Chapter 12

Oscillatory Motion

Periodic motion is motion of an object that regularly repeats

If the force is always directed toward the equilibrium position, the motion is called simple harmonic motion

Restoring Force and Hooke’s Law

The block is displaced to the right of x = 0 The position is

positive The restoring force

is directed to the left

Hooke’s Law states Fs = - k x

2

2netd xF ma m kxdt

= ⇒ = −

A solution is x(t) = A cos (ωt + φ)or x(t) = A sin (ωt + φ)

where kwm

=

cos(30 90 ) cos(120 ) sin 30cos(30 ) cos(90 60 ) sin 60

o o o o

o o o o

+ = = −

= − =

Simple Harmonic Motion –Graphical Representation A solution is x(t) = A cos (ωt + φ) A, ω, φ are all

constants A cosine or sine

curves can be used to give physical significance to these constants

Period and Frequency The frequency and period equations

can be rewritten to solve for ω

The period and frequency can also be expressed as:

Motion Equations for Simple Harmonic Motion

max2

max

v Aa A

ω

ω

=

=

SHM Example 1: cosine

Initial conditions at t = 0 are x (0)= A v (0) = 0

This means φ = 0

SHM Example 2: sine Initial conditions at

t = 0 are x (0)=0 v (0) = vi

This means φ = − π/2

2 2 2 21 1 1 12 2 2 2total o oE mv kx mv kx= + = +

Conservation of Total Energy

Chapter 13

Mechanical Waves

Terminology: Amplitude and Wavelength The crest of the wave:

This distance is called the amplitude, A The point at the negative

amplitude is called the trough

The wavelength: λThe distance from one crest to the next

y = f(x) y = f(x-a)

f(x) f(x-a)

a

Traveling Wave The brown curve

represents a snapshot of the curve at t = 0

The blue curve represents the wave at some later time, t

Wave Equations

The wave y(x,t) = A sin (k x – ωt) The speed v = λƒ If x ≠ 0 at t = 0, the wave function can

be generalized toy = A sin (k x – ωt + φ)

where φ is called the phase constant

Waves on a String

This is the speed of a wave on a string It applies to any shape pulse

Speed of Sound WavesCompression and rarefaction regions of the gas continue to move

Sound Waves as Displacement or Pressure Wave A sound wave may

be considered either a displacement wave or a pressure wave

The pressure wave is 90o out of phase with the displacement wave

Chapter 14

Superposition and Standing Waves

Superposition of Sinusoidal Waves y1 = A sin (kx - ωt) y2 = A sin (kx - ωt + φ)

y = A {sin a + sin b} == 2A cos [(a-b)/2] sin [(a+b)/2]

y = y1+y2= 2A cos (φ/2) sin (kx - ωt + φ/2)

Sinusoidal Waves with Destructive Interference

When φ = π, then cos (φ/2) = 0 Also any even multiple of π

The amplitude of the resultant wave is 0 Crests of one wave

coincide with troughs of the other wave

The waves interfere destructively

Standing Waves Assume two waves with the same

amplitude, frequency and wavelength, traveling in opposite directions in a medium

y1 = A sin (kx – ωt) and y2 = A sin (kx + ωt) They interfere according to the

superposition principle

Superposition of Two Identical Traveling Waves in Opposite Directions = Standing Wave y1 = A sin (kx - ωt) moving to the right y2 = A sin (kx + ωt) moving to the left

y = A {sin a + sin b} == 2A cos [(a-b)/2] sin [(a+b)/2]

y = y1+y2= [2A cos (ωt)] sin (kx )

Nodes and Antinodes, Photoy = y1+y2 = [2A cos (ωt)] sin (kx )

Nodes and Antinodes, cont

The diagrams above show standing-wave patterns produced at various times by two waves of equal amplitude traveling in opposite directions

In a standing wave, the elements of the medium alternate between the extremes shown in (a) and (c)

Standing Waves in a String Consider a string fixed

at both ends The string has length L Standing waves are set

up by a continuous superposition of waves incident on and reflected from the ends

There is a boundary condition on the waves

Standing Waves in a String

1/2λ1 = L so λ1 = 2L

Standing Waves in a String, 4 The second

harmonic λ = L

The third harmonic λ = 2L/3

Standing Waves on a String, Summary

n = 1, 2, 3, …

n is the nth harmonic These are the possible modes for the string

22

nn

Ln Ln

λ λ= ∴ =

vv f fλλ

= ∴ =

Standing Waves in an Open Tube

Standing Waves in a Tube Closed at One End

Beats

Temporal interference will occur when the interfering waves have slightly different frequencies

Beating is the periodic variation in amplitude at a given point due to the superposition of two waves having slightly different frequencies

Beats

The beat frequency is ƒbeat = |ƒ1 – ƒ2|

Average frequency heardƒbeat = (ƒ1 + ƒ2)/2

Quality of Sound –Tuning Fork A tuning fork

produces only the fundamental frequency

Quality of Sound –Flute The same note

played on a flute sounds differently

The second harmonic is very strong

The fourth harmonic is close in strength to the first

Quality of Sound –Clarinet The fifth harmonic is

very strong The first and fourth

harmonics are very similar, with the third being close to them

Analyzing Nonsinusoidal Wave Patterns If the wave pattern is periodic, it can be

represented as closely as desired by the combination of a sufficiently large number of sinusoidal waves that form a harmonic series

Any periodic function can be represented as a series of sine and cosine terms This is based on a mathematical technique called

Fourier’s theorem

Fourier Series A Fourier series is the corresponding

sum of terms that represents the periodic wave pattern

If we have a function y that is periodic in time, Fourier’s theorem says the function can be written as

ƒ1 = 1/T and ƒn= nƒ1 An and Bn are amplitudes of the waves

Fourier Synthesis of a Square Wave

Fourier synthesis of a square wave, which is represented by the sum of odd multiples of the first harmonic, which has frequency f

In (a) waves of frequency fand 3f are added.

In (b) the harmonic of frequency 5f is added.

In (c) the wave approaches closer to the square wave when odd frequencies up to 9f are added.