Waves



Waves

Waves account for an enormous amount of our everyday experience. Imagine if we couldn’t see or hear. Imagine a world with no light or sound. There would be no radio, TV, movies, music, or pictures of any kind without waves. Reading would be impossible, and so would talking. Virtually every form of communication we know of would be non-existent.

Fortunately, waves do exist. Light and sound help us to sense the matter in the universe. (And matter is required to produce sound and light, so there’s a nice symmetry between waves & matter.) We will be studying many of the properties of two types of waves: sound and light, so let’s begin...

Simple Harmonic Motion

As we saw in the classroom demonstrations with the mass bouncing on a string and the pendulum, simple harmonic motion (also called periodic motion) is when an object moves ‘evenly’ back and forth between two points. This type of motion results when a restorative force is present, which means that there always is a force present that points back toward the equilibrium position.

Objects in simple harmonic motion will create waves in the surrounding area and beyond, and we will consider them as the source of the wave. Remember, in real life, no oscillator is perfect, and the amplitude would slowly diminish, or dampen, as time went by, but we will ignore the effects of dampening in most cases. Describe the demonstrations you will see in the space below. What is the source? What similarities do you notice?

Slinky:

Marker on dryboard:

Wave Vocabulary

(a) medium - The material through which a wave travels (for sound its often the air. For light, we’ll use air, glass, and water; for some demonstrations we’ll use springs).

(b) crest - The high point on a wave.

(c) trough - The low point on a wave.

(d) amplitude - The “height” of the wave, representing the amount of energy carried by mechanical waves. With sound waves, we’ll correlate amplitude with volume.

(e) wavelength (λ) - The distance from one crest to the next; the length of one complete cycle.

(f) frequency (f) - The number of complete cycles (oscillations) produced each second (the unit for this is the Hertz, Hz).

Types of Waves

(a) transverse : These are waves for which a point on the wave vibrates in a direction that is perpendicular to the direction in which the wave is traveling. Think about what happens to you when you’re in the ocean and a wave passes by you... you represent a point on a transverse wave! You bob vertically (up and down) while the wave moves horizontally (toward the shore)!

Imagine yourself in the ocean, well out beyond the beach, when a series of many waves pass by you. Represent your motion as a function of time (make a graph of your y position vs. t) in the space below:

(b) longitudinal : These are waves for which a point on the wave vibrates parallel to the motion of the wave. Sound is a longitudinal wave. Evidence for this is the way that your ear drum works. Warning: when we represent sound graphically, it will appear as a transverse wave. Be sure to remember we are only showing the displacement of a point in the medium as a function of time. You will need to determine if that displacement is perpendicular (for a transverse wave) or parallel (for a longitudinal wave) to the direction the wave itself is traveling in.

We demonstrated the two different types of waves in class. Draw diagrams to help you remember the difference in the space below:

Transverse wave on a slinky looks like this:

Longitudinal wave looks like this:

WAVELENGTH and FREQUENCY

The illustration below represents the relationship between wavelength and frequency for a wave in a given medium. If one increases, the other decreases.

Which wave has the greater frequency? Which has a greater wavelength?

In order to test your understanding of some basic vocabulary, try the example below. You are to draw a new wave form, with on the adjustment listed, based on the wave form given. You’ll need a ruler and a pencil.

WAVE SPEED

Speed is (in general) distance covered divided by time, speed = distance / time.

For a wave we know that during the period T the wave will travel a distance of one wavelength λ. That makes the equation v = λ / T.

Since f = 1 / T, we can represent the speed of a wave as

v = λf

The speed of a wave in a given medium is fixed by the properties of the medium.

Usually, physical waves travel faster through denser materials. Think about how the vibrations must move through the material. A molecule must push on its neighbor to set it in motion. Denser, more rigid materials are more ‘tightly bound’ and take less time to ‘push’ on each other.

REFLECTION AND REFRACTION

Reflection and refraction are formally known as boundary behavior. When a wave meets a boundary, it can either totally reflect (think bounce) from the boundary, or it can partially reflect and partially transmit across the boundary into a different medium.

Can you think of examples for each? List one reflection and one refraction example here:

Why do we say that in refraction the wave must transmit into a different medium? Can you re-explain that in your own words?

Representation of Reflection

Amplitude, speed and wavelength remain the same (if you discount dampening), and frequency is seen to remain unchanged.

Representation of Refraction

In the incident (initial) medium the wavelength and speed of the reflected portion of the wave stay the same as they were before the wave met the boundary. The amplitude of the reflected pulse is less.

In the refracted medium (other side of the boundary) the wavelength and speed either both become larger or they both become smaller. The amplitude will be smaller than the incident wave.

SUPERPOSITON

When two waves cross at a point in space, their amplitudes will add to form a single wave form. Afterwards, they resume their original shape and direction as if they had never ‘met’ in the first place. This amazing property of waves certainly seems to make waves much different from matter huh?

When two waves add to give a resultant amplitude greater than either one, its constructive interference. When a diminished amplitude results, that’s called destructive interference.

Constructive Interference Destructive Interference

Observations of interference you have seen (or will see): slinkies, ripple tank, beats

Wave Motion Problems

1. For the wave form shown below, sketch :

2. For the wave form shown below, sketch :

3. The speed of an ocean wave at the shoreline is 1 m/s. If its wavelength is 3.2 meters, find the frequency with which it hits the shore.

(0.31 Hz)

4. The speed of sound in air is 340 m/s. What is the wavelength of a sound that has a frequency of 453.3 Hz?

(0.75 m)

5. Radio waves travel at the speed of light...3 x 108 m/s. Find the length of the wave broadcast by 750 AM (750,000 Hz) and by 105 FM (105,000,000 Hz).

(400 m ; 2.86 m)

6. A standing wave consisting of 4 nodes and three antinodes is 12 m long. If its frequency is 0.5 Hz, find its speed. Draw a sketch of this pattern.

(4 m/s)

7. A standing wave in a pipe with an end closed off covers a distance of 7 m. If its frequency is 85 Hz, find the number of antinodes in the pattern. Draw a sketch of the pattern.

(4 A’s)

8. A standing wave in an open pipe produces 8 antinodes in a distance of

6 meters. Find the frequency of the wave and draw a sketch of the standing wave pattern.

(199 Hz)

9. A standing wave in a string travels at 10 m/s when its frequency is

2 Hz. If the pattern covers 15 meters, how many antinodes are in the pattern?

Sketch the wave form.

(6 A’s)

10. A standing wave in a closed pipe has 3 nodes. If the frequency of the sound produced by the pipe is 226.7 Hz, find the length of the pipe. Oh yeah, would you please sketch the pattern also?

(1.87 m)

11. A standing wave in an open pipe has a frequency of 113.3 Hz and covers a distance of 4.5 meters. How many nodes are in this pattern? While you’re at it, sketch the pattern as well.

(3 nodes)

12. the waves A & B shown below, SKETCH & EXPLAIN what the waves look like when

(a) the front of A meets the front of B

(b) A & B overlap

(c) the back of A meets the back of B

front of A

meets front of B

total overlap

back meets back

-----------------------

Wave A

Wave B

given wave

A wave with twice the amplitude

A wave with twice the frequency

A wave with twice the wavelength

“Free-End” wave reflections not inverted

“Fixed-End” wave reflections are inverted

Wave pulse approaching boundary between mediums

After contacting boundary, some of the wave travels into the new medium, and some reflects back into the incident medium.

Note there is no change in the wavelength (() of the reflected pulse, but the refracted pulse in this case has a smaller wavelength! What does that imply about the vwave of the refracted medium?

Also note that the amplitude of the original wave is larger than either the reflected or refracted pulse. In fact, if we neglect dampening, the sum of the reflected and refracted pulses will equal the amplitude of the original wave! This is a prelude to the Conservation of Energy….

Two pulses approach each other

|ko¯±Ùçèì C N W c h i lv{Two pulses approach each other

When they overlap, they create a lower amplitude (in this case zero amplitude)

When they overlap, they create a higher amplitude, by combining

After overlapping, they continue on their original direction, as if nothing happened!

After overlapping, they continue on their original direction, as if nothing happened!

a. a wave with twice the amplitude

b. a wave with twice the frequency

c. a wave with twice the wavelength

a. a wave with 1/3 the amplitude

b. a wave with 1/2 the frequency

c. a wave with 1/4 the wavelength

A

B

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