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# Waves

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## Zachary McAllister

on 17 March 2015

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#### Transcript of Waves

Waves
Waves
When you hear the word "waves", many things may come to mind. You may think of the waves one sees at the beach or you may think of radio waves or sound waves. In this section, we will learn about the fundamentals of waves. Later, we will discuss different types of waves such as light, sound, electromagnetic, and others.
Linear & Harmonic Motion
When you travel from one place to another, you use linear motion. Linear motion goes from one place to another without repeating. Harmonic motion, on the other hand, is motion that repeats over and over. For example, our four seasons are caused by Earth's harmonic motion.
Motion in Cycles
To describe harmonic motion, we need to learn how to describe a repeating action or motion. A cycle is one unit of harmonic motion. This motion can be back-and-forth motion or cycles that repeat over and over again.
A pendulum is a device that swings back and forth. We can use a pendulum to better understand a cycle. Each box in the diagram above is a snapshot of the motion at a different time in one cycle.
Cycles
Cycles
The cycle starts with (1) the swing from left to center. Next, the cycle continues with (2) center to right, and (3) back from right to center. The cycle ends when the pendulum moves (4) from center to left because this brings the pendulum back to the beginning of the next cycle. Once a cycle is completed, the next cycle begins without any interruption in the motion.
Oscillators
An oscillator is a physical system that has repeating cycles (harmonic motion). A child on a swing is an oscillator, as is a vibrating guitar string. A wagon rolling down a hill is not an oscillator.
Equilibrium
Systems that oscillate move back and forth around a center or equilibrium position. You can think of equilibrium as the system at rest, undisturbed, with zero net gravity. This is why the wagon rolling down the hill is not an oscillator. The gravity causing it to accelerate down the hill is not balanced by any other force.
Restoring Forces
A restoring force is any force that always acts to pull a system back toward equilibrium. Restoring force is related to the force of gravity or weight and the lift force (tension) of the string of a pendulum. If a pendulum is pulled forward or backward, gravity creates a restoring force that pulls it toward equilibrium.
Systems with restoring forces become oscillators.
Inertia
The motion of an oscillator is the result of the interaction between a restoring force and inertia. For example, the restoring force pulls a pendulum toward equilibrium. But, because of Newton's first law, the pendulum does not just stop at equilibrium. The pendulum has inertia that keeps it moving forward so it overshoots its equilibrium position every time.
Restoring Force
Lift Force (Tension)
Frequency & Period
Harmonic motion can be fast or slow, but we don't use speed to tell the difference. This is because the speed of a pendulum constantly changes during its cycle. We use the terms period and frequency to describe how quickly cycles repeat themselves. The time it takes for one cycle to occur is called a period. A clock pendulum with a period of one second will complete one full back and forth swing each second.
Frequency & Period
The frequency is the number of complete cycles per second. The unit of one cycle per second is called a hertz (Hz). For example, something that completes 10 cycles each second has a frequency of 10 Hz. A guitar string playing the note A vibrates back and forth at a frequency of 220 Hz. Your heartbeat has a frequency between 0.5 and 2 cycles per second (0.5-2.0 Hz).
Frequency & Period
Frequency and period are inversely related. The period is the number of seconds per cycle and the frequency is the number of cycles per second. For example, if the period of a pendulum is 2 seconds, then its frequency is 0.5 Hz.
Period =
1
Frequency
Period
Frequency =
1
When to use period
or frequency
While both period and frequency tell us the same information, we usually use period when cycles are slower than a few per second. A simple pendulum has a period between 0.9 and 2 seconds. We use frequency when cycles repeat faster. For example, the vibrations that make sound in musical instruments have frequencies between 20 and 20,000 Hz.
Curved Motion
Curved motion is caused by sideways accelerations. Sideways accelerations cause velocity to change direction, which results in turning. As an example, imagine a soccer ball kicked into the air. The ball starts with an upward angle. The acceleration of gravity bends the velocity vector more towards the ground during each second the ball is in the air. Near the end of the motion, the ball's angle is now pointing down toward the ground.
Amplitude
The "size" of a cycle is called amplitude. The picture below shows two pendulum with different size amplitudes. With a moving object like a pendulum, the amplitude is often a distance or angle. With other types of oscillators, the amplitude may be expressed in voltage or pressure.
Measuring amplitude
The amplitude is measured as the maximum distance the oscillator moves away from its equilibrium position. For the pendulum to the right, the amplitude is about 22 degrees. This means the pendulum will move approximately 22 degrees a way from the equilibrium position before the restoring force brings it back towards equilibrium.
Damping & Friction
Friction slows a pendulum down, just as it slows all motion. That means the amplitude gets reduced until the pendulum is hanging straight down, motionless. We use the word damping to describe the gradual loss of amplitude.
Graphing
harmonic motion
It is easy to recognize cycles on a graph of harmonic motion. The picture below shows the difference between a graph of linear motion and a graph of harmonic motion. The most common type of harmonic motion graph places time on the horizontal (x) axis and position on the vertical (y) axis.
Finding the period
In the graph below, the pattern repeats every 1.5 seconds. This repeating pattern represents the period of the pendulum, which is 1.5 seconds. If you were to cut out any piece of the graph and slide it left or right 1.5 seconds, it would line up perfectly.
Using positive and negative
positions
Harmonic motion graphs often use positive and negative values to represent motion on either side of a center (equilibrium) position. Zero usually represents the equilibrium position. Notice on the vertical axis that zero is in the middle so as to allow room for negative and positive motion.
Finding the Amplitude
The amplitude of harmonic motion can also be seen on a graph. The graph below shows that the pendulum swings back and forth from +20cm to -20cm. The equilibrium position is represented as the zero line. Therefore, the amplitude is 20 centimeters.
Natural Frequency
An oscillator will have the same period and frequency each time you set it moving. This phenomenon is called natural frequency, the frequency at which a system naturally oscillates. Musical instruments use natural frequency. For example, guitar strings are tuned by adjusting their natural frequency to match musical notes.
Natural Frequency
An oscillator, like the child's swing in the video below, have a natural frequency. When a person pushes another person on a swing, they use a push that matches the natural frequency of the swing. In the beginning of the video, the man uses a push that matches the natural frequency of the swing. At the end, he uses a push that does not match the natural frequency and the swing does not move properly.
Changing Natural Frequency
The natural frequency of an oscillator changes according to its length. In the case of a vibrating guitar string, you can shorten the string to increase the force pulling the string back toward equilibrium. Higher force means higher acceleration so the natural frequency is higher and the period is shorter. Lengthening an oscillator results in a lower frequency and a longer period.
Changing the natural frequency
of instruments
How mass
affects oscillators
For oscillators with side-to-side movement, increasing the mass means the oscillator moves slower and the period gets longer. This is because of Newton's second law of motion- as mass increases, the acceleration decreases proportionally. However, for a pendulum, changing the mass does not affect its period.
Periodic Force & Resonance
A force that is repeated over and over is called a periodic force. A periodic force supplies energy to an oscillator and has a cycle with an amplitude, frequency and period. Resonance happens when a period force has the same frequency as the natural frequency.
What is a wave?
A wave is an oscillation that travels from one place to another. A musician's instrument creates waves of sound that move through the air to your ears. When you throw a stone into a pond, the energy of the falling stone creates waves in the water that carry energy to the edge of the pond.
Why do waves travel?
When you drop a ball into water, some of the water is pushed aside and up by the ball (A). The higher water pushes the water next to it (B). The water that has been pushed then pushes on the water next to it, and so on. The waves spread or propagate through the connection between each drop of water and the water next to it (C).
Energy & Information
Waves are a traveling form of energy because they can cause changes in the objects they encounter. Waves also carry information, such as sound, pictures, or even numbers. Waves are used in many technologies because they quickly carry information over great distances. All the information you receive in your eyes and ears comes from waves.
Waves are oscillators
Like all oscillators, waves have cycles, frequency, and amplitude. The frequency of a wave is a measure of how often it goes up and down at any one place. The frequency of one point on the wave is the frequency of the whole wave. A wave carries its frequency to every place it reaches. Like other frequencies, the frequency of a wave is measured in hertz (Hz). A wave with a frequency of 1 Hz causes everything it touches to oscillate at one cycle per second.
Wavelength
You can think of a wave as a moving series of high points and low points. The high point of a wave is called the crest, and the low point is called the trough. The wavelength is the distance from any point on a wave to the same point on the next cycle of the wave. You can measure the distance between one crest and the next or one trough and the next to determine the wavelength.
Wavelength
Amplitude
You have learned that the amplitude of an oscillator- such as a wave- is measured as the maximum distance it moves away form its equilibrium position. For a wave, equilibrium is the average, or resting, position. You can measure the amplitude as 1/2 the distance between the crest and trough of a wave.
Wave motion is due to the spreading of the wave from where it begins. For a water wave, the water itself stays in the same average place. Therefore, to gauge the speed of a wave you measure how fast the wave spreads, not how fast the water surface moves up and down.
Measuring Wave Speed
The graphic below shows what happens in water when you begin a wave in one location. You can measure the speed of this spreading wave by timing how long it takes the wave to affect a place some distance away. The speed of a typical water wave is about 1 m/s. Light waves, on the other hand, are quite fast- 300,000 km/s or 186,000 mi/s. Sound waves travel at about 1,000 km/hr or 660 mph.
Speed Formula
In one complete cycle, a wave moves one wavelength. The speed is the distance traveled divided by the time it takes. We can also calculate the speed of a wave by multiplying frequency and wavelength. These formulas work for all kinds of waves, including water, sound, light, and even earthquake waves!

Speed = Frequency x Wavelength
Solving wave speed
problems
The wavelength of a wave is 0.5 meter and its period is 2 seconds. What is the speed of this wave?

If the period of a wave is 15 seconds, how many wavelengths pass a certain point in 2 minutes?
Wave Motion & Interference
Sometimes your car radio fades out. Why? It's because radio waves are affected by objects. If you enter a tunnel, some or all of the radio waves get blocked. In this section, we will learn how waves move and what happens when they encounter an object or collide with other waves.
Wave Fronts
A wave front is the leading edge of a moving wave and is often considered to be a wave crest rather than a trough. You can make waves in all shapes but we will be discussing plane and circular waves. The crests of a plane wave look like parallel lines. The crests of a circular wave are circles. A plane wave is started by disturbing water in a line. A circular wave is started by disturbing water at a single point.
The direction of the wave
The shape of the wave front determines the direction the wave moves. Circular waves have circular wave fronts that move outward from the center. Plane waves have straight wave fronts that move in a line perpendicular to the wave fronts.
The four wave interactions
Both circular and plane waves eventually hit surfaces. Four interactions are possible when a wave encounters a surface- reflection, refraction, diffraction, or absorption.
Wave reflection
When a wave bounces off an object we call it reflection. A reflected wave is like the original wave but moving in a new direction. The wavelength and frequency are usually unchanged. An echo is an example of a sound wave reflecting from a distant object or wall.
Boundaries
A boundary is an edge or surface where one material meets a different material. The surface of a glass window is a boundary. A wave traveling through the air experiences a sudden change when it encounters the boundary between the air and the glass of a window. Each or the wave interactions usually occur at boundaries.
Wave refraction
Refraction occurs when a wave bends as it crosses a boundary. We say the wave is refracted as it passes through the boundary. The process of refraction of light through eyeglasses helps people see better. The lenses in a pair of glasses bend incoming light waves so that an image is correctly focused within the eye.
Wave diffraction
The process of a wave bending around a corner or passing through an opening is called diffraction. We say a wave is diffracted when it is changed by passing through a hole or around an edge. Diffraction usually changes the direction and shape of a wave. When a plane wave passes through a small hole, diffraction turns it into a circular wave. Diffraction explains why you can hear sound through a partially closed door. Diffraction causes the sound wave to spread out from any small opening.
Wave absorption
Absorption is what happens when the amplitude of a wave gets smaller and smaller as it passes through a material. The wave energy is transferred to the absorbing material. Theaters often use heavy curtains to absorb sound waves so the audience cannot hear backstage noise. The tinted glass or plastic in the lenses of your sunglasses absorbs some of the energy in light waves.
Wave pulses
A wave pulse is a short "burst" of a traveling wave. A pulse can be produced with a single up-down movement. The illustrations in the following slides show wave pulses in springs. You can see the difference between two basic kinds of wave- transverse and longitudinal- by observing the motion of a wave pulse.
Transverse waves
The oscillations of a transverse wave
ARE NOT
in the direction the wave moves. For example, the wave pulse in the illustration below moves from left to right. The oscillation (caused by the boy's hand) is up and down. Water waves are an example of a transverse wave.
Longitudinal waves
The oscillations of a longitudinal wave are in the same direction that the wave moves. A sharp push-pull on the end of the spring makes a traveling wave pulse as portions of the spring compress then relax. The direction of the compressions are in the same direction that the wave moves. Sound waves are longitudinal.
Constructive interference
Suppose you have a long elastic string being held between you and a buddy. If you pull the string and let it go, a pulse will be created. You and your buddy both create a wave pulse coming from each direction. When the two pulses meet, they combine to create a single large pulse. Constructive interference happens when waves combine to make a larger amplitude.
Destructive Interference
Using the same situation described in the previous slide, suppose the two pulses created this time are coming from opposite directions and are on opposite sides of equilibrium. This means one pulse is moving under and the other is moving over the equilibrium line. When these two meet, one will pull the line upward and the other will pull downward. The result is that the string will flatten and both pulses vanish for a moment. In destructive interference, waves add up to make a wave with smaller or zero amplitude.
Real World Applications
with Waves
The process that allows a cell phone to communicate is the same as for a radio or walkie-talkie. All of these devices use electromagnetic waves of within a specific frequency range to send information. Cell phones use a frequency between 800 and 1900 MHz.

Sound is translated into an electromagnetic wave at the desired frequency by a transmitter through a process called encoding. Once the sound has been encoded, it is sent to the intended recipient.
When a cell phone sends out a signal, it travels at the speed of light (300,000 km/s) to the recipient. An antenna detects the wave because it causes electrons to move in the antenna.

Once the correct signal is detected, the information is taken from the signal, called decoding. An electric current is then sent to the speaker, where it is translated back into a sound wave.
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