091 Waves

14 October 2009

Equipment

• Super ball or other high bounce ball such a space ball.
• Super ball or other high bounce ball such a space ball.
• Stopwatch
• Very light chain
• Meter stick
• Calculator

An introduction to the concepts of oscillations

A ball being bounced is a good example of a repeating system. In physics, repeating systems are described by special terms. When the ball repeatedly returns to a previous position and velocity, each repeat is called a cycle. A system that repeats is also known as an oscillating system.

The duration in time for one cycle is called a period τ. Period is often calculated by timing many cycles. Divide the time by the number of cycles to get the period. The units for period are seconds per cycle.

The mathematical reciprocal, dividing the number of cycles by the time, is called the frequency f. The units for frequency are cycles per second. Cycles per second has a special name: Hertz. One Hertz is one cycle per second.

If the distance the ball is dribbled is small (short), the then frequency f is high.

If the distance the ball is dribbled is large (long), the then frequency f is low.

Instructional note: Demonstrate using a stop watch, timing ten to twenty cycles (dribbles). Calculate both the period and frequency.

Waves on a chain

• The distance from one circle to the next circle on the diagram above is one wavelength λ. In the above diagram one wavelength is 48 centimeters. The symbol λ is the lower case Greek letter "lambda." Wavelength is a space measurement.
• The distance from the middle of the wave to the crest, triangle to triangle, is the amplitude a. In the above diagram the amplitude a is 20 centimeters. Amplitude is a space measurement.

In the diagram above:

• Wave A has an amplitude a of 5 cm
• Wave B has an amplitude aof 10 centimeters
• The wavelength λ of wave A is 20 cm.
• The wavelength λ of wave B is 8 cm.

Amplitude and wavelength are measures of space. Waves have a time component as well. The time component of a wave is the frequency.

• The number of wavelengths (think: crests) that pass a point per unit time is the frequency f. The frequency is also the rate in cycles per second for the wave - the same concept as further above. Frequency is a time measurement and is not shown in the graphs above.
• The generic equation of a basic wave is y = sin(x) where sin is the trigonometric function sine. The actual function for the equation in the first graph in this section is:

$y=20\sin \left(\frac{2\mathrm{\pi }x}{48}\right)$

When time and space combine linearily, the result may be a velocity. Waves have a velocity. The velocity v for a wave is calculated by multiplying the wavelength λ by the frequency f.

$v=\lambda ƒ$

Velocity of a water wave in shallow water where g is the acceleration of gravity and d is the depth of the water.

$v=\sqrt{gd}$

Instructional notes: Using a meter stick and stopwatch demonstrate the relationships between frequency f and wavelength λ. Diagram the wavelength and amplitude on the board. Calculate the velocity v of a wave on a chain. Cover the connections to sound waves, water waves.