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Figure 3. Depiction of a parametric array, created with radiation of ultrasound by a loudspeaker in the ceiling, used to produce an audio spotlight of highly directional speech. The amplitude-modulated ultrasound radiated by the loudspeaker, represented in the time and frequency domains in the top panel, self-demodulates into audible speech, represented in the bottom panel, during nonlinear propagation toward the listener.
absorption increases with frequency, the medium acts as a low-passfilterandultimatelyonlyalow-frequency(fD)wave will remain. Nonlinearity thus makes it possible to create an acoustic beam at low frequency using a high-frequency source. Nonlinear acoustic sources based on this principle were proposed in the 1950s to 1960s by Westervelt in the United States and independently by Zverev and Kalachev in the Union of Soviet Socialist Republics (USSR), and they are referred to as parametric arrays. Like many studies in under- water acoustics during the era of the Cold War, the initial studies were classified (see Zverev, 1999, for a review of early research on parametric arrays in the USSR; Westervelt, 1960,
1963, for the first open publications on this subject).
One might question why one would choose to excite a low- frequency beam in such a complicated and comparatively inefficient way when it would be easier to radiate the low-
frequency wave directly from the transducer. The main advantage is the ability to create very directional low-fre- quency sources or receivers having dimensions considerably smaller than required by linear acoustics. The volume of fluid in the medium insonified by the high-frequency primary waves at f1 and f2 acts as an antenna for fD. The length of this antenna is limited only by the propagation distance (L) of the primary waves, which is determined by their attenuation due to absorption and diffraction. The insonified region cor- responds to a traveling-wave antenna that emits a fD beam with an angular width on the order of √λD /L radians, where λD is the wavelength at fD. In addition to this significant size advantage over conventional sound sources, another impor- tant property is that the high directivity is maintained over a wide bandwidth relative to fD. Also, sidelobes typically asso- ciated with the high-frequency pump waves are significantly suppressed. The main drawback of the parametric array is its low efficiency in converting energy in the pump waves into the fD wave.
Parametric arrays were the first practical devices based on nonlinear acoustics, initially with application to sonar (Esipov et al., 2010). In recent years, parametric arrays have been used to create highly directional beams of audible sound in air. The first measurements of audible frequen- cies produced by a parametric array in air were reported in the mid-1970s (Bennett and Blackstock, 1975). However, engineering challenges connected with the development of transducers capable of producing sufficiently intense beams of ultrasound in the air prevented commercialization of the audio spotlight until the early 2000s.
The principle of the audio spotlight, which has been understood since the 1960s (Berktay, 1965), follows from recognition that using a parametric array to generate a low frequency fD due to the interaction of two high frequencies (f1 and f2) is a special case of self-demodulation. If the time waveform is expressed as E(t)sin(2πf0t), where E(t) is an amplitude modulation that varies slowly in time relative to f0, then the time waveform produced by the paramet- ric array is proportional, or nearly so, to the second time derivative of the square of the modulation function (d2E2/ dt2). Creation of such a waveform in the fluid is referred to as self-demodulation.
In the context of the audio spotlight, f0 is typically around 60 kHz, well above the range of human hearing. The frequency is sufficiently high to create a directional beam but not so
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