ripple tank Spring 2004
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Capstone (Math 451H) Spring 2004:

Theory design and use of a ripple tank to illustrate linear wave
phenomena

A large-size ripple tank, 48 in x 30 in with water depth
up to 2 in, was set up to study linear wave phenomena.

The main purpose of the project was to analyze and demonstrate diffraction -
loosely, the ability of a wave to `slightly bend around a corner'
or other sharp edge in its path, or - for light - to blur the light-dark
boundary at the edge of a shadow.

Diffraction can occur in different types of linear waves, but the ripple
tank uses waves or ripples on the surface of water below air to demonstrate
it. First, we wanted to understand the behavior of this particular type of
wave, and began by studying the equations that govern the motion of a fluid,
and the conditions that hold at a water-air interface. From this, we found
the `dispersion relation' that must be satisfied by time-harmonic, linearized
waves propagating on the surface of water, which gives their speed as a
function of their wavelength and the water depth.

This introduced the distinction between `dispersive waves', such as
those at the water-air interface of the ripple tank, and `non-dispersive'
waves, such as those governed by the classical wave equation of, for example,
acoustics. Non-dispersive waves have a propagation speed that is independent
of their wavelength, while for dispersive waves speed depends on wavelength.
We also found out how, by selecting the water depth appropriately, the ripple
tank could be operated in such a way that dispersion of waves was minimal.
So, in our studies of diffraction, while the analysis was often based on the
classical wave equation - because of its relative mathematical simplicity,
and with its non-dispersive waves - we used the ripple tank (with its
dispersive waves) to observe the effect experimentally.

The distinction between dispersive and non-dispersive waves turned out not
to matter for our studies of diffraction - because our old-and-borrowed
wave generator could reliably produce a good, single-frequency
(or `monochromatic') harmonic wavetrain. Here is an example, below. The
dark corner in the foreground is showing that the point light source
we used would not illuminate the whole area of the tank well - if we moved
the light, we saw waves there.

We looked at the `Sommerfeld problem' for diffraction from the edge of a
half-plane. The picture below shows the effect experimentally using the
ripple tank, with the half-plane barrier at top-right.

Below are two views showing diffraction by a single slit or aperture of width
about 7 wavelengths. We used approximate analytical methods to describe
`Fresnel' diffraction, which occurs around the light-dark boundary or
beam-edge near the source, and `Fraunhofer' diffraction, which occurs far
from the source. The diffraction fringes of theory can be seen in these
images fairly well.

To understand dispersion, we used the dispersion relation to compute water
surface profiles at a sequence of times after the release of an initial
square-shaped hump of water. (See, for example: Mathematics Applied to
Continuum Mechanics, by Lee Segel. Dover Publications, 1987, Chapter 9.)
The sequence of profiles (below) shows ripples propagating away from the
initial disturbance and `breaking up' or dispersing into longer waves that
travel faster, and therefore appear at the front of the disturbance, and
shorter waves that travel more slowly and stay at the back.

We tried to demonstrate this using the ripple tank, with the initial
disturbance made by pressing a ruler down into the water. The results after
a few attempts were mixed - but perhaps if you download the movie
and watch it slowed down, you will see longer waves passing through or
overtaking shorter waves as they travel from the right to the left of the
picture.

Movie: Ripple Tank Demonstration of
Dispersion.