Standing Water Waves Research Articles

Internal structure of extreme standing waves on deep water

A special procedure is applied to remove the non-physical distortion that always appears when numerically calculating velocity and related quantities in the close vicinity of a discretized vortex sheet. The procedure is used in conjunction with a semi-Lagrangian boundary integral method to study the flowfield beneath free surface standing water waves. Pressure, velocity and acceleration are calculated below the wave and some aspects of crest formation are investigated. At large depths results confirm the existence of an unattenuated oscillatory pressure component, giving behavior consistent with previously known theory.

The form of standing waves on finite depth water

A stable and accurate numerical method to calculate the motion of an interface between two fluids, the lower fluid being of a finite depth, is used to calculate two-dimensional standing water waves. Such standing waves may occur within fluid tanks or containers. Particular attention is paid to the resonant depths where multiple solutions are possible.

On the modulational stability of traveling and standing water waves

Asymptotically exact evolution equations are derived for trains of small amplitude counterpropagating water waves over finite depth. Surface tension is included. The resulting equations are nonlocal and generalize the equations derived by Davey and Stewartson for unidirectional wave trains. The stability properties of stationary standing and quasiperiodic waves are determined as a function of surface tension and fluid depth for both long wavelength longitudinal and transverse perturbations.

Sand bed configurations under 2‐D standing waves

Abstract Laboratory experiments conducted in a wave basin with movable sand bed have shown that, under the action of 2‐D standing water waves, longitudinal sand bars with small ripples riding on their tops are formed beneath the nodes of standing waves. Wavelengths of the sand bars are approximately equal to one‐half of those of standing water waves. The formation of sand bars under standing waves over an initially horizontal sand bed is explained by the suction and impinging effects of the wave action. A mathematical model for describing the induced density of sediment‐water mixture due to wave action is proposed. By employing the induced‐density model and mass conservation of sediment‐water mixture, a numerical simulation model is constructed. Bed configurations resulting from the actions of 2‐D standing waves are studied by using the simulation model. The simulation results of sand bed configuration under the action of 2‐D standing waves are in good agreement with those obtained from laboratory experim...

Standing waves in deep water: Their stability and extreme form

A stable and accurate numerical method to calculate the motion of an interface between two fluids is used to calculate two-dimensional standing water waves. The general method calculates arbitrary time-dependent motion of an interface, possibly including interfacial tension and different density ratios between the fluids. Extremely steep standing waves are determined, significantly steeper than has been previously reported. The peak crest acceleration is used as the determining parameter rather than the wave steepness as the wave steepness is found to have a maximum short of the most extreme wave. Profiles with crest accelerations up to 98% of gravity are calculated (a sequence of raster images of this profile as it evolves in time over one period may be obtained upon application to the authors: e-mail gmercer@spam.ua.oz.au or aroberts@spam.ua.oz.au), and the shape of these extreme standing wave profiles are discussed. The stability of the standing waves is examined and growth rates of the unstable modes are calculated. It is found that all but very steep standing waves are generally stable to harmonic perturbations. However, standing waves are typically unstable to subharmonic perturbations via a sideband-type instability.

Highly nonlinear short-crested water waves

The properties of a fully three-dimensional surface gravity wave, the short-crested wave, are examined. Linearly, a short-crested wave is formed by two wavetrains of equal amplitudes and wavelengths propagating at an angle to each other. Resonant interactions between the fundamental and its harmonics are a major feature of short-crested waves and a major complication to the use at finite wave steepness of the derived perturbation expansion. Nonetheless, estimates are made of the maximum steepnesses, and wave properties are calculated over the range of steepnesses. Although results for values of the parameter θ near 20° remain uncertain, we find that short-crested waves can be up to 60% steeper than the two-dimensional progressive wave. At limits of the parameter range the results compare well with those for known two-dimensional progressive and standing water waves.

Simple Experiments on Sound Velocity and Standing Waves in Water

Simple Experiments on Sound Velocity and Standing Waves in Water

Higher Order Calculation of Finite Periodic Standing Gravity Waves by Means of the Computer

A computer-aided calculation is made of the standing water waves which can be produced in liquid contained between two parallel vertical plane walls. The periodic solution is obtained analytically up to the eighth order of an amplitude by the use of the formula manipulation language REDUCE-2. This result agrees with that of Penney and Price (Phil. Trans. Roy. Soc. A 244 (1952) 254) up to the fourth order, but not in the fifth order. It is clue to their algebraic error in this order. The present solution gives 0.138λ as the maximum crest height of the greatest stable standing wave, where λ is the wavelength. The wave profile from our solution is in better agreement with experiments than that from Penney and Price.

Capacitance measuring circuit for water waves

A simple, reliable, integrated circuit is described which is used to measure standing water waves of amplitudes ranging from less than 1 mm. The circuit has a high dynamic range and is linear for changes in water depth of several centimeters. The sensitivity of this capacitance measuring device is 38 mV/mm of wave amplitude, enabling small amplitude waves to be readily detected.

On the natural frequencies of standing water waves in a canal of arbitrary shape

The small amplitude natural oscillation of liquid in a partially filled canal of arbitrary shape is investigated. The method is based on potential theory. The field equation subject to the appropriate boundary conditions is transformed into an integral equation, which is then solved numerically. Two specific geometries were chosen for the numerical examples; namely, a circular canal with arbitrary depth of liquid, and half full elliptical canals with various eccentricities. The results are presented and discussed.

A Theory of Microbaroms

summary A theory analogous to the Longuet-Higgins theory on the generation of microseisms explains the generation of microbaroms by standing water waves associated with marine storms. The spectral characteristics and the amplitude order-of-magnitude of microbaroms that are predicted by this theory agree well with observations. The theory is based on the oscillations of the centre of gravity of the air above the Ocean surface on which the standing waves appear (or of the water below, to explain microseisms). These oscillations are of twice the ocean wave frequency and thereby explain the observed frequency-doubling common to both microbaroms and microseisms. The theory is expanded by statistical methods to predict the microbarom-generating effect of more realistic ocean waves, whose phases vary randomly over the Ocean surface. In addition, the effect of the widespread source on microbarom coherence and resolvability at the receiving array is discussed.

Excitation of seiches in channels by seismic waves

Description of seiches which occurred along the Gulf Coast of the United States and in Norway at the times of the Alaskan earthquake of 1964 and the Assam earthquake of 1950 are reviewed. An analytical expression is derived for the effect of seismic surface waves on a water channel of width L and depth H. The expression is used to explain many of the observations of Norwegian seiches and also to compute a theoretical marigram which is compared with an actual marigram recorded in a channel at Freeport, Texas. The horizontal components of seismic surface waves are shown to be the dominant cause of seiches due to teleseisms. The maximum seiche amplitudes vary as (1) the seismic wave amplitude, (2) (H)1/2 where H is the channel depth, and (3) the degree of resonance between the seismic waves and the natural periods of oscillation of standing water waves in the channel. The seismic surface waves from the Alaskan earthquake provided enough energy to produce all of the observed seiches in the Gulf Coast area. The seiche observations were limited to the Gulf Coast region because the thick sediments unique to this area are necessary to provide sufficient seismic wave amplification for the production of seiches.

An experimental study of the pressure variations in standing water waves

Although the first-order pressure variations in surface waves on water are known to decrease exponentially downwards, it has recently been shown theoretically that in a standing wave there should be some second-order terms which are unattenuated with depth. The present paper describes experiments which verify the existence of pressure variations of this type in waves of period 0·45 to 0·50 sec. When the motion consists of two progressive waves of equal wave-length travelling in opposite directions, the amplitude of the unattenuated pressure variations is found to be proportional to the product of the wave amplitudes. This property is used to determine the coefficient of reflexion from a sloping plane barrier.