Properties of Waves
Jevaughn J. Scott
Wave Behavior
Key Terms
Displacement
•
Displacement in the context of waves refers to the distance a particle in the medium moves from its
equilibrium (rest) position. It is usually represented on the vertical axis in wave graphs.
Amplitude
•
Amplitude is the
maximum displacement of
particles from their
equilibrium position. It
represents the wave's
energy; larger amplitudes
indicate more energetic
waves. In a graph,
amplitude is the peak
height or depth from the
central line (equilibrium
position).
Period (T)
•
The period is the time taken for
one complete cycle of the wave
to pass a given point. It is the
inverse of frequency and is
measured in seconds (s). On a
graph, it is the distance between
two successive crests or troughs
along the time axis.
1
𝜆
•𝑇= =
𝑓
𝑣
Wavelength (𝜆)
•
Wavelength is the distance
between two successive crests or
troughs in a transverse wave, or
two successive compressions or
rarefactions in a longitudinal
wave. It is measured in meters
(m).
Period vs
Wavelength
Frequency (f)
•
Frequency is the number of wave cycles
that pass a given point per second. It is
measured in hertz (Hz) and is inversely
related to the period (f = 1/T)
Velocity (v)
•
Wave velocity is the speed
at which the wave
propagates through the
medium. It is given by the
equation:
𝜔
𝜆
• 𝑣 = 𝑓𝜆 = =
𝑘
𝑇
•
However for a wave on a
string or chord:
•
𝑣=
•
𝜇=
𝐹𝑇
𝜇
𝑚
𝑙
Transverse vs. Longitudinal
Mechanical Waves
•
In physics, mechanical waves are disturbances that travel
through a medium (solid, liquid, or gas), transferring
energy from one point to another without transporting
matter. These waves can be classified into two main types:
transverse and longitudinal, distinguished by the direction
of particle displacement relative to the direction of wave
propagation.
Transverse Waves
•
In transverse waves, the particles of the medium move
perpendicular to the direction of wave propagation.
Imagine a wave traveling along a string: as the wave
moves horizontally, the particles of the string oscillate
up and down.
Key Characteristics:
•
Perpendicular Motion: Particle displacement is at right
angles to the direction of wave travel.
Wave Components:
•
Crest: The highest point of the wave.
•
Trough: The lowest point of the wave.
•
Examples: Water Waves, Light Waves, String Vibrations.
Movement of Particles in the Medium
•
Particle Motion: In transverse waves, particles of the medium oscillate perpendicular to the
direction of wave propagation. For instance, if a wave travels horizontally along a string, the
particles of the string move up and down.
•
Each particle moves from its equilibrium position to a maximum displacement (amplitude) in
one direction, returns to equilibrium, and then moves to maximum displacement in the
opposite direction.
Longitudinal Waves
•
In longitudinal waves, the particles of the medium move
parallel to the direction of wave propagation. Imagine a
slinky: as a compression travels through it, the coils of
the slinky move back and forth along the direction of the
wave.
Key Characteristics:
Parallel Motion: Particle displacement is in the same
direction as the wave travel.
Wave Components:
Compression: Regions where particles are close together.
Rarefaction: Regions where particles are spread apart.
Examples: Sound Waves, Seismic P-waves, Spring
Oscillations
Movement of Particles in the Medium
•
Particle Motion: In longitudinal waves, particles of the medium oscillate parallel to the direction of wave
propagation.Imagine a sound wave moving through air: air molecules compress and then spread apart in the
direction the wave is traveling.
•
This creates regions of compression (where particles are close together) and rarefaction (where particles are spread
apart).
•
Each particle moves back and forth around its equilibrium position.
Polarization of Waves
• Polarization refers to the orientation of
the oscillations of a wave in relation to
its direction of propagation. It is a
phenomenon that applies to transverse
waves, where the direction of particle
oscillation can vary. Longitudinal waves,
such as sound waves, cannot be
polarized because their oscillations
occur in the same direction as the wave
propagation.
Polarization in Transverse Waves
•
Linear Polarization: The wave oscillates in a single plane. For
example, in light waves, if the electric field oscillates vertically,
the wave is vertically polarized.
Polarization in Transverse Waves
•
Circular Polarization: The electric field rotates in a circular motion as the wave propagates. This
occurs when two perpendicular components of a wave have equal amplitude but a phase difference
of 90 degrees.
Polarization in Transverse Waves
•
Elliptical Polarization: A more general form where the electric field describes an ellipse. It occurs
when the amplitudes of the perpendicular components are different or the phase difference is not
exactly 90 degrees.
Mechanisms of Polarization
Natural Polarization:
•
Light can become partially polarized when it
reflects off surfaces like water or glass. This is
because certain orientations of the electric
field are preferentially absorbed or reflected.
Mechanisms of Polarization
Artificial Polarization:
•
Polaroid Filters: These are materials that
allow only light waves with a specific
orientation of the electric field to pass
through. Commonly used in sunglasses to
reduce glare.
•
Birefringence: Some materials, like calcite,
have different refractive indices for light
polarized in different directions, which can
split light into two polarized beams.
Examples of Polarization
•
Polarized Sunglasses: These sunglasses use polarizing filters to
block horizontally polarized light, reducing glare from reflective
surfaces like roads or water.
•
Television and Radio Signals: Electromagnetic waves used in
broadcasting can be polarized. For example, TV antennas are
designed to receive either vertically or horizontally polarized
signals.
•
Optical Devices: Polarization is used in various optical devices
like cameras and microscopes to enhance contrast and reduce
reflections.
•
3D Movies: In 3D cinema, two images are projected with
different polarizations. Glasses with corresponding polarized
lenses ensure that each eye sees only one image, creating a
stereoscopic effect.
Intensity of a wave
•
The relationship between the intensity of a wave and its amplitude is fundamental in understanding
how the energy of a wave is distributed.
•
Intensity (I) is defined as the power transmitted per unit area. For a wave, this corresponds to how
much energy passes through a given area in a specific amount of time.
•
𝐼=
•
Where P is the power and A is the área through which the power is passing.
𝑃
𝐴
Amplitude and Energy in Waves
•
he amplitude (A) of a wave is the maximum displacement of particles from their equilibrium
position.
•
The energy carried by a wave is directly related to the square of its amplitude. This is because both
kinetic and potential energy in the oscillating medium increase with the square of the amplitude.
•
𝑥 = 𝐴 sin cot +𝜙 or the cos variation
Intensity-Amplitude Relationship
•
For a wave, the intensity is proportional to the square of the amplitude. This relationship can be
written as:
•
𝐼∝𝐴
This means that if the amplitude of a wave doubles, its intensity increases by a factor of four.
Phase and Phase Difference in Waves
•
Phase:
•
The phase of a wave refers to the position of a
point within one complete cycle of the wave,
usually expressed in terms of an angle (in
degrees or radians). It indicates where the
particle is within its oscillation at a particular
point in time.
For instance, if you have a sinusoidal wave, the
phase determines whether the particle is at a
maximum, minimum, or zero displacement.
• If the phase is θ, it describes the angle between the
point of interest and the reference point.
•
Phase and Phase Difference in Waves
•
Phase Difference:
•
Phase difference refers to the difference in
phase between two points on the wave or
between two waves at the same point in space at
the same time.
If two waves are in phase (i.e., they have the same
phase), their maximum and minimum points occur at
the same time.
• If two waves are out of phase, their maximum and
minimum points occur at different times. A phase
difference of 180∘∘ or π radians means that the two
waves are exactly out of phase and will cancel each
other if they meet (destructive interference).
•
Standing vs Progressive Waves
•
Progressive Waves:
•
Definition: Progressive waves move through the medium, transferring energy from one point to another
without transferring matter.
•
Behavior: The displacement of particles propagates from one point to another.
•
Example: A sound wave traveling through air, light waves, and water waves.
Standing vs Progressive Waves
•
Stationary Waves:
•
Definition: Stationary waves (also called standing waves) are formed when
two waves of the same frequency, amplitude, and wavelength travel in
opposite directions and interfere with each other.
•
Behavior: In stationary waves, some points (called nodes) do not move,
while others (called anti-nodes) oscillate with maximum amplitude.
•
Nodes and Anti-nodes:
•
Nodes: Points in a stationary wave where there is no oscillation,
i.e., zero displacement. These occur where the waves are in
complete destructive interference.
•
Anti-nodes: Points in a stationary wave where the displacement is
at its maximum, i.e., where the waves constructively interfere.
Properties of Stationary Waves
•
Stationary waves have distinct properties:
•
Amplitude: Varies between zero (at nodes) and a maximum
(at anti-nodes).
•
Energy Distribution: Energy is not transferred along the
wave as in progressive waves. Instead, it is confined to
specific regions of the wave.
•
Wavelength: The distance between two consecutive nodes
or anti-nodes is half the wavelength of the original waves.
Example of Stationary Waves in
Practical Applications:
•
Microwaves:
•
Stationary Waves in Microwaves: Microwaves in a
microwave oven create stationary waves inside the oven
cavity. Food placed in areas of maximum amplitude (antinodes) heats up faster than in regions of minimum
amplitude (nodes).
•
Resonance: The wavelength of the microwaves is such that
they form stationary waves with nodes and anti-nodes in the
oven, affecting how heat is distributed.
Waves on Strings:
•
A wave on a string can form stationary waves when the
string is fixed at both ends. The fundamental frequency
of vibration is determined by the length of the string,
the tension, and the mass per unit length of the string.
•
The string vibrates with nodes at both ends and antinodes in between.
Closed and Open Pipes (Resonance
Tube):
•
Closed Pipe: One end of the pipe is closed, and
the other is open. In this case, there is a node at
the closed end and an anti-node at the open
end.
•
Open Pipe: Both ends of the pipe are open, and
there are anti-nodes at both ends.
•
Resonance Tube: A tube open at both ends or at
one end and closed at the other can
demonstrate resonance when it vibrates at its
natural frequencies, forming stationary waves.
The tube vibrates with nodes and anti-nodes
based on the frequency and length of the tube.
Practical Applications of Sound
Waves in Industry
•
Sonar Waves (Depth of the Sea):
•
Sonar (Sound Navigation and Ranging)
uses sound waves to measure the distance
to objects underwater, such as the seafloor
or submarines.
•
Sonar waves are emitted, and the time it
takes for the waves to reflect back to the
source allows the calculation of the depth
of water or the distance to an object.
Practical Applications of Sound
Waves in Industry
•
Medical Applications (Foetal Imaging):
•
Ultrasound uses high-frequency sound
waves to create images of the inside of
the body. In fetal imaging, ultrasound
waves are used to capture images of a
developing fetus.
•
The waves are reflected by tissues of
different densities, allowing the creation
of a clear image of internal organs.
Application of Sound Waves to
Musical Instruments
•
Percussion Instruments (e.g., Steel
Pan):
•
Steel Pan: The steel pan is a percussion
instrument where the vibrations of the
metal surface create sound waves. The
sound depends on the shape, size, and
tension of the metal. The vibrations on
different parts of the pan produce
different pitches.
•
The stationary wave patterns formed on
the pan's surface give rise to the different
musical notes.
Application of Sound Waves to
Musical Instruments
•
Stringed Instruments (e.g., Guitar):
•
Guitar: In stringed instruments like the
guitar, the vibration of strings forms
stationary waves. The length, tension,
and mass per unit length of the string
determine its frequency. The sound
produced depends on which string is
plucked and how it vibrates.
•
The resonating body (e.g., the guitar's
body) amplifies the sound produced by
the vibrating strings.
Application of Sound Waves to
Musical Instruments
•
Wind Instruments (e.g., Flute):
•
Flute: In wind instruments, air is blown
through a tube to create sound. The air
inside the flute forms a stationary wave,
with nodes and anti-nodes based on the
length of the flute and the position of
finger holes.
•
The pitch of the flute changes by altering
the length of the vibrating air column
(by opening or closing finger holes).