There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

Origins of Newton's laws of motion

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

## Thursday, March 13, 2008

### Water waves

When a person mentions "waves", water waves usually come to mind ... those long lines of undulating waves rolling into the shore. Of course there are many other types of waves: sound waves, light waves, and seismic waves, to name a few, but water waves embody the very essence of waves for most of us. In spite of their being common, water waves are one of the more complex types of waves. They are dispersive, often non-linear, and involve both longitudinal and transverse motions. We will explore these concepts in this posting.

Fig. 2 below is an animation of water waves, showing marker dots and their trails to emphasize the motion of the water inside the wave. A few of the dots are red just to help your eye focus on the motion of a single dot. (Try to focus on a particular blue dot away from the red ones to appreciate how visually confusing a school of fish is to its predators.) Mouse over the animation to start it and off to suspend it. Click on it to restart it.

#### Key to numbered buttons in animation

The buttons at the bottom of the animation represent options to cause different types of waves to appear. An explanation of the waves for each button follows. While we are interested primarily in buttons 1, 2 and 3 that involve water waves, we include the rest, so that you can compare water waves with other types of waves.
1. A normal water water, of the traveling wave type. Note the dots move in circles. The water near the surface moves in larger circles while the water further down moves in smaller circles.
2. This is a larger amplitude traveling wave. The amplitude has been adjusted to make the wave have very sharp cusps at its crests. Real water waves produce spray at the crests at this amplitude. The animation's math becomes incorrect at greater amplitudes than this.
3. This button adds a reflection, causing a standing wave version of a water wave, once the wave has propagated across the screen and back again. The amplitude is large enough to make the wave crests into sharp cusps.
4. This is a traveling wave version of a compression or sound wave. It is not a regular water wave. It could be a very slowed-down representation of sound propagating in air, water, or a solid.
5. This is a standing wave version of a compression wave. Wait for the wave to propagate across the screen and back for the standing wave to form.
6. This is a traveling wave on a string of beads. You see only the beads ... the string is invisible. Note that the motion of the dots is purely up and down, as opposed to the combination of both horizontal and vertical motion for the water waves.
7. This is a standing wave version of a wave on a string.

Fig. 1. Images of water waves from Wikipedia and morguefile.

Fig. 2. Dynamics of water waves and other types of waves.

### Some properties of water waves

• If you examine the various types of waves in the animation above, you will see that:
• Water waves involve motion in both the x and y directions.
• Compression or sound waves involve motion in only the x direction.
• Waves on a string involves motion in only the y direction. This is somewhat of a simplification, but it is approximately true.

• Comparing the shape of the surface of water waves to that of waves on a string, we see that while waves on a string have the normal rounded sinusoidal shape, water waves can have spike-like crests. This is due to the combination of both x and y motions. The x motion of the water converges at the crests to sharpen them. In the valleys of the waves, the x motion is moving water away from the center of the valleys, broadening the valleys. These effects are most apparent on option 2 of the animation, showing larger amplitude traveling waves. For this amplitude, the water surface makes a perfect cycloid shape. This same shape has long been studied by mathematicians and was named by Galileo in 1599. It is the shape that a spot on the rim of a wheel makes when the wheel is rolled along (the shape made by the wheel is actually upside down from the shape of the water surface.) Figure 3 below is an animation illustrating this. At lower amplitudes, a water wave does not form the sharp points, or cusps, but instead forms the more rounded shape of a curtate cycloid which is the path of a point on the wheel closer to the wheel's center than the rim, as shown with the blue and green traces on the animation. These rounded shapes are similar to the normal sinusoidal shapes seen in waves on a string and previously posted animations on this blog. The previously posted animations of water waves show sinusoidal profiles to be consistent with the material posted there, and are not exactly correct for real water waves.
 Fig. 2b. Close up of the phasor dials in Fig. 2 above.

 ← Fig. 3. Animation showing the generation of a cycloid shape by a rolling wheel. Mouse over the figure to see the animation and off it to suspend the animation. Click on it to restart it. The red, blue, and green lines are made from "lights" mounted on the wheel at three different radii (measured from the center of the wheel). The generated shapes are cycloids, the red being a true cycloid while the blue and green lines are curtate cycloid and do not have the sharp cusps of a true cycloid.

• When the amplitude of the waves is large enough to form the sharp peaks, the waves motion becomes non-linear and dissipative effects set in. These result in energy loss from the wave. Such a wave is hard to mathematically analyze. We will do the math for the non-dissipative waves in the next posting. Hyperphysics delves into this in more depth.

• In option 1 of Fig. 2 above, the traveling waves involve water parcels moving in circles and ellipses as mentioned above. The animations in Fig. 4 below further illustrate this. In deep water, these are circles while in shallow water, they are ellipses. Near the bottom, the motion becomes entirely horizontal and the ellipses flatten into lines. In option 3 of Fig. 2 above which features a standing wave, the water parcels move in slanted straight lines.

 Fig. 4a. In deep water, parcels of water through which a wave is traveling move in circles. These circular paths are smaller for water farther from the surface which means the wave action is pretty much confined to a region near the water surface. More specifically, at a depth equal to one wavelength, the circles are e-2π = 1.9×10-3 = 0.2% of their size at the surface and continue to decrease exponentially the deeper one goes. Mouse over the figure to see the action. Fig. 4b. In shallow water, parcels of water through which a wave is traveling move in ellipses. These elliptical paths are smaller and flatter for water farther from the surface. At the very bottom, the water motion is entirely horizontal and the elliptical paths are completely flattened. Mouse over the figure to see the action. Fig. 4c. In very shallow water, parcels of water through which a wave is traveling move in flat ellipses. The wave involves more slushing of water to and fro in the horizontal direction and less vertical motion. The width of the ellipses do not decrease much as one moves to the bottom. Their height, however, does reduce to zero at the very bottom. Mouse over the figure to see the action. Fig. 4d. Animation of "natural phasors" built into water waves. The motion of a water parcel can serve as a phasor, since in the case of water waves, water parcels trace a similar motion as do the tips of phasors. For waves moving towards the right, the water parcels move clockwise around their circular paths. The phasor we draw above is similar to the normal phasor except that we rotated the axes 90 degrees so that the real axis points upwards, and we flipped the imaginary axis so that its positive direction points to the right. This later flip is necessary to make a phasor that rotates clockwise, as opposed to the normal phasor rotation of counterclockwise.

• Figure 2 above shows phasor diagrams for the waves. These phasors show the amplitude and phase of one aspect of each of the waves shown. As shown in Figs. 2 and 2b, options 1, 2, and 3, we have linked the phasors to the variations in height of the water surface due to the wave. We have also rotated the phasors so that the positive real axis is pointing upwards and the positive imaginary axis is pointing towards the left. In this orientation, a phasor will be pointing straight up if the wave immediately above it is at its maximum upward displacement. In the case of the water waves (options 1, 2,and 3), the phasors show the amplitude and phase of the height aspect of the waves, y(x,t) (Remember water waves involve both vertical and horizontal motion.) We could equally well have had the phasors show the amplitude and phase of the horizontal displacement of the wave. In the case of the compression wave (options 4 and 5), the phasors show the amplitude and phase of the compression of the "atoms". In the case of the chain of beads (options 6 and 7), the phasors show the vertical displacement from the average height, similar to the water wave case.

• In the case of traveling water waves, the motion of the parcels of water actually trace out natural phasor diagrams by virtue of their circular motions. To make the correspondence with the phasors, we see that the maximum wave height occurs with the water parcels at the top of their circular paths. Thus, the top of their paths corresponds to the direction of the real axis in the complex phasor, the same orientation as the phasors in the animation. In this animation, the traveling waves are propagating towards the right which makes the parcels move in a clockwise direction, whereas convention has phasors rotating in a counterclockwise direction. Because of this difference in rotational directions, a parcel in the animation will move as though it were the mirror image of the phasor tip just below it (look at a red dot and the nearest phasor). Fig.4d shows the superposition of a phasor on the water paths, where we have also flipped the imaginary axis to make the phasor rotate along with the wave parcel. In the case of water parcels tracing out elliptical paths, these natural phasor dials are somewhat flattened, perhaps similar to the distorted clock faces in a Salvador Dali painting. See my earlier postings for more on phasors. Actually, a number of the previous postings deal with the subject of phasors. For example, there is a posting on phasors for waves.

• Gravity is the main restoring force for water waves. The momentum of the waves forces the surface from its initial flat shape, while gravity tries to restore the water's initial shape. It is the interplay of these two effects that allows the wave to propagate. Water waves are also called gravity waves by physicists.

• A water wave is a surface wave. It occurs at the interface between the water and the air. In a deep body of water, at a depth below a few wavelengths, there is little motion of the water due to the wave. The wave is confined to a region near the interface. You can see this effect in the animations above by the decreasing size of the circles the water parcels trace out.

• Wind causes most water waves. Wind exerts a force on the wave crests and pumps energy into the waves that happen to be moving in the same direction as the wind, causing them to grow. Windless days often result in mirror-like waveless seas, while strong winds result in large waves. Strong winds interacting with waves over long expanses of water usually cause very large waves. Most often, there is a mix of waves of varying wavelengths (traveling at varying velocities, we shall see below). This causes the complex surface we often see at the ocean.

• Another source of waves are underwater earthquakes which can cause giant, destructive waves called tsunamis. Hyperphysics has a good discussion of tsunamis.

← Figs. 5a and b. Altocumulus undulatus clouds and satellite image of internal ocean waves (center photo) near the Straits of Gibraltar. Both are examples of internal waves occurring at the interface between two layers of a fluid. Images from NOAA and ESA
 ← Fig. 5c. Animation showing interior waves at the interface of warm surface water and deeper cold water, excited by the flow of the surface water. The circles and ellipses indicate the motion of the water parcels caused by the internal wave (ignoring the steady flow of the surface water). The wavelength of interior waves ranges from a few meters to kilometers long. They typically propagate much more slowly than surface waves. Mouse over the figure to see the action. The regular surface wave propagation is not animated.

• Similar waves will occur at the interface between any two fluids of different densities. Examples of these internal waves are waves in altocummulus and altostratus clouds formed at the interface between air of two different temperatures. See the photo above for one example. The wave motion takes place at the interface between air layers of different temperature and affects cloud formations also occurring at this interface. In a sense, the clouds are the markers of the waves. See more on this type of internal wave look at cimss.ssec.wisc.edu, usra.edu, weathervortex.com, taylor.math.ualberta.ca, and absoluteastronomy.com.

• Other examples are the internal waves that occur at the interface between different layers of water deep in the ocean, where there is an abrupt change in temperature or salinity. These do not result in large undulations in the surface of the water, but may cause very small slowly moving undulations in surface height, in the water color, or in the surface roughness, any of which can allow satellites to observe them. They also can be detected by underwater probes that record the undulations in temperature and salinity layers. The wave motion is confined to a few wavelengths on either side of the interface. These underwater, internal waves are thought to be responsible for much of the mixing between the layers of water in the ocean. See pictures and discussions at the U. of British Colombia website and the Woods Hole website.

•  Fig. 6. Graph of the velocity, c, of water waves versus their wavelength, λ, for various water depths, h. In deep water, waves with longer wavelength travel faster than shorter ones. In shallow water (where the water depth is small compared to a wavelength) the wave velocity does not vary with wavelength, but it does vary with water depth.
Another property of water waves is that they are dispersive. This means that their velocity varies with wavelength. Waves of longer wavelength travel faster than those with short wavelength. This means that very long waves like tsunamis can travel at very great speeds while waves in a mud puddle travel at a few tens of cm/s. The graph at the right shows the dependence of the speed on wavelength and water depth as given by the equation

,

where g = 9.8m/s2 is the acceleration of gravity, λ is the wavelength of the water waves in meters, h is the water depth in meters, and κ = 2π/λ is the wavenumber in radians per meter. The average depth of the oceans is about 3800m, with the deepest trench being about 10,000m deep, which corresponds to the upper most part of the graph. We might remember that the equation has been derived assuming a flat ocean bottom and is not valid for propagation where an underwater trench is important.

• The theory behind the equation ignores surface tension, which is important for short waves, i.e. ripples, of wavelength less than 10cm. Physics of waves by Elmore and Heald covers surface tension effects in Chapter 8. Hyperphysics does also. Surface tension makes the velocity of waves increase below a certain wavelength. The minimum velocity is at a wavelength of 1.7cm with a velocity of 24cm/s for pure water. Waves of wavelength shorter than this will travel faster. We need to remember that surface tension is strongly affected by impurities such as soap so that these will strongly affect the wave velocities at these short wavelengths.

• The above equation for the velocity of water waves was derived assuming small amplitude waves. The wavelength can be short, medium, or long, but the amplitude should technically be small for the equation to be valid. Large amplitude waves generally travel faster than predicted by the equation and the graph.

• A useful approximation for the case of deep water, i.e. where the water depth is much greater than a wavelength, is given by:

.

In the case of very shallow water, where the water depth is much less than a wavelength, then the speed of water waves can be approximated as:

.

 Fig. 7. Poseidon, Greek god of the seas.

The animations in this posting can be downloaded free from George Mason University Archival Repository. Please read the fair use policy for this work.