8. Two Dimensional Waves.


So far these simulations have only shown one dimensional waves. Even though there is motion perpendicular to the direction the wave travels for a transverse wave, the function describing the wave is a function of only one spacial variable, x. Waves can exist in two or three dimensions, however. One example is a plane wave where the wave front or crest of the wave makes a line (in two dimensions) or a plane (in three dimensions). Cirular waves (in two dimensions) and spherical waves (in three dimensions) also exist. These simulations can only show two dimensional waves.

The simulation below starts by showing you a plane wave in two dimensions traveling in the x-y plane, in the x direction, viewed from above. In these simulations the ligher regions represent the bottom of the waves, the darker regions represent the tops of the waves.
Note: Animations may take a few seconds to load.



wavelength =  period =  s

side view. (Amplitude plotted in the z-direction.)

Questions:

8.1. Click on the 'play plane wave' button. Experiment with wavelength and period. Is the simulation accurate in representing a fixed speed for a real wave? How do you know?

8.2. Can this representation be used to describe longitudinal waves as well as transverse waves? Why or why not?


The 'side view' box shows the amplitude of the wave in the z-direction but keep in mind the wave shown is really a function of just two variables, x and y and so is not a true three dimensional wave. For a plane wave traveling in the x direction the amplitude is the same for any value of y so the equation describing it is exactly the same as the one dimensional wave. The equation for a plane wave traveling in an arbitrary direction in the x- y plane is given by where z is the height of the wave at location (x,y) at time t, kx = k cos q  and ky= k sin q where q is angle between the probagation direction and the x axis. 

8.2. Click in the 'side view' check box and then 'play plane wave' to see the side view. You can grab and rotate the image by holding the mouse button down and moving it. In this view the waves appear to be transverse where the amplitude of the transverse motion is plotted in the z-direction. What would be plotted in the z-direction if this was a representation of a longitudinal wave?

8.3.  Suppose the simulation above represents waves in a large, flat pan of water. Given a pan of water and other equipment of your choice in the laboratory, describe how you could create plane waves.


Circular and spherical waves are also examples two and three dimensional waves. The waves produced by dropping a rock into a lake are and example of cirular two dimensional waves. A source of nearly spherical waves is a light bulb which emits light in almost all directions at once. Fortunately we can still describe the wave by the same sine function if we switch to polar coordinates. For a cicrular wave we use so that the equation describing the wave is .

8.4. Uncheck the 'side view' box and click on the 'play circular wave' button. What is happening to the curvature of the waves as they move away from the source? Is it becoming more or less like a plane wave?

8.5. This simulation is unrealistic in one sense because the amplitude of the circular wave does not change as the diameter gets bigger. Why is this unrealistic?

8.6. Click in the check box and then 'play circular wave' to see the 2-D version. Suppose this circular wave simulation above represents waves in a pan of water. Given a pan of water and other equipment of your choice in the laboratory, how could you create these waves?

Credits.
Go To: IUS Physics Top Page.
Contact Dr. K. Forinash, for comments/suggestions/corrections.