Traveling electromagnetic wave mastering physics

Wave Motion

And a bit on damped oscillations

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Phase Constant

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Cosine reflection

Sine reflection

Tangent reflection

Determining Quadrant:

X varies with cos, and velocity with negative sin, so take a look at those graphs to see which quadrant your initial conditions are in.

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Damped Oscillations

Our model so far has excluded all non-conservative forces

So what’s the rub?

Here’s one model of a damped oscillator:

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FBD:

b = damping constant

This is not an easy differential equation to guess a solution for… we’ll just jump to the solution.

The Answer

The solution for this model of a damped oscillator is a decaying exponential:

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oscillating part

decaying part

The angular frequency

of the undamped

oscillator.

Whiteboard Problem 15-8

A 250g air track glider is attached to a spring with spring constant 4.0N/m. The damping constant due to air resistance, b, is 0.015kg/s. The glider is pulled out 20cm from equilibrium and released.

How many oscillations will it make during the time in which the amplitude decays to e-1 of its initial value? (LC)

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Driven Oscillations & Resonance

Nonconservative forces can take energy out of an oscillator, but they can also put energy in

The driving force should be applied with the same frequency as the oscillator; the oscillator’s natural frequency

When this condition is met, it is called resonance

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Examples include swinging, shattering glass, and the Tacoma narrows bridge

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Wave Motion

What if the oscillator wasn’t fixed in place?

The oscillation could travel through space

A wave is an organized disturbance that travels at a well defined speed.

A wave is not the same thing as the medium

Like particles, waves can carry energy and momentum

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‘Dex Entries

There are 3 main categories of waves, and 2 types

Mechanical Waves:

Require a medium to propagate

Wave speed is determined by the medium

Oscillations can be transverse or longitudinal

Electromagnetic Waves:

Medium not required

Wave speed fixed ~ 3x108 m/s

Matter Waves:

Light is both a particle and a wave? That means particles can be waves too

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Wave Speed

Consider a single traveling down a string

The wave moves through the medium, displacing the string from it’s equilibrium position

The text goes through a derivation of the speed of a wave on a string:

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Where:

Whiteboard Problem 16-1

The wave speed on a string is 150m/s when the tension is 75N. What tension will give a speed of 180m/s? (LC)

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The Sine Wave

A sinusoidal (harmonic) disturbance creates a sinusoidal travelling wave

At a given point in space, a single particle undergoes simple harmonic motion (a); for a snapshot in time, the whole wave is a sine wave (b)

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Where D(x,t) is the general disturbance from the equilibrium state. Note: it is a function of two variables.

Modeling Sine

With a snapshot in time, we can describe a few things about a wave:

Many of these variables are fundamentally related:

With that relationship, we can describe the motion of a traveling wave…

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Travelling in +x direction

Wave number and spring constant are different!

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One more thing…

The differential equation for the harmonic oscillator gave us a clue as to what ω was:

Waves have a similar differential equation:

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Expression for a Travelling Wave

Wiggle on over to Mastering Physics and finish the assignment that’s the same as the title

The goal is to become more familiar with the pieces and parts of the wave equation

When you’re done, consider starting your homework, or playing with the wave on a string PhET

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Whiteboard Problem 16-2

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Whiteboard Problem 16-3

Write the displacement equation for a sinusoidal wave that is traveling in the negative y-direction with a wavelength of 50cm, speed of 4.0m/s, and an amplitude of 5.0cm. Assume the phase constant is zero.