1. Vibrations and Waves
-
Upload
kiara-de-leon -
Category
Documents
-
view
239 -
download
2
description
Transcript of 1. Vibrations and Waves
Hooke’s Law
• Fs = - k x
• Fs is the spring force
• k is the spring constant• It is a measure of the stiffness of the spring
• A large k indicates a stiff spring and a small k indicates a soft spring
• x is the displacement of the object from its equilibrium position• x = 0 at the equilibrium position
• The negative sign indicates that the force is always directed opposite to the displacement
Simple Harmonic Motion
• Motion that occurs when the net force along the direction of motion obeys Hooke’s Law• The force is proportional to the displacement and always
directed toward the equilibrium position
• The motion of a spring mass system is an example of Simple Harmonic Motion
Spring- Mass System
When a body is displaced fromits equilibrium position, thespring force tends to restore itto equilibrium. (Restoring Force)
Amplitude, Period, Frequency and Angular Frequency
• Amplitude - denoted by A, is the maximum magnitude of displacement from equilibrium. (SI unit is in m)
• Period – is the time to complete one cycle – say from A to –A and back to A. (SI unit is in seconds)
• Frequency – is the number of cycles per unit time. (SI unit is in hertz)
• Angular Frequency – denoted by ω, is equal to 2𝜋𝑓. It represents the rate of change of an angular quantity
• We encounter waves in many situations• Speech and hearing rely on wave propagation. • Modern telecommunications networks such as mobile
phones rely on waves.• Many key areas of Physics, Mathematics and Chemistry
are best described by waves and their interactions.• The way atoms bind together to form molecules can be
understood by the overlap of waves.
Types of waves
• There are several different types of wave that we must consider.
• Mechanical Waves:- These need a medium to propagate in - sound waves.
• Non-mechanical waves:-These waves do not need a medium in which to propagate - light waves.
• Matter waves:- Particles such as protons and electrons can betreated as waves. This forms the basis of quantum mechanics. We willnot be discussing this type of wave in this course.
Examples of Mechanical Waves
• Earthquake / seismic waves
• Water waves
• Sound waves
• Waves that travel down a spring or rope
Periodic Waves
Periodic waves consist of cycles or patterns that are produced over and
over again by the source.
In the figures, every segment of the slinky vibrates in simple harmonic
motion, provided the end of the slinky is moved in simple harmonic
motion.
In the drawing, one cycle is shaded in color.
The period is the time required for one complete cycle.
The frequency is related to the period and has units of Hz, or s-1.
The angular frequency is the rate of change of an angular quantity.
Tf
1
Periodic Waves
T=1
𝑓= 2𝜋/𝜔
𝜔 = 2𝜋f
Graphing Wave Functions
33
Amplitude A is
the maximum
excursion of a
particle of the
medium from
the particle’s
undisturbed
position.
Wavelength
is the horizontal
length of one
cycle of the
wave.
Crest
Trough
Example
The distance between the crest of a water wave and the
next trough is 2 m. If the frequency of a particular wave is 2
Hz, what is the speed of the wave?
(a) 4 m/s
(b) 1 m/s
(c) 8 m/s
(d) 2 m/s
(e) impossible to determine from the information given
Find the amplitude, wavelength, speed and period of the wave if
it has a frequency of 8.00 Hz. Δx = 40.0 cm, Δy = 15.0 cm
A wave with a frequency of 12.3 Hz is traveling from left to right
across a rope as shown in the diagram at the right Positions A and B
in the diagram are separated by a horizontal distance of 42.8 cm.
Positions C and D in the diagram are separated by a vertical distance
of 12.4 cm. Determine the amplitude, wavelength, period and speed of
this wave.
• During wave motion, the particle
are displaced some distance y in
the direction perpendicular to the
x-axis.
• The motion of the particle on the
right lags behind the motion of the
particle on the left by an amount of
time proportional to the distance
between the particles.
Particle Velocity and Acceleration in a
Sinusoidal Wave
• A disturbance can propagate as a wave along the x-
axis with wave speed v.
• Electric and Magnetic fields satisfy the wave equation,
the wave speed turns out to be the speed of light.
• At which time is point A on the
string moving upward with
maximum speed?
• At which time does point B on
the string have the greatest
upward acceleration?
• At which time does point C on
the string have a downward
acceleration but an upward
velocity?
Energy in Wave Motion
• Power is the instantaneous rate at which energy is
transferred along the string. (It depends on x and t)
• For a sinusoidal wave (sine function), the power is either
positive or zero.
• Energy is transferred to the direction of wave
propagation.
Wave Intensity
• The time average rate at which energy is transported by
the wave per unit area, across a surface perpendicular to
the direction of propagation.
• SI Unit : W/m2 (watts per square meter)
Interference of Waves
• Two traveling waves can meet and pass through each
other without being destroyed or even altered
• Waves obey the Superposition Principle
– When two or more traveling waves encounter each other while
moving through a medium, the resulting wave is found by adding
together the displacements of the individual waves point by point
– Actually only true for waves with small amplitudes
Standing Waves on a String
• It is called a standing wave
because the wave pattern
remains in the same position
along the string and its
amplitude fluctuates.
• Nodes – points that never
move at all
• Antinodes – points where the
amplitude of motion is
greatest
• The picture on the left is a
travelling wave. The wave
does move along the string,
with a speed equal to the
wave speed.
Standing Waves
• A standing wave does not
transfer Energy from one end
to the other.
• The two waves that form
it would individually carry
equal amounts of power
in opposite direction.
• A standing wave can only
exist if its wavelength satisfies
Fundamental Frequencies, Harmonics and
Overtones
o This is the fundamental frequency. The smallest frequency that corresponds
to the largest wavelength 2L.
o The other frequencies, f2, f3, f4, …. Are all multiple integer multiples of the
fundamental frequency. These frequencies are called harmonics. (shown
below)
o f2, f3, f4 and so on are what musicians call overtones. The second harmonic
f2 is called the first overtone.