Simple Harmonic Motion
• Simple harmonic motion (SHM) refers an
oscillatory, or wave-like motion that describes the
behavior of many physical phenomena:
– a pendulum
– a bob attached to a spring
– low amp. waves in air, water, the ground
– vibration of a plucked guitar string
Objects undergoing SHM trace
out sine waves where the d is pos
and neg with time.
Simple Pendulum
displacement– time graph w
worksheet
Velocity and acceleration in SHM
• The position of an object undergoing SHM changes with
time, thus it has a velocity
• The velocity of an object is the slope of its graph of
position vs. time. Thus, we can see that velocity in SHM
also changes with time, and so object is accelerating:
Vibrations & Waves
Energy Transfer by Waves
Waves are an energy disturbance
propagates through material or
empty space.
Waves Transfer Energy
Matter is not transferred
Ex: Cork on water or buoy
Waves start with vibration
Some Types of Energy that
travel as Waves
• Sound – vibrating tuning fork, string, wood
etc.
• Light (EM) – vibrating charges.
• Earthquake – vibration of Earth’s crust
How can we prove that waves
transfer energy?
-Can waves do work?
-Give examples.
Mechanical waves need medium through
which to travel. Mediums include:
Gasses - air
liquids/water
Solids - wood
Ex: Sound/Earthquake waves
Non mechanical – no medium
required!
Electromagnetic Waves (EM) need
no medium
*EM waves can also propagate
through a medium
Two Main Types of Waves
Transverse (all EM waves),
seismic S waves
Longitudinal (Compressional)
Sound, seismic P waves
Transverse Wave Pulse
One disturbance
Transverse Periodic Wave
Pulses Pass at
Regular Intervals
Particles vibrate perpendicular to energy
transport. Trace out sine wave.
Particle motion transverse wave.
Longitudinal/Compressional Wave
Particles compressed and expanded
parallel to energy propagation.
Another View
Sound Waves ex of mechanical
wave. Need medium to propagate.
Vibrations
in air molecules
from vibrating
tuning fork or
vibrating string.
Eardrum
sound waves do work on eardrum
Water Waves
Combination of 2 Types
Parts of a Wave
Wavelength (l) = distance btw
Crests or Troughs
Midpoint = Equilibrium Position
l
Crests
d
d
Wave Pulse
• Single disturbance
Periodic Wave
• Many pulses with regular l and
period
1. State the difference between a
mechanical and non-mechanical
wave.
Longitudinal Wave Parts
Transverse & Longitudinal Waves can
be represented by sine waves.
Longitudinal Waves can be graphed
as density of particles vs time. Then
will graph as sine wave.
Period (T) & Frequency (f)
Period = time to complete one cycle
of wave crests or troughs. Time for
disturbance to travel 1l.
Usually measured in seconds.
T = 0.5 s/cycle.
Frequency = Number of cycles in
unit time. Inverse of period.
Usually number per second
called Hertz (Hz)
Ex: 3 crests or cycles per second
= 3s-1 or 3 hz
f = how often T = how long
f = a rate
T = a time
T & f are inverse
f = 1/T
or T = 1/f.
2. A wave has a period T of 5.0
seconds. What is its frequency?
0.2 hz
3. A wave has a frequency of
100 Hz. What is its period?
0.01 s.
4. The wave below shows a “snapshot”
that lasted 4.0 seconds. What is the
frequency of the wave?
4.0 seconds
2 cycles/4 s
=
0.5 Hz
Wave Speed
Speed/Velocity = d/t
If a crest (or any point on a wave) moves
20m in 5 sec, v = 20m/5s = 4 m/s.
Relationship of wave speed to
wavelength(l) and frequency(f).
v = d/t but for waves
d = 1l occurs in time T (1period)
so
since freq
v = l/T
f =1/T
v =lf
5. A piano emits from 28 Hz to 4200
Hz. Find the range of wavelengths in
air attained by this instrument when
the speed of sound in air is 340 m/s.
l = 0.081 m to 12 m
What determines wave speed?
Only the medium through which it travels!
• Wave speed is constant if medium is uniform.
• Air at constant T and P.
• Homogenous solids.
• Water with constant T.
7. A tuning fork produces a sound with f = 256 Hz
and l in air of 1.35 m.
• What is the speed of sound in air?
• What would be the wavelength of this tuning fork is
sound travels through water at 1500 m/s?
• 346 m/s
• 5.86 m
Velocity depends on medium’s properties:
-EM waves all travel at c in a vacuum.
- EM waves slower through materials.
-Vibrations travel faster on tighter
strings - slower on loose strings.
-v sound constant in air but
depends on temp/density of air.
8. What determines the wave’s
frequency?
• Vibrational Rate
Wave song
http://www.youtube.com/watch?v=EzU79Egl3-c
Example Problems & Hwk.
Read Text 12 - 3
• Read Text Chap 12-3
• Do pg 470 #23- 32, 35, 36.
• Write all out will collect.
Do Now Text Pg 457 #2
Quiz
• 1. What is only factor that determines wave speed.
• 2. Give a real life example of:
– A longitudinal wave
– A transverse wave.
• Sketch a transverse wave. Label the
– Wavelength
– Amplitude
– Equilibrium position
What is the motion of points on a
wave?
Up & down motion of particle on wave.
Another View
Given a wave moving to the left as
below, what will be the motion of the red
beach ball just after the time shown?
•
•
•
•
Up
Down
Right
Left
Down
Wave Behaviors & Interactions
Boundary Behavior & Wave
Superposition
When a wave enters a material with
new properties it:
•
•
•
•
Goes through it without noticing
Slows down
Speeds up
Accelerates
Wave Behaviors and
Interactions
Reflection- a wave incident on a
boundary (new material), part
bounces off, part transmitted.
Example Echo: A sound wave is traveling in
air at STP. The echo is heard 2.6 second
later. How far away is the reflecting object?
•
•
•
•
•
•
Time to object = 1.3 seconds.
Speed sound STP = 331 m/s
v = d/t
tv = d
(1.3s)(331 m/s)
430.3 m
Reflection off Fixed Boundary –
pulse inverts
Reflection off Fixed Boundary –
pulse inverts
Pulse Passing into new medium from less
dense material. What happens to pulse?
Changes in:
Velocity and Amplitude change with
medium.
No Change - frequency.
More dense into less dense
How do multiple waves combine?
• Waves can overlap and occupy the same
space at the same time.
• How they do it depends on the position or
phase of the crests and troughs.
• Superposition – constructive destructive
interference.
Phase of Particles in Wave
• “in phase” = points in identical position. Whole
number of l apart.(A,F B,G E,J C,H)
• 180o out of phase = equal displacement fr
equilibrium but moving opposite directions.
• Odd number of ½ l apart. (A,D)
Superposition /Interference–
2 or more waves or pulses
interact/superimpose & combine. Their
amplitudes add or subtract.
The resultant wave is the sum of the two.
Constructive Interference
– waves superimpose with displacement in
same direction + or -, amplitude increases.
Constructive Interference.
Destructive Interference- waves or pulses
meet with opposite displacement. Waves
partially or totally cancel.
Points on waves that meet “in
phase” interfere constructively.
Points that meet “out of phase” interfere
destructively. Below is total destructive
interference.
Standing Waves
Wave pattern that results when 2 waves, of
same f, l, & v travel in opposite directions.
Often formed from pulses reflected off a
boundary. Waves interfere constructively &
destructively at fixed points.
Standing Wave – wave appears to be
standing still. No net transfer of energy.
Standing wave formed from wave
pulses in same medium.
Nodes are points of max. destructive
interference.
Antinodes = points of max. constructive
interference.
Standing Wave Formation from 2
waves.
Standing waves have no net transfer
of energy – no propagation of energy.
Sketch Resultant on wksht from
RB pg 271.
Hwk Text pg 471 #33 - 44, 48-49
Hwk Text pg 471
#36 - 44, 48-50
Relation of Wavelength to String
Length for Standing Waves
Standing waves form only when the
string length allows a whole number of
half wavelengths to fit.
½ l = L or 2L = l.
l = L.
l= L
General expression relating
wavelength to string length for
standing waves:
n ( ½ l) = L
n is a whole number
Mechanical Universe “Waves”