Ch13. Vibrations and Waves

HW# 1, 5, 9, 13, 19, 29, 35, 37, 39, 41, 43, 47, 51, 53, 61

If you displace a system that obeys Hooke’s Law,It will follow “simple harmonic motion”.

The system will oscillate.

A graph of position versus time, will look like a sinusoidal function.

Example: mass attached to a springswinging pendulum (for small oscillations)

Important Definitions

For an object moving with simple harmonic motion (SHM) we can look at Newton’s 2nd Law.

ΣF = m a

F = -k ∆x

a= F/m = -k(∆x)/m

The acceleration depends on the current amount of displacement.

See quick quiz 13.2

See pictures for x, v, and a as a function of time on page 458

Simple pendulum

Damped OscillationsPreviously we looked at systems that would oscillate indefinitely.

lack of frictionsmooth pivot pointsno external forces that impede motion

For real mechanical systems, this is not the case.Friction will “damp” the motion.

Other examples:pushing a kid on a swing, out of phase with the motionair resistance on a pendulum

Damped OscillationsAny mechanical system with some resistive force will not oscillate forever.The motion is damped.

See figures 13.19 and 13.20

This is how the shock absorbers on your car work.

They are slightly underdamped.

WavesRelated to the energy of a vibrating object.

All waves (sound waves, waves on a string, seismic waves, water waves, electromagnetic waves,…) have a vibrating object as their source.

We can apply SHM to describe the waves.

Let’s first define a wave: The motion of a disturbance.

Example: the surface of water after dropping a stone in the water. After the stone is dropped, we can observe the propagation of the disturbed water.

Waves need…

1) A source of disturbance.

2) A medium that can be disturbed. Electromagnetic waves don’t need a medium, they are special.

3) Some physical connection through which the adjacent portions of the medium influence each other.

Types of wavesDescribe the waves based on the motion of the disturbance.

Traveling wave – wave in which the disturbance movesfigure 13.21

Standing wave – wave where the disturbances do not travel across the medium. Result of superposition of traveling waves. Will use these in chapter 14.

Types of WavesDescribe waves based on direction of disturbances.

Transverse waves – the medium is disturbed perpendicular to the direction of wave motion.

example: waves on a guitar string.

Longitudinal waves – the medium is disturbed parallel to the direction of wave motion.

example: sound waves

See figure 13.23

http://en.wikipedia.org/wiki/Transverse_wave

http://en.wikipedia.org/wiki/Longitudinal_wave

Water waves are a combination of transverse and longitudinal waves.

http://www.acs.psu.edu/drussell/Demos/waves/wavemotion.html

Waves have alternating maxima and minima separated by nodes.

At a node, there is no displacement (no disturbance).

In transverse waves, the maxima and minima are crests and troughs.

In a longitudinal wave, the maxima and minima are regions of high density and low density.

See figure 13.25

Light and sound are two important waves to be studied.Both light and sound are waves.

Biggest difference is that light (electromagnetic wave) does not need a medium to be disturbed. Light can travel through the vacuum of space.

Sound – longitudinal wave.Light – transverse wave made up of two waves, varying electric field, and magnetic field perpendicular to each other.

Compare frequencies of light and sound waves IF the wavelengths are the same.Example 13.9

Waves on a stringPluck a string perpendicularly to the direction of the string’s orientation.Important quantities:

TensionMass per unit length.

Higher tension – quicker response to motion => higher frequency=> higher speed

Higher mass per unit length – more inertia => vibrates slower=> lower speed.

This is why different guitar strings sound differently.

The strings have different mass per lengths……and why you tune a guitar by adjusting the tension.

Interference of Waves

• When two traveling waves meet. They pass through each other without changing.

• When they overlap, they add up.

• This is a superposition principle.

• You just have to add up the individual displacements of each waves point by point.

• See figures on pages 470, 471

ReflectionWhen a wave encounters a change in medium, the wave reflects, or bounce back.

Examples: string tied to a polelight hitting a mirrorocean wave reaching a shore

How the reflection occurs depends on the boundary conditions.See page 472.

To complicate things we can have situations where some of the wave is reflected, while the rest of the wave is transmitted into a new medium.