Ms. Wai In LAU
S.K.H. Tsang Shiu Tim Secondary School
Dr. Ho-Fai CHEUNG
Department of Physics and Materials Science, City University of Hong Kong

Have you ridden a boat or a ferry? Guess what kind of wake would a boat or ferry left? The following diagrams (Fig.1 and Fig.2) show two possibilities.

(Fig.1 An illustration showing the possible pattern of the wake produced by a small boat.)		(Fig.2 An illustration showing another possible pattern of the wake produced by a small boat.)

Fig.1 shows long lines of wave in the wake generated by the boat. Fig.2 shows many shorter lines of wave, each of them at a certain angle to the wave front in the wake. You may guess right, picture 2 is a more correct description of the wake. Please see the photo (Fig.5) at the end of this article. As a science student, we always look for simple understanding of a phenomenon. In fact, this phenomenon is closely related to one important characteristic of surface waves on water. This article will give you a brief introduction to surface waves on water, or simply water waves.

A wave is a disturbance that propagates in space. The usual examples taught in secondary school textbooks are sound waves and light waves. Sound waves are pressure disturbances in air, while light waves are electric and magnetic disturbances in space. Have you wonder why water wave is not often used as an example for detailed discussion? Water waves are simple in the sense that you can see it, and you have probably seen it many times. The physics behind is simple (water waves are driven by gravity), but associated phenomena are more complicated. For example, it is not easy to explain why water waves broke.

In describing the properties of waves, the simplest type and at the same time a very special type of wave is often adopted. They are the plane travelling waves or simply the plane waves. Figure 3 illustrates the appearance of a plane wave on water. To see the motion of the wave, you have to double-click the picture. On seeing its motion, it should be obvious why this is called travelling wave. Once we know the properties of these plane waves, we can use the knowledge to understand more complicated waves. So knowing the properties of plane waves is an important first step.

( Fig.3. A video showing a plane travelling water wave. To see the motion of the wave, you have to click the "start" button. )

(Fig.4. A plane water wave showing the crest, trough, and 1 wavelength.)

Letˇ¦s define some technical terms to describe plane waves. Take the water wave shown in Fig.4 as an example. The highest point of this water wave is called the wave crest, and the lowest point is called the wave trough. Starting from this, we want you to note that ˇ§all plane travelling waves are characterized by their amplitude, wavelength, frequency, period, and wave velocityˇ¨. For this particular water wave, amplitude is the height of the wave crests above the average water level. Wavelength () is the length between two consecutive wave crests. To measure the wavelength, we have to freeze the motion of the plane wave (for example, by imagination). Next, the frequency (f), which is the number of crests passing over a fixed point in space within a unit of time. The period (T) is the time lapse for the next crest to move up to the position where the first crest was at initial. The velocity (v) is the distance that the crests travelled within a unit of time.

As we have mentioned before, the usual examples taught in secondary school textbooks are sound waves and light waves. Surface wave on water is usually not discussed in any details. The basic reason is ˇ§sound wave and light wave is simpler in the sense that their velocity is roughly constantˇ¨. For example, the velocity of light is roughly equal to 300,000,000 meters per second and the velocity of sound is roughly equal to 330 meter per second. These numbers are independent of the wavelength or frequency of the plane waves. In contrast, there is no single velocity for water waves. The velocity of plane water waves is highly dependent on both their wavelength and the water depth. The following table shows the values measured in our laboratory (see reference 2).

Table 1: Water wave velocity for water depth = 0.025m
[(*)Remark: Within experimental error of about 10%, these value are very much equal]

One should distinguish between deep-water waves and shallow-water waves. If the wavelength of the water waves is smaller than 10 times the water depth, then they can be classified as deep-water waves. Most water waves we see everyday are deep-water waves. In the above table, the last 5 rows are for deep-water waves. Among them, those with longer wavelength actually travel faster. This somewhat "strange" behavior of deep-water waves is the basic starting point to understand why the wake of boat or ferry looks like that in Fig.5.

If the wavelength of water waves is larger than 10 times the water depth, then they can be classified as shallow-water waves. In the above table, the first 4 rows are for shallow-water waves. For a fixed water depth, all shallow-water waves travel with the same velocity, a value that is independent of the wavelength. When the water depth is changed, shallow-water waves travel faster in deeper water. This is the starting point to explain why water waves break when they come ashore.

The most horrifying water waves occurring in nature are the seismic sea waves (or Tsunami). These are huge water waves of very long wavelength created by earthquakes or volcanic activities. The seismic sea wave caused by the 15 November 1994 Mindoro earthquake is reported in website 3. This water wave was at least 6 meters high. At least 41 persons died of drowning and 1530 houses were destroyed. The period of this wave is known accurately. (It is approximately 20 minutes) With this information, we can estimate its wavelength in the Pacific Ocean. The average depth of the Pacific Ocean is about 4 km. Then the velocity of this seismic sea waves would be 710 kilometer per hour, and its wavelength would be 240 km. Although they travel slower in shallower water, they are still hard to escape from.

As I have mentioned before, water waves are driven by gravity. You may be able to guess that water waves generated on the moon would travel slower. You may even realize that strange phenomena will occur if we try to generate water waves on a space station, where the "gravity" is zero. Indeed one will find unexpected phenomena.

(1) In a space station orbiting around earth, the force of gravity is completely "cancelled" out. The shape of water in space will be dominated by surface tension. For example, water will form spherical drops of varying sizes. Water inside a container will have inward or outward curved surface depending on the surface tension at the container surface.

(2) Surface waves can be generated on these water surfaces, and they will be driven by surface tension. Although you may not know anything about surface tension, you can feel that these water waves will have velocity different to the usual surface waves. I will not go into the details any further. Please read the references if you want more.

(Fig.5. Photo of the wake created by a ship. The image at the above right was found at the US Navy's Digital Image site. The photo was taken by Photographer's Mate 2nd Class Christian Eskelund, is a US Navy photo and free for public use.)