Vibrations and Waves Summary Sheet
Chapters 11 and 12
Simple Harmonic Motion
Amplitude (A), max PE
m
m
m
Equilibrium (x = o), max KE
m
m
Amplitude (A), max PE
m
2
2 1/2
x = A sin(2πft) v = v max(1-x /A )
1/2
vmax = A (k/m)
or
x = A cos(2πft) a = a cos(2πft)
max
amax = kA/m
xmax = A
T = 2π(m/k)
f = 1/T
1/2
2
Etotal = ½ kx + ½ mv
1/2
2
F = kx
T = 2π(l/g)
Note: period not
dependent on mass
or amplitude
Anatomy and Types of Waves
v = λf
wavelength crest
wavelength compressions
amplitude
Transverse Wave
v=
E
ñ
v=
FT
m
l
extensions
trough
Longitudinal Wave
m
m
Notation
A = amplitude
f = frequency
T = period
x = displacement
v = speed
a = acceleration
t = time
m = mass
F = force on spring
k = spring constant
l = pendulum length
2
g = 9.8m/s
Wave Speed
v = wave speed
λ = wavelength
f = frequency
Using material
properties
E = elastic modulus
(longitudinal)
ρ = density
FT = tension in string
l = length of string
Waves on a
string
m = mass of string
general equation
(transverse)
Wave Properties
Reflection – upon encountering a new medium, a pulse or wave may “bounce” back
Example:
Pulses on a
string
Fixed end
Free end
Example: water
waves hitting a
barrier
θi
θr
Angle of incidence
equals angle of
reflection
Transmission –upon encountering a new medium, a portion of the wave continues into the new medium
Example:
Pulses on string
Less dense to more dense: amplitude decreases, wave slows
More dense to less dense: amplitude increases, wave speeds up
Interference – occurs when multiple waves interact
Principle of superposition – to find the resulting wave, the displacements of each wave are added
Constructive – resulting amplitude is greater than either
pulse’s/wave’s amplitude
Destructive – resulting amplitude is less than
either pulse’s/wave’s amplitude
Refraction – wave changes direction due to a
change in the medium through which it travels
Diffraction – wave bends when it encounters
a barrier
Vibrations and Waves Summary Sheet
Chapters 11 and 12
Standing Waves; Resonance
Damped Harmonic Motion
In class we saw that if you fix one end of a string or long
spring and send a wave down from the other end the
wave reflects and interferes with the wave being sent. At
particular frequencies we observed a special event – the
wave appeared to be standing rather than moving.
Fundamental or
st
1 harmonic
st
1 overtone or
nd
2 harmonic
Curve A represents overdamping – system is
brought to equilibrium over a long period of time.
Curve B represents critical damping – system is
brought to equilibrium over a shorter period of time.
Curve C represents underdamping – system
undergoes several oscillations before reaching
equilibrium.
Often we design for critical damping – situation that
brings the system back to equilibrium in a short
period of time without any oscillations. Example:
the car shock absorber.
nd
2 overtone or
rd
3 harmonic
Doppler Effect
Observers are stationary; sound source is moving
f(=
f
v
1' s
v
f
f(=
v
1+ s
v
Sound source is stationary; observer is moving
&
f ( = $1 +
%
&
f ( = $1 '
%
vo #
!f
v "
vo #
!f
v "
f ! = new frequency
f = source frequency
v s = source speed
v o = observer speed
v = wave speed