Waves
Chapter 12
12-1 Simple Harmonic Motion
• Simple Harmonic Motion (SHM)- Any periodic
motion that is the result of the restoring force
that is proportional to the displacement.
• The mass-spring system and the simple
pendulum are examples of SHM
Mass-Spring System
• Mass-spring system- A mass oscillating on a
spring undergoes simple harmonic motion.
• Hooke’s Law Fspring = -kΔx
Fspring = Force on the spring (N)
K = spring constant (N/m)
Δx = change in displacement (m)
Fspring is negative
because it always goes
against the force
applied to stretch or
compress the spring
Simple Pendulum
• Simple pendulum- an object that swings back
and forth.
• The object is called the bob.
• The restoring force is equal to the bob’s weight
• Simple pendulum works for small angles only
(13≥θ≥0˚)
• θ = angular displacement = amplitude
• GPE increases as θ increases.
• Energy changes from GPE→KE→GPE →etc.
Simple Pendulum
L= arm length
Θ = amplitude
T = tension forces
mg = weight
S = arc length
Time Period
Time period (T)- the time it takes to complete one
full cycle of SHM
Mass-spring system
Simple Pendulum
𝑚
𝐿
𝑇 = 2Π
𝑇 = 2Π
𝑘
𝑔
m=mass
L= Length
k= spring constant
2
g=
gravity
9.81
m/s
T= time period for one
T= time period for one
full cycle
full cycle
Frequency
• Frequency-number of cycles that occur per
time.
• Frequency units are Hertz (Hz)
•
•
1
Hertz=
sec
1
1
f=
T=
T
f
, frequency and time period are
inverse of each other
12-3 Wave Properties
• Wave – motion of a disturbance
• Medium – material the disturbance travels
through. The particles do not move with the
wave.
• Mechanical Waves – waves that require a
medium.
• Pulse Wave – a single pulse sent up and back
• Periodic Wave – repeated series of pulses.
Pulse Wave
Periodic Waves
Parts of a Wave
•
•
•
•
Crest – highest point
Trough – lowest point
Amplitude – maximum displacement
Wavelength – distance traveled during one
cycle(λ)
• Frequency(f) – number of cycles per time period.
• Period – time required for one complete vibration
of particles in a medium
• Wave Speed: v=fλ
Parts of a Wave
Period
(seconds)
time
(meters)
12-3 Types of Waves
• Transverse Wave – particles of the disturbance
move perpendicular to the wave motion. Ex.
Baseball game wave
• Longitudinal Wave – particles of the
disturbance move parallel to the wave motion.
Ex. Sound wave, Density wave
Types of Waves
Types of Waves
• Longitudinal waves can also be called density
waves.
• Compressed particles are under high density
• Stretched (rarefaction) particles are under low
density.
• Density waves can be represented as a sine
wave. Compressions are crests and
rarefactions are troughs.
Wave Interference
• Constructive Interference – When two waves
are added together and the resultant wave
amplitude is larger.
• Destructive Interference - When two waves
are added together and the resultant wave
amplitude is smaller.
Wave Interference
Constructive Interference
Destructive Interference
Wave Reflection
• Reflection – The process of waves bouncing back.
– Fixed boundary – waves pulses will reflect inverted
from its original position.
– Free boundary - waves pulses will reflect in its original
position.
• Standing Wave – wave pattern consisting of
pulses and reflected pulses, that does not appear
to move along a medium. Only happens at
certain frequencies.
– Nodes – points of complete destructive interference.
– Antinodes – points of complete constructive
interference.
Reflection
Standing Wave
N = Nodes
A = Antinode
2 nodes, 1 antinode
3 nodes, 2 antinode
4 nodes, 3 antinode
Chapter 13 Sound
• Sound is a longitudinal wave.
• Frequency determines pitch.
• Wave speed depends on the medium. Faster in
denser material. Ex. Speaking underwater.
• Infrasonic sound waves – low frequency (<20Hz)
• Range of human hearing 20Hz→20,000 Hz
• Ultrasonic sound waves – high frequency (>20,000Hz)
• An ultrasound is a procedure where high frequency
sound waves are transmitted through a section of
your body. Waves are partially reflected back during
density changes. A sensor takes the reflected waves
and makes an image. Ex. Pregnant woman
Ultrasound
Sound Waves
• Sound travels in 3-D
• Ex. Drop a rock in a
pond
• S= Source of the
disturbance
λ
S
Doppler Effect
Doppler Effect- relative motion
creates a change in frequency of a
wave. The frequency is higher in
the direction the source moves and
lower when moving away.
The Doppler Effect explains why
sirens on emergency vehicles
sound higher pitch when travelling
toward you, but lower pitch while
traveling away from you. The same
process is used to track high
density vs. low density clouds to
predict storm movement.
Lower λ, Higher f
λ
S
Lower λ, Higher f
v
v = velocity of the source
λ
Sound Intensity
• 𝑆𝑜𝑢𝑛𝑑 𝐼𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 =
𝑃𝑜𝑤𝑒𝑟
𝐴𝑟𝑒𝑎
𝑃𝑜𝑤𝑒𝑟
4Π𝑟2
• 𝑆𝑜𝑢𝑛𝑑 𝐼𝑛𝑡𝑒𝑛𝑠𝑖𝑡𝑦 =
for a spherical
wave
• Decibel Level – Sound intensity relative to the
threshold of hearing
Sound Intensity
Resonance
• Resonance- the natural frequency of a
material.
• Ex. Two identical tuning forks, same length
pendulums, opera singer and a crystal glass,
Tacoma Bridge
Homework Problems
• Chapter 12 Problems 1, 3-10, 12-13, 15-16,
18-22, 24-28, 30-32, 35-37, 39-45