Time-Compressing Infrasonic Recordings to Discover New Sounds, by Clark
Surface Waves Versus
visibility and relatively low speed makes water surface waves a common analogy for sound. But no
is perfect. Since one goal of the Infrasonicon is to convert the former
into the latter (see Overview Part 4), this page outlines some similarities and differences
between the phenomena. They affect transducer design requirements and
help predict time-compressed playback results.
waves on water and sound waves in air have much in common:
The relative difference in speed between sound and water surface
waves can be considered quantitative. But there are key qualitative
- Both convey
(and information) without changing the average structure of their
- Their power is
proportional to amplitude squared, so decibel (dB)
scales may be applied to water waves, as they are to sound.
- Both wave types
be reflected, refracted, absorbed, and/or diffracted by boundaries.
- Like all wave
phenomena, the relationship
between frequency (F), wavelength (λ), and propagation speed (c; also called phase
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
"x" axis can be time, distance, or phase angle. While physical sound
longitudinal, their oscillographic display is transverse; one can envision the axis of particle
motion rotated 90 degrees to represent the waveform in two dimensions.
- 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.
- Air particle
in sound is longitudinal (co-axial with propagation rays); water
motion in surface waves is circular (deep water) or elliptical
orbiting in vertical planes parallel to rays.
- The restoring
force, acting to return displaced particles to their
equilibrium (average) position, is elasticity for sound waves; depending on
it is gravity and/or surface tension for water surface waves.
sound is independent of frequency, but in deep water, surface wave
depends on wavelength and thus frequency, making surface waves dispersive.
- Within the dynamic range
perception (over 120 dB), air (near sea level) does not limit amplitude
audible wavelength. But the amplitude of water surface waves is limited to a
fraction of their wavelength.
In contrast, the
circular particle motion in water surface waves has both longitudinal
transverse components. Sinusoids thus cannot represent water surface height
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.
circle on the right (radius = a)
depicts the counter-clockwise path of imaginary water surface particle
circle's center is P's average position, which is stationary. Wave
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
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:
Notice the asymmetry of the cycloid in Figure A.
Crests are pointed and troughs
are broad, like a cartoon water waveform.
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
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.
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
asymmetrical. But symmetry increases as amplitude decreases,
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
wavelength decrease; amplitude increases, making crests steeper.
the crests become unstable and "break." The
velocity of a wave's leading phase is less than that of its
causing crests to curl forward while breaking.
Phase Velocity Equations. In deep
water, phase velocity (c) depends on
wavelength and restoring force according to
the wavelength determinant (a = λ/2π; see Figures A and B), G is
acceleration due to gravity (980
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:
plots phase velocity versus frequency obtained by solving Equation 2 in
of Equation 3, with Equation 4 providing the depth-limited plateaus.
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
of higher frequency. Figure D plots approximate frequency ranges for examples
of water surface wave phenomena.
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
majority of surface waves originate from atmospheric energy transferred
wind. A "fully developed sea"
is in equilibrium with wind. Its spectrum is asymmetrical, with a sharp
below a wind speed-dependent peak frequency, and a broad skew toward
frequencies. Oceanographers use "significant
wave height" in spectral studies. This is the average height
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
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
37 Hz corresponds to the demodulator's dynamic limit (See Data Converter's tech page)--which appears unlikely to be
in nature. (Actually, however, the practical cut-off is likely at a lower frequency,
how fast the antenna can shed water.) At best, my WT's domain is
small wavelets. A transducer designed for ocean swell would require at
50 dB greater headroom!
My research for this page relied on the following resources:
surface wave equations and physical oceanography:
wave spectra; significant wave height versus frequency:
Infrasonicon Home Page
Example Page Water Theremin Overview Water Theremin Tech Page
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