Tuesday, January 28, 2014

By the end of the day today, IWBAT…• Analyze characteristics of waves (velocity, frequency, amplitude, wavelength).

Do Now: Which of the following would be thebest physical demonstration of a transverse wave?A a grandfather clock chiming the hourB a jump rope lying on the ground being oscillated side to sideC the energy in a Slinky® traveling back and forth along its lengthD a large number of marbles rolling down a flight of stairs

Monday, January 27, 2014

Do Now: Which of the following would be thebest physical demonstration of a transverse wave?B a jump rope lying on the ground being oscillated side to side

T.Q.

Why it matters in THIS CLASS: Unit #9 Exam this Friday, January 31, 2014.

BE PREPARED!

Why it matters in LIFE:We interact with WAVES every day: your music is delivered to your ears via sound waves. Waves even quake the earth beneath our feet.

By the end of the day today, IWBAT…• Analyze characteristics of waves (velocity, frequency, amplitude, wavelength).

Transverse

Longitudinal

• The Unit 9 Cover Sheet goes into your journals to keep track of

your completed daily work prior to the exam. Record all

assignments and keep it together.

Unit 9: Mechanical Waves

Tuesday, 01/28/14TEKS:

P.7B Investigate and analyze

characteristics of waves, including

velocity, frequency, amplitude, and wavelength…

By the end of today, IWBAT…Analyze different

wave characteristics,

including damping and diffraction. Essential

Question:What is damping?

What is diffraction?

Topic:Mechanical

Wave Equations

C-Notes!

Simple Harmonic Motion

A simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a mass m, which experiences a single force, F, which pulls the mass in the direction of the point x=0 and depends only on the mass's position x and a constant k.

Harmonic Waves

m1

m2

m3

m4

m5

Imagine a whole bunch of equal masses hanging from identical springs. If the masses are set to bobbing at staggered time intervals, a snapshot of the masses forms a transverse wave. Each mass undergoes simple harmonic motion, and the period of each is the same. If the release of the masses is timed so that the masses form a sinusoid at each point in time, the wave is called harmonic. Right now, m4 is peaking. A little later m4 will be lower and m3 will be peaking. The masses (the particles of the medium) bob up and down but do not move horizontally, but the wave does move horizontally.

m6

m7

m8

m9

m10

wave direction

Making a Harmonic Wave

A generator attached to a rope moves up and down in simple harmonic motion. This generates a harmonic wave in the rope. Each little piece of rope moves vertically just like the masses on the last slide. Only the wave itself moves horizontally. The time it takes the wave to move from P to Q is the period of the wave, T. The distance from P to Q is the wavelength, . So, the wave speed is given by: v = / T = f (since frequency and period are reciprocals).

Since the generator moves vertically in SHM, the vertical position of the black doo-jobber is given by: y(t) = A cos t. The doo-jobber’s period is given by T = 2 / . This is also the period of the wave.

wave direction

P Q

Damping Damping is an influence within or upon an

oscillatory system that has the effect of reducing, restricting or preventing its oscillations. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation.

Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators.

Damping What in the world

did the last slide even mean?!

Damping is like “turning down a radio…” Or a mute button.

This is an “under-damped spring,” by the way.

Interference

Interference Animation

Wave Interference

Like force vectors, waves can work together or opposition. Sometimes they can even do some of both at the same time. Superposition applies even when the waves are not identical.

Constructive interference occurs at a point when two waves have displacements in the same direction. The amplitude of the combo wave is larger either individual wave.

Destructive interference occurs at a point when two waves have displacements in opposite directions. The amplitude of the combo wave is smaller than that of the wave biggest wave.

Superposition can involve both constructive and destructive interference at the same time (but at different points in the medium).

Constructive & Destructive Interference

Constructive Interference

Waves are “in phase.” By super-position, red + blue = green. If red and blue each have amplitude A, then green has amplitude 2A.

Destructive Interference

Waves are “out of phase.” By superposition, red and blue completely cancel each other out, if their amplitudes and frequencies are the same.

Diffraction

Diffraction is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings.

Diffraction

This diagram shows straight wave fronts passing through an opening in a barrier.   The spreading (or bending) at the slit is referred to as diffraction.

Diffraction

Diffraction

Numerical approximation of diffraction pattern from a slit of width equal to wavelength of an incident plane wave in 3D spectrum visualization

DiffractionWhen waves bounce off a barrier, this is reflection. When waves bend due to a change in the medium, this is refraction. When waves change direction as they pass around a barrier or through a small opening, this is diffraction. Refraction involves a change in wave speed and wavelength; diffraction doesn’t.

Diffraction of water happens as waves bend around a boat in a harbor. This is different than the refraction of waves near shore because the depth of water does not decrease around the boat like it does near shore. Diffraction is most noticeable when the wavelength is large compared to the obstacle or opening. Thus, no noticeable diffraction may occur if the boat in the harbor is very big.

The sound waves from an owl’s hoot travel a greater distance in the forest than a song bird’s call, because a low pitch owl hoot has a longer wavelength than a high pitch songbird call, and the owl’s waves are able to diffract around trees. Pics on next slide

Diffraction PicsWhen waves pass a barrier they curve around it slightly. When they pass through a small opening, they spread out almost as if they had come from a point source. These effects happen for any type of wave: water; sound; light; seismic waves, etc.

Diffraction & BatsBats use ultrasonic sound waves (a frequency too high for humans to hear) to hunt moths. The reason they use ultrasound is because at lower frequencies much of the sound waves would have a wavelength close to the size of a moth, which means much of the sound would diffract around it.

Bats hunt by echolocation—bouncing sound waves off of prey and listening for the echoes, so they need to emit sound with a wavelength smaller that the typical moth, which means a high frequency is required. High frequency sound waves reflect off the moths rather than diffracting around them. If bats hunted bigger prey, we might have emitted sounds that we could hear.

We’ll learn more about diffraction when we study light.

Standing WavesWhen waves on a rope hits a fixed end, it reflects and is inverted. This reflected waves then combine with oncoming incident waves. At certain frequencies the resulting superposition yields a standing wave, in which some points on the rope called nodes never move at all, and other points called antinodes have an amplitude twice as big as the original wave.

A rope of given length can support standing waves of many different frequencies, called harmonics, which are named based on the number of antinodes.

1st Harmonic ( The Fundamental )

4th Harmonic

continued

2nd Harmonic

3rd Harmonic

Animations:

Standing Waves (cont.)

It is important to understand that a standing is the result of the a wave interfering constructively and destructively with its reflection. Only certain wavelengths will interfere with themselves and produce a standing wave. The wavelengths that work depend on the length of the rope, and we’ll learn how to calculate them in the sound unit. (Standing waves are very important in music.)

Wavelengths that don’t work result in irregular patterns. A standing wave could be simulated with a series of masses on springs, as long as their amplitudes varied sinusoidally.

Standing Wave with Incident & Reflected Waves Shown Separately (scroll down)

Standing Wave with Superposition Shown

(scroll down)

Animations:

ResonanceObjects that oscillate or vibrate tend to do so at a particular frequency called the natural frequency. For example, a pendulum will swing back and forth at a certain frequency that only depends on its length, and a mass on a spring will bob up in down at a frequency that depends on the mass and the spring constant. It is possible physically to grab hold of the pendulum or mass and force it to swing or bob at any frequency, but if no one forces them, each will swing of bob at its

m

M

own natural frequency. If left alone, friction will rob the masses of their energy, and their amplitudes will decay. If a periodic force, like an occasional push, matches the period of one of the masses, this is called resonance, and the mass’s amplitude will grow. (continued)

Resonance Animation

Resonance (cont.)Tarzan is swinging through the jungle, but he can’t quite make it across the river to the next tree. So, he asks Jane for a little help. She obliges by giving him a push every time he’s just about to swing away from her. In order to maximize his amplitude to get him across the river, her pushing frequency must match his natural frequency. This is resonance. When resonance occurs her applied force does the maximum amount of positive work. If she mis-times the push, she might do negative work, which would diminish his amplitude. The moral of the story is: Resonance involves timing and matching the natural frequency of an oscillator. When it happens, the oscillator’s amplitude increases.

F

x

Jane does positive work

F

x

Jane does negative work

Resonance QuestionExplain how you could get a 700 lb wrecking ball swing with a large amplitude only by pulling on it with a scrawny piece of dental floss. answer:

Wrecking Ball

Give the ball a little tug, as much as you can without breaking the floss. The ball with barely budge. Continue giving it tugs every time the ball is at its closest to you. If you match the natural frequency of the ball, its amplitude will slowly increase to the desired amount. In this way you are adding energy to the ball very slowly.

Tacoma Narrows BridgeEven bridges have resonant (natural) frequencies. The Tacoma Narrows bridge in Washington state collapsed due to the complicated effects of wind. One day in 1940 the wind blew at just the right speed. The wind was like Jane pushing Tarzan, and the bridge was like Tarzan. The bridge twisted and shook

violently for about an hour. Eventually, the vibrations caused the by wind grew in amplitude until the bridge was destroyed.

Click the pic to see the MPEG video clip.