Chapter 11

Vibrations and Waves

Simple Harmonic MotionSimple Harmonic Motion

A restoring force is one that moves a system A restoring force is one that moves a system back to an equilibrium position.back to an equilibrium position.

ExampleExample: : mass on mass on frictionless frictionless table, table, attached to attached to spring.spring.

Example: gravity acting on a mass hanging from a string.

Example: gravity acting on a mass hanging from a spring.

kxFelasticHooke’s LawHooke’s Law

When the restoring force is linearly proportional When the restoring force is linearly proportional to the amount of the displacement from to the amount of the displacement from equilibrium, the force is said to be a Hooke’s Law equilibrium, the force is said to be a Hooke’s Law force. force.

Vibration: in a general sense, anything that switches back and forth, to and fro, side to side, in and out, off and on, loud and soft, or up and down is vibrating. A vibration is a wiggle in time.

Wave: a wiggle in both space and time is a wave. A wave extends from one place to another.

Vibrations and waves: the source of all waves is something that is vibrating. Waves are propagations of vibrations throughout space.

General definitions of vibrations and waves

When oscillations are small, the motion is called simple harmonic motion (shm) and can be described by a simple sine curve.

Wave PropertiesWavelength

Wavelength,, is the distance between two consecutive peaks.

Wave PropertiesAmplitude

Amplitude is the height of the wave above or below the equilibrium point.

Wave PropertiesPeriod

The wave period, P, this the time it take one wave to pass the observer.

Wave PropertiesFrequency

Frequency, f, is the number of waves passing a particular point in one second.

What is the frequency of the second hand of a clock?

Examples of Frequency

Frequency = 1cycle/60 secFrequency = 1cycle/60 sec Period = 60 secPeriod = 60 sec

What is the frequency of US What is the frequency of US Presidential elections?Presidential elections?

Frequency = 1 election/4 yrsFrequency = 1 election/4 yrs Period = 4 yrsPeriod = 4 yrs

T 1T

f

T

f

T

fT

f Tf fTf

In symbolic formIn symbolic form

oror

f

Waves to transfer energy, not matter, from one place to another

A Vibrating source transfers a disturbanceSpeed depends on type of vibrating source and

medium through which it travelsWave speed = f x The same type of wave moves at the same speed regardless of f or For any wave, f is inversely proportional to

Wave Motion, Speed, Type

VIBRATION OF A PENDULUM

glT 2

What does the period (T) depend upon? Length of the pendulum (l). Acceleration due to gravity (g).

Period does not depend upon the bob mass or the amplitude of the swing.

Vibration of a pendulum. The to-and-fro vibratory motion is also called oscillatory motion (or oscillation).

Transverse waves vibrate across from direction of travel

Longitudinal waves vibrate along the direction of travel (as in a spring)

Wave Type

Sound Waves

Molecules in the air vibrate about some average position creating the compressions and rarefactions. We call the frequency of sound the pitch.

Wave Interference

When two wave pass each other their superposition

causes reinforcement or cancellation.

Reinforcement when the crest of one wave overlaps the crest of another

Their individual effects adds together, resulting in a wave increased in amplitude

Constructive interference

Cancellation when crest of one wave overlaps trough of another reducing their individual effects

Water waves show these best Out of phase- the crest of one wave arrives at a point at the same time as a trough of the second wave arrives, effects cancel each other In phase- two waves crests and troughs arrive at a place at the same time, effects reinforce each other

Destructive Interference

Sound Wave Interference

•Interference occurs when two sounds of difference frequency are heard superposed. •Constructive interference causes louder sound and destructive inference cause fainter sound.•This alternating pattern produces a beat. A piano tuners listens for beats to disappear.

Water Wave Interference

Left side is theoretical drawing of an interference pattern.

Right side is the actual interference pattern.

Occurs when a wave reflects upon itself and interference causes the pattern

Nodes remain stationary Anti nodes-occur half way between nodes

Standing Waves

Standing WavesChange the frequency in a standing wave and more nodes/antinodes appear in the event

We know that waves travel through mediums.

But what happens when that medium runs out?

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Wave Behavior

The behavior of a wave when it reaches the end of its medium is called the wave’s BOUNDARY BEHAVIOR.

When one medium ends and another begins, that is called a boundary.

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Boundary Behavior

One type of boundary that a wave may encounter is that it may be attached to a fixed end.

In this case, the end of the medium will not be able to move.

What is going to happen if a wave pulse goes down this string and encounters the fixed end?

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Fixed End

Here the incident pulse is an upward pulse. The reflected pulse is upside-down. It is

inverted. The reflected pulse has the same speed,

wavelength, and amplitude as the incident pulse.

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Fixed End

Fixed End Animation

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Another boundary type is when a wave’s medium is attached to a stationary object as a free end.

In this situation, the end of the medium is allowed to slide up and down.

What would happen in this case?

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Free End

Here the reflected pulse is not inverted. It is identical to the incident pulse, except it

is moving in the opposite direction. The speed, wavelength, and amplitude are

the same as the incident pulse.

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Free End

Free End Animation

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Our third boundary condition is when the medium of a wave changes.

Think of a thin rope attached to a thin rope. The point where the two ropes are attached is the boundary.

At this point, a wave pulse will transfer from one medium to another.

What will happen here?

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Change in Medium

In this situation part of the wave is reflected, and part of the wave is transmitted.

Part of the wave energy is transferred to the more dense medium, and part is reflected.

The transmitted pulse is upright, while the reflected pulse is inverted.

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Change in Medium

The speed and wavelength of the reflected wave remain the same, but the amplitude decreases.

The speed, wavelength, and amplitude of the transmitted pulse are all smaller than in the incident pulse.

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Change in Medium

Change in Medium Animation

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