STFT to improve the frequency resolution to 0.5 Hz. The reason for performing this STFT-based time-frequency analysis of the recorded signals in this experiment is to evaluate whether observed frequency changes of the primary and secondary waves were affected by the relative positions of the moving ultrasound oscillator and microphone.

FIG. 10.

(Color online) Microphone positions in experiment 4.

FIG. 10.

(Color online) Microphone positions in experiment 4.

Close modal

The results for v = 150 mm/s are presented in Fig. 11. For each observing position (Pos.1 to -5), the spectrogram around the frequency of the parametric sound (4 kHz) and that around the driving frequency of the moving source (44 kHz) were displayed. The most important fact displayed here is that the spectrograms of the primary waves much differed across the observation positions, whereas those of the secondary parametric waves did not and were consistently composed of two significant peaks across all observation positions, as expected in our model.

FIG. 11.

(Color online) Spectrogram of sound pressure measured at positions 1–5 (Pos.1 to -5, respectively), around 4 and 44 kHz, for experiment 4. The values are normalized by the maximum in the graph. All figures are obtained for a sound source moving at 150 mm/s.

FIG. 11.

(Color online) Spectrogram of sound pressure measured at positions 1–5 (Pos.1 to -5, respectively), around 4 and 44 kHz, for experiment 4. The values are normalized by the maximum in the graph. All figures are obtained for a sound source moving at 150 mm/s.

Close modal

The frequency of the parametric sound was theoretically shifted by Δ f diff = 44 kHz × [ ( 0.15 m / s ) / ( 340 m / s ) ] = 19.41 Hz, as indicated in experiment 0-2 (Fig. 5). As presented in the spectrogram obtained at around 4 kHz in Figs. 11(a), 11(b), 11(c), and 11(h), the amplitude of the peak at about 4.02 kHz was significantly large and stable over time for all those positions. This indicates that the main frequency component of the difference sounds was consistently observed regardless of the observation position. The magnitude of the amplitude can be time-varying. For Pos.4, the peak around 3.98 kHz was observed in addition to the peak around 4.02 kHz, which was even more prominent [Fig. 11(g)]. The reason for this is discussed in Sec. IV.

At the same time, for the spectrogram at approximately 44 kHz, we can see that the peak frequency gradually decreases in Figs. 11(d) and 11(e). The frequency shift of the primary wave is due to the conventional Doppler shift and, hence, is determined by the source's moving velocity component toward the microphone. Therefore, the time-varying frequency shifts illustrated in Figs. 11(d) and 11(e) are attributed to the fact that the positional relationship between the moving sound source and the microphone was time-varying, and thus, the Doppler effect observed by the microphone correspondingly varied across time. The time-varying frequency component also observed at Pos.2 exhibits a smaller ratio of change than that at Pos.1. We consider this to be because Pos.1 is closer to the source's trajectory, which makes its velocity component toward the microphone drastically change during the travel of the source, whereas Pos.2 is away from the right side, which results in less change in the relative velocity component of the source. The frequency change observed at Pos.3 was even smaller, with its peak almost constant at 43.99 kHz. This corresponds to the fact that the traveling direction of the moving source is almost constant in this case. Figures 11(d), 11(e), 11(f), 11(i), and 11(j) clearly demonstrate that frequency changes of the primary wave depend on observation positions. Note that the peak at approximately 44.02 kHz was stably observed at Pos.1, 2, and 3, as illustrated in Figs. 11(d), 11(e), and 11(f), and it exactly corresponded to the speed of the sound source. This does not seem to be the sound wave directly traveling from the sound source to the microphone, but rather the Doppler-shifted primary wave emitted by the moving source and reflected by an object surrounding the experimental environment, such as the fixed source's surface. As demonstrated in Fig. 11(i), the sound field reaching directly from the approaching source was dominantly observed at Pos.4, and the peak at 44.02 kHz was not remarkable. As presented in Fig. 11(j), various components of the primary wave traveling on a respective path were observed at Pos.5, located much farther from the source, compared with Pos.1 to -4. The frequency shifts depended on the individual sound paths to the microphone. Remarkably, the frequency composition of the parametric sound was not significantly different from that of Pos.1 to -4, even under these conditions with the spread observed frequencies of the primary waves. It is noteworthy that velocimetry utilizing a secondary parametric wave measured far from the sound source gave precise results, even when the frequency shift of the primary wave was spread and unreliable for velocimetry.

As mentioned in Sec. III B, we focused on the frequency components predicted adopting Eq. (9) and did not consider others as targets of analysis. In this section, those subordinate frequency components are discussed.

In Figs. 5(b) and 5(c), several peaks can be seen, including the primary ones indicated by orange arrows. For the case of v = 150 mm/s, they are 4019.2 Hz (indicated by the orange arrow), 3983.5 Hz (yellow arrow), 4022.6 Hz (green arrow), and 4057.0 Hz (blue arrow). As already discussed, the frequency component indicated by the orange arrows is f diff ( v / c ) f 1, derived from the interaction between f2 and f 1 ; f1 is shifted because of the movement of the sound source.

As mentioned above, the frequency components indicated by the yellow arrows can be explained by considering a primary wave reflected from the emitting surface of the moving sound source. The primary wave originating from the fixed source driven by f2 is Doppler-shifted when reflected by the moving source to the front. The shifted frequency, f 2 , is
f 2 = c + v c v f 2 .
(11)
Note that we must consider both the effects of the reflector: one as a moving observer and the other as a moving source. Here, the interaction between the primary waves of frequencies f 2 and f 1 generated a parametric sound. Its frequency shift from f diff to Δ f diff 2 is
Δ f diff 2 = ( f 1 f 2 ) ( f 1 f 2 ) v c f 1 2 v c f 2 .
(12)
When we substitute 0.15 m/s for v, Δ f diff 2 = ( 44 kHz 2 × 40 kHz ) × [ ( 0.15 m / s ) / ( 340 m / s ) ] = 15.9 Hz , which indicates a correspondence between Eq. (12) and the yellow component.
The frequency components for the green arrows are obtained by considering that the Doppler-shifted primary wave f 1 is reflected by a fixed sound source (a frequency shift does not occur here) and is reflected by the moving source. The shifted frequency of this wave, f 1 , after the two-stage reflection is
f 1 = c + v c v f 1 = c ( c + v ) ( c v ) 2 f 1 .
(13)
Here, the parametric sound arises from the interaction between the primary waves of f 1 and f 2 . Its frequency shift from f diff, which is equal to Δ f diff 3, is
Δ f diff 3 = ( f 1 f 2 ) ( f 1 f 2 ) 3 v c f 1 2 v c f 2 .
(14)
When we substitute 0.15 m/s for v, Δ f diff 3 = ( 3 × 44 kHz 2 × 40 kHz ) × [ ( 0.15 m / s ) / 340 m / s ) ] = + 22.9 Hz, which indicates a correspondence between Eq. (14) and observation. The frequency components for the blue arrows are generated by the interaction between a primary wave, f 1 , and f2. Its frequency shift from f diff to Δ f diff 4 is
Δ f diff 4 = ( f 1 f 2 ) ( f 1 f 2 ) 3 v c f 1 .
(15)
When we substitute 0.15 m/s for v, Δ f diff 4 = 3 × 44 kHz × [ ( 0.15 m / s ) / ( 340 m / s ) ] = + 58.2 Hz, which indicates a correspondence between Eq. (15) and observations.

When we estimate the velocity of the moving source from the parametric sound spectrum, knowing that there are multiple reflections between the sources, utilizing these multiple peaks may lead to a more robust estimation because they are all predictable from the source velocity. In Fig. 11(g), compared to the peak around 4.02 kHz, which was supposed to be the principal component, the one around 3.98 kHz was larger and more stable. We assume that this corresponds to a frequency shift Δ f diff 2.

We also discuss the possibility of applying this phenomenon to practical velocimetry. As experimentally demonstrated, a larger ratio of the frequency change for the moving velocity of the source is a fundamental advantage of the investigated phenomenon compared to the conventional Doppler effect for the same “carrier” frequency. Thus, source movements as slow as 10 mm/s can be distinctly and precisely detected for the observation frequency of 4 kHz with 0.2 s STFT-based signal analysis. Because the ratio of measured frequency change per moving velocity of the object can be magnified, a more accurate velocity estimation can be done with a measurement of a shorter duration.

In addition, as expected from Sec. II of the paper, the frequency changes of the parametric sound were steadily measured irrespective of the observation position. Our experiments also show that observed the primary wave frequency varied while the relative position of the moving object and the microphone was changed. This result means that the conventional (i.e., primary wave based) method cannot stably measure the moving velocity of the sound-emitting object with respect to a certain fixed spatial axis when the direction of the object from the microphone drastically changes during measurement. In contrast, the observed parametric frequency shifts always corresponded to the velocity component of the moving oscillators with respect to the normal value of the fixed oscillators throughout our experiments.

Our method has two disadvantages. One is the need for causing nonlinear acoustic interaction, meaning that considerably intense ultrasound emission is required. The other is that the emitting frequency of the moving object must be known prior to measurement.

The current experiments only explain the case with a moving planar source and finite emission apertures with a fixed relative posture. Therefore, the frequency changes in parametric sounds with different aperture sizes and shapes or moving trajectories of the source may differ from those in our investigation. A more generalized theory describing the whole-space parametric source distribution should be constructed to handle these cases.

Current velocimetry methods can only perform unidirectional velocity measurements. In many practical cases, the traveling direction of a moving object is unknown in advance. These situations require multi-axis measurements of object velocity in a fixed coordinate system. To this end, utilizing multiple sources with driving frequencies that differ according to the measurement axes may be a good strategy. In addition, measurements based on ultrasound reflection on the surfaces of moving objects remain an important issue for a more practical velocimetry method.

The experiments conducted here cannot completely exclude the possibility of a difference sound being generated by the conventional Doppler effect on the incident wave, which is reflected on the periodically vibrating source surface. This phenomenon and the mechanism of parametric sound generation owing to the nonlinearity of air are known to have quite similar effects (Mujica , 2003; Wunenburger , 2004). However, the results of experiment 4 indicate that the frequency shifts of the primary waves due to the (conventional) Doppler effect are dependent on the positions of the microphone and source. Therefore, the possible difference in sound owing to the surface Doppler effect should depend on these positions. In the observed frequency components around the difference frequencies of both sources, such position-dependent components might exist but are not as prominently observed as the position-invariant components resulting from the parametric sound sources in the air. Therefore, we conclude that the observed difference in sound in our experiments mainly resulted from nonlinear acoustic effects in the air.

There are some factors that might yield frequency components around the difference frequency, whose origin was left unclear in the above discussions. We could not completely get rid of the following two factors because of the experimental circumstances. One is the effect of grating lobes that might cause nonlinear acoustic interactions that could generate those unidentified frequency components. Nevertheless, the effect of the source terms in the model is considered to be very limited because the intersection areas of the grating lobes are much smaller than those of the main beams that stretched between the oscillator arrays. Another is the effect of multiple paths due to the ultrasound reflection in the experimental environment. We additionally performed a complementary experiment, where acoustic absorbers were employed to suppress the reflection paths to the microphone and a short-duration burst waves were adopted as a driving signal for the moving oscillators to exclusively capture the initial interaction between the primary waves. Unfortunately, both attempts did not result in a better measurement: not all reflection paths were eliminated, and the burst waves were too instantaneous to provide a frequency resolution in observed signals that is sufficient for identifying the frequency shifts. To clearly explain the origins of those unidentified frequency components still remains as an essential question to be solved in future research.

Regarding the acoustic phenomenon called the parametric array, which is a nonlinear interaction between ultrasound waves with two different frequencies, we discovered a peculiar frequency shift mechanism of parametric sound generated by moving and fixed sound sources. We modeled this phenomenon assuming the two sound sources were plane sources with a limited width. Thus, we demonstrated that the frequency shift of the parametric sound is proportional to the product of the velocity in the normal direction of the emitting surface and the radiation frequency of the moving sound source and that the observed shift is larger than that caused by the conventional Doppler effect for a moving source with the same audible driving frequency and velocity. In addition, a parametric sound was observed outside the intersection area of the primary waves, and its frequency shift, independent of the microphone's position, was observed stably. As discussed in Sec. IV, the results here can be applied to velocimetry, which has a distinct advantage over the conventional method based on the Doppler effect in that a sound source moving at a slow speed can be measured regardless of the observation positions while maintaining a low carrier frequency. This study attempts to analyze the phenomenon, and current velocimetry presented in this paper has many limitations. Therefore, utilizing the phenomenon discussed in this paper, a more practical method for noncontact velocimetry of moving objects is supposed to be devised as our future works. For example, we expect that utilizing nonplanar intense ultrasound wave fields (Hasegawa , 2020; Mukai , 2021) will broaden the application scenarios of the proposed technique, such as multi-axis simultaneous measurement of multiple targets.

The authors thank support from Takeru Momoki for additional measurements. This research was supported by the Japan Society for the Promotion of Science, Kakenhi (Grant No. 23H03473).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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