Class_Notes

February 9, 1999.... Physics 201B... Spring 1999

Waves

I believe we all know what a wave "looks" like

This picture of a wave shows the amplitude of the wave above and below the normal position of whatever is "waving". Here the amplitude is 2 units...

The wave is composed of peaks called CRESTS and low points called TROUGHS. There is a repetition of crests and troughs.

The horizontal line can represent EITHER time OR distance...

If it represents time, then the time between adjacent crests is called the PERIOD (T). In the figure, the period is 1 second per cycle. The reciprocal of the period (1/T) is called the FREQUENCY (f) so in this case it would be 1 cycle per second. If period were 0.1 second per cycle, then frequency would be 10 cycles/second. We can think of the wave in time as the position of a single point in the wave as time goes on... the oscillation of a single point in a rope or a single molecule in a water wave.

If it represents distance, then the time between adjacent crests is called the WAVELENGTH (l). In the figure, the wavelength is 1 meter/cycle. We can think of the wave in position as a snapshot of the entire wave at a single instant in time... like taking a picture of a wave.

The velocity (v) of the wave... the speed you would move if you "sat" on a crest and moved with the wave. I like to think of it as "surfing"... you are passed from one molecule at its peak to the next molecule which is just reaching the peak so we have to move with the same speed as the crest is moving. This velocity is calculated by

So, for example,

if we are moving with a sound wave which travels at 330 m/s, then a wave of frequency 440 cycles/sec (440 Hertz) would have a wavelength of 3/4 meters.

if we are moving with a light wave which travels at 300,000,000 m/s then a wave of frequency 600,000,000,000,000 cycles/sec would have a wavelength of 0.0000005 meters... this would be green light... this would be the length obtained by cutting 1 millimeter into 2000 pieces... one of those pieces would be the wavelength of green light.

But to understand waves we need to start with a single harmonic oscillator which has a period and frequency but no wavelength.

Observe the Video... SHM.mov

We can compare simple harmonic motion with circular motion. As the radius of the circle rotates... the tip's x and y values are shown. Note that it starts with x maximum and y zero. In one cycle of the rotation it produces a complete wave. To see this in reality, have someone whirl a ball on a string in a vertical circle and observe the balls motion by lining your vision with the plane of rotation.

Observe the Video... CircOsc.mov

Resonance between two oscillators... driving oscillator matches natural oscillation frequency...

ex. Memorex... ball on elastic rope.

A wave is just a series of oscillators along a line where each oscillator is causing the next oscillator to go through the same motion but slightly delayed as seen in the diagram below.

The arrows on each oscillator shows the direction in which it is moving as the "wave" moves to the right... ex. ball #5 will be at the crest so that the crest would have moved from ball 4 to ball 5.

Types of Waves...

Transverse... oscillators move back and forth perpendicular to the direction of wave motion.

ex. slinky, rope, electromagnetic

Longitudinal... oscillators move back and forth along the direction of wave motion

Circular... oscillators move in circle lined up with direction of wave motion.

ex. water

Wave Interaction with Matter...

Reflection:

Law of Reflection... Do you know how to dribble a basketball?... Same Law...

ANGLE OF INCIDENCE = ANGLE OF REFLECTION

How could you determine the path light would take from point a to point b after hitting the mirror?

Note that if we had reversed the direction of light in either case, the paths taken by the light would be the same... this is called the "reversibility principle".

IMAGES FORMED BY MIRRORS

The eye traces a ray back along the line of entry... and thinks that the source is somewhere along that line...... if we use both eyes and trace two rays back along the rays entering the eye... we call the point of intersection of those rays as the image point....

We can see that the image is NOT IN the mirror but behind the mirror... Can you show that, in a plane mirror, the image is just as far behind the mirror as the object is in front?

Suppose the object is off the edge of the mirror... will it still have an image?

Full-length mirror... what did you find from the exploratory?...

Does it depend on your distance from the mirror?

How long must the mirror be?

What about the wall behind you?

Double mirror...

Where are the images? What are the rules?

How do you see the images? Draw a line from each image to the eye then trace it back, obeying the law of reflection till you get back to the object.

Curved Mirror...

Applying the law of reflection to a curved mirror... the first step is to identify a special point called the focal point, the point at which parallel rays of light will be meet after reflecting off the mirror... in the diverging mirror we must trace back the reflected waves to find the focal point.

Note that a ray going through the focal point will come out parallel to the axis of the mirror.

Now we can find the image of a real object.... and see why the rear-view mirror on the car has the message: "Objects are closer than they APPEAR to be"... Draw two rays (one parallel to the axis and one through the focal point) to find the image of the the tip of the arrow.

Refraction...

Are you ready to take the lifeguard test?... if you pass this test you know the law of refraction.

You are the lifeguard at point A on the sandy beach. A swimmer at point B in the water calls for help... Find the path you would take from point A to point B.

And, what principle of "lifeguarding" did you use?