The speed of capillary waves on water

Paul McKeag,: 2006

Introduction

There are two types of waves on water: gravity waves and capillary waves.  Gravity waves are longer wavelength and travel slowly, whereas capillary waves have very short wavelengths and travel faster than gravity waves.
 
The restoring force for gravity weaves is provided only by the weight of water displaced. The velocity equation for gravity waves is:

Waves that have a wavelength of less than or equal to ~1 cm are defined as capillary waves.  Capillary waves are strongly influenced by surface tension because of their small size. The velocity of capillary waves spreading across water is higher than gravity waves. The velocity of capillary waves increases as wavelengths shorten.
 

Research Question
 
How does the velocity of short wavelengths on water depend on wavelength?
 
Hypothesis
 
The capillary waves will not fit with the gravity wave curve.  They will decrease slightly, and then they will increase once the frequency is increased to a point where gravity overpowers the force of surface tension.

Materials and procedure
 
A speaker (shown at right) was connected to a sine wave signal generator and amplifier. A plastic playing card holder was taped securely to the center of the speaker diaphragm.

Fig 2 The experimental setup. Black paper was put under the clear card container to make the waves more visible.

Five mm of water was added to the tank and black paper was placed underneath to increase the contrast in images of ripples on the water. A ruler was placed close to the tank and two small beads were placed underneath to reduce friction.

Fig 3 - beads in place below the tank.

Photographs

Higher frequency waves were photographed inside, because the waves were more easily seen on the reflection of ceiling lights. Low frequency waves were more easily seen out-doors with the sky as a reflection.  Several pictures were taken of each wavelength, generated by a different frequency. 

 
Data

Table 1

Frequency, wavelength and velocity (fl) for short wavelength standing waves on water.

The data in Table 1 is plotted in Graph 1. The calculated gravity wave curve is shown on the bottom in comparison to the velocities over the wavelengths of the waves in the experiment. The red line shows the sharp split between capillary waves and gravity waves. The changeover point is found to be at about 6 mm.  

Analysis:

Graph 1 shows that the clear differentiation between higher frequency waves and lower frequency waves.  The lower frequency waves of 10, 15, and 20 Hz follow almost parallel to the gravity curve, but the higher frequencies from 50 Hz and up don't follow the gravity curve.  They are more affected by the surface tension of the water.

The velocity of the equivalent gravity waves was calculated with the formula....

  

Discussion
 
The waves were hard to see, especially for the higher frequencies.  The amplitude had to be adjusted so that there were no waves resonating from the sides of the container.  The higher frequency waves were easier to see inside a room with fluorescent lighting, while the lower frequency waves were better seen outdoors.  There were limitations as to how high and low the frequencies could be measured. The highest was 350 Hz because that was the minimum visible wavelength. The lowest was 10 Hz because the speaker was unable to move at any lower frequency.
 
Evaluation
 
The setup of the experiment was simple and efficient.  At first, the experiment was tried without the beads under the water dish. It was found that measuring the higher frequencies like this was more difficult.  Lower frequencies wouldn't work at all without the beads. Perhaps if a student wanted to measure lower frequencies, they could use a sub woofer, or even a woofer which is designed to reverberate at much lower frequencies than the small speaker used in the experiment. The small container used for the water was made of plastic and was small enough that waves were vibrating along the sides and creating crossing standing waves.  A student could redo the experiment similarly with a stationary container and a small wave wall inside that is connected to the speaker. This would eliminate the side wave interference caused by the vibrations of a single object. It is possible that this technique in a k larger tank would give more accurate data for wavelengths in the 1-3 cm region. Future continuations of this lab could include using Alcohol or Mercury instead of water.