Infrasonicon icon. Click here for home page.INFRASONICON: Time-Compressing Infrasonic Recordings to Discover New Sounds, by Clark Huckaby


Water Surface Waves Versus Sound

Their visibility and relatively low speed makes water surface waves a common analogy for sound. But no analogy is perfect. Since one goal of the Infrasonicon is to convert the former into the latter (see Overview Part 4), this page outlines some s
imilarities and differences between the phenomena. They affect transducer design requirements and help predict time-compressed playback results.

Similarities.
Surface waves on water and sound waves in air have much in common:
  1. Both convey energy (and information) without changing the average structure of their respective media.
  2. Their power is proportional to amplitude squared, so decibel (dB) scales may be applied to water waves, as they are to sound.
  3. Both wave types can be reflected, refracted, absorbed, and/or diffracted by boundaries.
  4. Like all wave phenomena, the relationship between frequency (F), wavelength  (λ), and propagation speed (c; also called phase velocity) is:
F = c/λ            (Equation 1).

Differences. The relative difference in speed between sound and water surface waves can be considered quantitative. But there are key qualitative differences, including:
  1. Sound waves propagate in 3-dimensional space while surface waves propagate on a plane. In other words, a set of sound wave rays (i.e., vectors perpendicular to wave fronts, indicating direction of propagation) need three dimensions, but surface wave rays need only two.
  2. Air particle motion in sound is longitudinal (co-axial with propagation rays); water particle motion in surface waves is circular (deep water) or elliptical (shallow), orbiting in vertical planes parallel to rays.
  3. The restoring force, acting to return displaced particles to their equilibrium (average) position, is elasticity for sound waves; depending on wavelength, it is gravity and/or surface tension for water surface waves.
  4. The speed of sound is independent of frequency, but in deep water, surface wave speed depends on wavelength and thus frequency, making surface waves dispersive.
  5. Within the dynamic range of sound perception (over 120 dB), air (near sea level) does not limit amplitude of any audible wavelength. But the amplitude of water surface waves is limited to a certain fraction of their wavelength.
Simplest Theoretical Waveforms. The elementary sound waveform is the familiar and symmetrical sinusoid (sine wave). The "y" axis can stand for either air pressure, individual air particle displacement, or an analogous voltage or current. The "x" axis can be time, distance, or phase angle. While physical sound waves are longitudinal, their oscillographic display is transverse; one can envision the axis of particle motion rotated 90 degrees to represent the waveform in two dimensions.

In contrast, the circular particle motion in water surface waves has both longitudinal and transverse components. Sinusoids thus cannot represent water surface height versus time, distance, or phase angle (except at zero amplitude). Instead, the simplest two-dimensional model for water surface waveforms is a trochoid. Every trocoid is related to a cycloid, the idealized waveform which needs to be described first.

The Cycloid.
Figure A shows a cycloid and how it can be generated. The "y" and "x" axes are vertical and horizontal dimensions in a wave cross-section along a propagation ray. The circle on the right (radius = a) depicts the counter-clockwise path of imaginary water surface particle P. This circle's center is P's average position, which is stationary. Wave propagation translates the x-axis from right to left; envision a scrolling x-axis gripping and turning the circle like a rack and pinion (suggested by arrows near intersection of circle and x-axis), with a chart recorder pen at point P. The waveform traced is a cycloid whose wavelength (λ) equals 2πa. Phase velocity (c) equals the speed of P in its orbit. By substitution into Equation 1, the frequency is:

F = c/2πa            (Equation 2).

Notice the asymmetry of the cycloid in Figure A. Crests are pointed and troughs are broad, like a cartoon water waveform.

Figures A and B: Construction of cycloid (A) and trochoid (B)

The Trochoid.
By definition, surface wave amplitude equals the radius (r) of particle orbit (analogous to a peak voltage). Wave height (h) equals the orbital diameter (2r) or vertical crest-to-trough distance (analogous to a peak-to-peak voltage). Real waves cannot exist as cycloids because steep crests are unstable. In deep water, the maximum amplitude (rmax) of stable waves is about a/2. When a > r > 0, the waveform is called a trochoid. This is the elementary form of all real water surface waves.

The solid curve in Figure B shows a trochoid with amplitude r = a/2. The orbit of surface particle W is shown by the solid circle with radius r. It is locked to the concentric outer (dashed) circle, whose radius equals a. Translation of the x-axis turns the circles through 2π radians per λ of horizontal distance. The trochoid traced by particle W therefore has the same wavelength, phase velocity and frequency as the superimposed imaginary cycloid (dashed curve). Like the cycloid, a trocoid is asymmetrical. But symmetry increases as amplitude decreases, approaching a perfectly symmetrical sine wave as amplitude approaches 0.

"Surface" Waves Have Depth.
Water particles below a wave's surface are also in circular motion. Amplitude decreases by 6 dB per λ/9 of depth. Oceanographers consider this "wave base" relevant to a depth of λ/2, where surface-referenced amplitude is -27 dB. If waves enter water shallower than λ/2, as in a gently sloping beach, particle orbits flatten into ellipses. The frequency and period do not change, but phase velocity and wavelength decrease; amplitude increases, making crests steeper. Ultimately, the crests become unstable and "break." The velocity of a wave's leading phase is less than that of its lagging phase, causing crests to curl forward while breaking.

Phase Velocity Equations.
In deep water, phase velocity (c) depends on wavelength and restoring force according to

c2 = Ga + T/Da            (Equation 3),

where a is the wavelength determinant (a = λ/2π; see Figures A and B), G is acceleration due to gravity (980 cm/sec2), T is surface tension (72.75 dynes/cm), and D is water density (1.00 g/cm3 for pure water). For gravity waves, λ > 10 cm, and the surface tension term (T/Da) may be neglected; for capillary waves, λ < 0.3 cm and the gravity term (Ga) may be neglected. As water gets shallow (depth < λ/2), phase velocity trades dependence on wavelength for limitation by depth (d), and is ultimately expressed by:

c2 = Gd            (Equation 4).

Figure C plots phase velocity versus frequency obtained by solving Equation 2 in terms of Equation 3, with Equation 4 providing the depth-limited plateaus. Clearly, surface waves in deep water are dispersive--their speed depends on their frequency. Effectively, deep water is a spectrum analyzer: Low-frequency gravity waves from a distant source arrive before those of higher frequency. Figure D plots approximate frequency ranges for examples of water surface wave phenomena.

Water surface wave (C) phase velocity versus frequency, (D) frequency range of examples, and (E) significant and maximum height versus frequency, with height  limit of WT response.

Natural Spectra.
Since a stable wave's amplitude is limited to rmax = a/2 in deep water (see above), maximum wave height hmax = a = λ/2π. The solid curve in Figure E is hmax versus frequency with height expressed in dB referred to one millimeter (0 dBmm = 1 mm). As frequency increases, the maximum theoretical height of gravity waves decreases at -12 dB per octave.

The vast majority of surface waves originate from atmospheric energy transferred via wind. A "fully developed sea" is in equilibrium with wind. Its spectrum is asymmetrical, with a sharp cut-off below a wind speed-dependent peak frequency, and a broad skew toward higher frequencies. Oceanographers use "significant wave height" in spectral studies. This is the average height within the tallest 1/3 of the waves in a sample. The dashed curve in Figure E is significant wave height at the peak frequencies of fully developed seas at various wind speeds. For any given frequency, significant wave height is about -15 dB less than the theoretical maximum height.

Implications for Surface Wave Transducer Design. The curves in Figure E help to define practical limits for water surface wave transducers, such as my "Water Theremin" (WT). As shown in Figure 3 of the WT's tech page, with 36-AWG antenna and five-stage frequency division, my WT-demodulator combination has a maximum crest-to-trough response of only 30 mm (29.5 dBmm). The dotted curve in Figure E shows this limit. This curve's -6 dB per octave roll-off beyond 37 Hz corresponds to the demodulator's dynamic limit (See Data Converter's tech page)--which appears unlikely to be reached in nature. (Actually, however, the practical cut-off is likely at a lower frequency, related to how fast the antenna can shed water.) At best, my WT's domain is ripples and small wavelets. A transducer designed for ocean swell would require at least 50 dB greater headroom!
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My research for this page relied on the following resources:
 
Cycloid/trocoid math:
Water surface wave equations and physical oceanography:
Ocean wave spectra; significant wave height versus frequency:


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