[Waves]

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Introduction

Do you notice that when you bend a ruler (assuming it is flexible of course) that it snaps back maybe several times when you release the force you applied on it? Or maybe the up and down motion of a spring. Well gee... it repeats itself again and again, so we can call that "harmonic motion."

Hooke's Law

When you consider the forces on...say the spring (the most common example of harmonic motion), there is a reason why an elastic object snaps back and forth. And it has to do with at least two forces: the one that set it in motion and the force of the spring itself...which wants to get back to equilibrium (called the "rest position"). It thus exerts an opposing force (hence the minus sign), which Robert Hooke expressed as:

F = -kx

where k is the spring constant and x is the displacement from rest position of the spring.

Thus we can say that simple harmonic motion occurs when there is a net force that is related to displacement.

Elastic Potential Energy

So a spring has force. Can it have work? Yeah it can =)

Well since there is...what's the big deal? Think about all the things that use springs for energy: guns, slinkys, clocks etc.

There seem to be more and more ways to express energy.

There is a force (shm force) when the spring is displaced...that means not at rest. This dependence on displacement is reminiscent of potential energy. It can give energy to objects by giving it a force and causing displacement and velocity.

Elastic Potential Energy is defined as:

PEs = 1 kx2
2

That is equal to kinetic energy (in magnitude). Well, there's also some sort of friction involved, and that would mean that some mechanical energy is not conserved. So the mechanical energy that is lost from non conservative forces must be the difference between the energy started with and the energy ended with (after the nonconservative act on it).

From the work-energy theorem:

Wnonconserved = (KE + PEspring +PEgravitational)f - (KE + PEspring + PEgravitational)i

Let's just take a quick analysis of the various key stages of simple harmonic motion in the spring:

  • At rest, there are no net forces acting on the spring
  • Compressed, there is an applied force on the spring (the spring is also exerting an opposite force on it)
  • Released, the applied force disappears and the opposing force pushes the spring back toward equilibrium position
  • The spring passes the equilibrium position and extends itself (the spring is also exerting an opposite force again)
  • Once the spring's momentum is all "used up," the opposing spring force comes into play and pulls it back toward equilibrium
  • And once again it overshoots and passes equilibrium...and so on.

Well so now we know the force involved in springs...how about being and wanting to know the velocity that a spring might impart on an object that is attached to it while it is oscillating?

v = ± [
k
(d2 - x2)] 1/2
m

where k is the spring constant, m is the mass of the object with the velocity we are trying to find out, d is the "stretch" of the spring (its initial displacement from rest position) and x is the final displacement from rest position. Because velocity is a measure of the speed between two points of displacement in time there are two "distances" (d and x) that the equation takes into account.

 Click the button to see the derivation of the formula

Waves

Keywords:

  • amplitude
  • crests
  • frequency
  • nodes
  • pulse
  • phase
  • polarization
  • resonance
  • troughs

Types of Waves:

  • longitudinal
  • transverse
  • electromagnetic
  • mechanical
  • standing waves

Transverse vs. Longitudinal Waves

There's a difference between them in the way waves transmit energy. Waves do NOT transmit mass...only the energy involved. The two ways waves propagate are through in-plane waves or perpendicular-to-the-direction-of-motion waves.

longitudinal: (in-plane --> wave propagates in the direction of motion)

example: sound waves

transverse: (perpendicular-to-plane --> wave propagates perpendicular to the direction of motion)

example:

<picture longitudinal vs. transverse>

The Wave Equation

Waves do a lot of useful transmitting. Radio waves, t.v. waves, etc. (things that some people can't seem to live without) So maybe you want to know how fast these waves transmit all that info you like. The most obvious relationship that exists is that between frequency, velocity, and wavelength.

v = f

On a string, you can find the velocity of a wave or pulse by this relationship:

v = [
   T  
] 1/2
(M/L)

Wave Reflection

Well...when waves hit a boundary...you'd expect them to bounce back or something. Waves can reflect in different ways though when they bounce back.

There are two cases that immediately come to the forefront in the reflection of waves. If you create a wave on a string and it reaches a boundary it will either reflect the wave upright or upside-down.

  • the string is fixed [anchored] to the boundary -- upside down reflection
  • the string is free at the boundary -- rightside up reflection

Why? Well if the string is fixed and is bringing a wave to the boundary, it is bringing a force to it. If it is exerting a force (in the direction of the crest) on the boundary, then the boundary is exerting an equal but opposite force: Newton's Third Law.

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