Superposition of Water Waves

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The sea surface motion is the superposition of numerous wave trains. Here one to four (linear) waves can be summed, travelling either in the +x direction (denoted with a +1) or the opposite direction (-1). You have the choice of examining one, two, three, or four waves together (by making the appropriate wave heights zero). The displayed quantities: x_max and time denote the width of the viewing panel in meters, and the elapsed time in seconds, repectively. Pressing Stop and then Calculate will reset the time. (Obviously the plot region is not to scale with the depth.) The waves that comprise the total wave field can be viewed in an unsummed form by choosing Components instead of Superpose with the Choice button; however, the animation is much slower and the buttons are less responsive.

A variety of phenomena can be examined with this applet:

• single wave trains, found by zeroing the heights of the other three waves. The speed of the wave, C, can be determined by noting the time that it takes a wave to traverse the entire screen: C=x_max/time.
• wave groups, which are very noticeable when two (or more) waves have nearly the same wave period. The group velocity can be determine empirically by timing the group over the distance, x_max. Conventionally, the waves travel in the same direction and the wave groups move at a velocity less than the individual waves velocity; however, try changing the direction of one of the waves. (change a +1 to a -1).
• standing wave systems, when two waves have the same characteristics, but are travelling in the opposite directions (change a +1 to a -1), such as would occur when an incident wave reflects from a wall. The wall would be located at the left side of the graphing panel.
• wave groups reflecting from a wall (at left boundary of plot). Use two waves to make a wave group (each wave with a slightly different period); then add two more waves with opposite direction, but otherwise the same characteristics. Note the long-term behavior.

The width of the plot window can be adjusted by the slide bar and hitting Stop and Calculate. Be careful that you don't plot too wide an area because the lack of plotting resolution can give spurious results.

The time step is taken as 1/30 of the period of the first wave.