Module-1-Lesson-1.3.pdf.Wave behavior and Types

INSTRUCTIONS: Answer the following questions:

• Relate a wave’s speed to the medium in
which the wave travels.
• Describe how waves are reflected and
refracted at boundaries between media.
• Apply the principle of superposition to
the phenomenon of interference

Waves transport energy by
oscillating a medium’s particles
without moving the medium
itself. From the echo of a shout
against a canyon wall to the
colorful rainbow arching
overhead, wave behaviors—
speed changes, reflections,
refractions, and interference—
shape much of what we observe.
INTRODUCTION

In this module, we explore how
medium properties govern wave
speed, how waves interact at
boundaries (fixed, free, or “no‐end”),
how overlapping waves combine,
and how one‐ and two‐dimensional
waves reveal patterns like standing
waves and ripple‐tank phenomena.
Each section builds deep conceptual
understanding alongside practical,
classroom‐ready examples and
problem solving.
INTRODUCTION

Medium
A material—solid, liquid, or
gas—whose constituent
particles transmit mechanical
disturbances by oscillating
around equilibrium positions.
Examples include air for
sound waves, water for
surface ripples, and metal
strings in instruments.
DEFINITION OF TERMS

Boundary Condition
Constraints at an interface or
end of a medium,
determining wave
displacement and phase
upon reflection or
transmission. Fixed ends
enforce zero displacement;
free ends allow maximal
motion.
DEFINITION OF TERMS

Incident Wave (Pulse)
The portion of a wave
train or single
disturbance traveling
toward a boundary before
any reflection or
transmission occurs.
DEFINITION OF TERMS

Reflected Wave (Pulse)
The portion of an incident
wave that reverses
direction upon meeting a
boundary, possibly
inverted (phase‐shifted
by 180°) or upright,
depending on boundary
mobility.
DEFINITION OF TERMS

Transmitted Wave (Pulse)
The part of the incident
disturbance that
continues into the second
medium, maintaining
original frequency but
adopting the new
medium’s speed and
wavelength.
DEFINITION OF TERMS

Superposition
The principle that when two or
more waves occupy the same
region, the net displacement at a
point is the algebraic sum of
individual displacements. The
principle that overlapping waves
combine their displacements
algebraically at each point:
𝑦total = 𝑦1 + 𝑦2 + …
DEFINITION OF TERMS

Interference
The phenomenon arising
from superposition,
producing regions of
reinforcement
(constructive) or
cancellation
(destructive).
DEFINITION OF TERMS

Diffraction
The bending and
spreading of waves
around obstacles or
through openings, most
pronounced when
obstacle dimensions are
comparable to the
wavelength.
DEFINITION OF TERMS

Refraction
The change in direction
and wavelength of a
wave as it crosses
obliquely from one
medium into another
with different
propagation speed,
governed by Snell’s law.
DEFINITION OF TERMS

Standing Wave (Stationary
Wave)
A pattern formed by two
waves of identical frequency
and amplitude traveling in
opposite directions, yielding
fixed nodes (zero
displacement) and antinodes
(maximum displacement).
DEFINITION OF TERMS

Node
A point in a standing
wave where the net
displacement is
always zero due to
perfect destructive
interference.
DEFINITION OF TERMS

1.1 Dependence on Air Temperature
Sound speed in dry air increases approximately
linearly with temperature because warmer air
molecules transmit pressure variations more
rapidly. An empirical relation valid near atmospheric
pressure is:
SPEED OF SOUND WAVES

SPEED OF SOUND WAVES
This arises from the exact thermodynamic formula on the
speed of sound in an ideal gas:

Mechanical impedance—rooted in density and elasticity—dictates
wave speed across materials. Stiffer and less compressible media
conduct sound faster.
DEPENDENCE ON MEDIUM

When the incident pulse reaches
a boundary, two things occur:
• A portion of the energy carried
by the pulse is reflected and
returns towards the left end of
the rope (reflected pulse).
• A portion of the energy carried
by the pulse is transmitted to
the wall
WAVE INTERACTION WITH BOUNDARY

• Constraint: End held
immobile (e.g., rope tied
to a rigid wall).
• Reflection: Pulse inverts
(180° phase shift)
because displacement at
boundary must be zero.
• Speed & Wavelength:
Remain equal to incident
values.
FIXED END

• Reflected wave is
inverted with the
incident wave
• Speed is the same
• Wavelength is the
same
• Reflected wave’s
amplitude is lesser
FIXED END

• Constraint: End can move
freely (e.g., rope over
frictionless ring).
• Reflection: Pulse remains
upright (no phase shift)
as boundary
displacement maximizes.
• Speed & Wavelength:
Unchanged from
incident.
FREE END

• Reflected wave is in
phase with the
incident wave
• Speed is the same
• Wavelength is the
same
• Reflected wave’s
amplitude is lesser
FREE END

• Constraint: Medium extends
indefinitely or terminates in
energy absorber (e.g., water
waves on a sandy beach).
• Reflection: None—incident
energy either continues or
dissipates.
• Illustration: Rope end
dipped in viscous fluid
shows pulse disappearing
without echo.
NO END

FIXED, FREE, AND NO-END CONDITIONS

• Waves will react differently with varying linear densities of
the strings.
WAVES IN VARYING DENSITY

• The transmitted pulse is
traveling slower than the
reflected pulse.
• The transmitted pulse has a
smaller wavelength than the
reflected pulse.
• The speed and the
wavelength of the reflected
pulse are the same as the
speed and the wavelength of
the incident pulse

• The transmitted pulse is
traveling faster than the
reflected pulse.
• The transmitted pulse has a
larger wavelength than the
reflected pulse.
• The speed and the wavelength
of the reflected pulse are the
same as the speed and the
wavelength of the incident
pulse

As a pulse travels from medium 1
into medium 2:
• Incident Pulse approaches the
boundary.
• Reflected Pulse returns into
medium 1 with reduced
amplitude.
• Transmitted Pulse continues
into medium 2 at speed 𝑣2 and
wavelength 𝜆2, sharing the
incident frequency 𝑓 .
INCIDENT, REFLECTED, AND TRANSMITTED PULSES

What happens when two waves
meet while they travel through
the same medium?
SUPERPOSITION AND INTERFERENCE

INTERFERENCE
• It is a phenomenon that occurs when two waves
meet while traveling along the same medium

PRINCIPLE OF SUPERPOSITION
• When multiple waves coexist in a medium, they do not
collide; rather, their displacements add algebraically at
each point:

PRINCIPLE OF SUPERPOSITION
• It states that the disturbance in a
medium caused by two or more
waves is the algebraic sum of the
displacements produced by the
individual waves.
• Waves can combine to form a new
wave
• Waves moving in opposite
directions can cancel or form a new
wave of lesser or greater amplitude.
• This principle underpins all
interference phenomena.

CONSTRUCTIVE INTERFERENCE
• In‐phase waves ( 0 °
difference) reinforce, yielding a
resultant amplitude 𝐴res = 𝐴1 +
𝐴2 .
• Two interfering waves have a
displacement in the same
direction.
• The amplitude of the resulting
wave is equal to the sum of
the individual amplitudes

• In‐phase waves ( 0 ° difference)
reinforce, yielding a resultant
amplitude 𝐴res = 𝐴1 + 𝐴2 .
• Two interfering waves have a
displacement in the same
direction.
• The amplitude of the resulting
wave is equal to the sum of the
individual amplitudes
• Waves pass each other.

Scenario 1: Same Phase, Unequal Amplitudes
• Description: Two waves are in phase, meaning their crests and
troughs align perfectly. But one wave has a larger amplitude
than the other.
• Result: The amplitudes add together, so the resultant wave has
a larger amplitude than either wave individually.
• Type of Interference: Constructive interference—even though
the amplitudes are unequal, the waves reinforce each other.
• Nickname: You could call this partial constructive interference,
but it's still fundamentally constructive.

DESTRUCTIVE INTERFERENCE
• Out‐of‐phase waves ( 180 °
difference) cancel, potentially
producing zero net displacement.
• The two interfering waves have a
displacement in the opposite
direction
• When waves overlap, the effect of
one of the pulses on the
displacement of a given particle of
the medium is destroyed or canceled
by the effect of the other pulse.
• Only a momentary condition-waves
pass each other

DESTRUCTIVE INTERFERENCE
Scenario 2: Out of Phase, Unequal Amplitudes
• Description: The waves are out of phase (e.g., crest
meets trough), but one wave is larger than the other.
• Result: The smaller wave partially cancels the larger one,
resulting in a wave with reduced amplitude, but not zero.
• Type of Interference: Destructive interference—because
cancellation occurs, even if it's not complete.
• Nickname: This is often called partial destructive
interference or incomplete cancellation.

INSTRUCTIONS: Identify what type of interference is asked

• G: Partial Constructive
(The waves are partially in phase,
resulting in a resultant wave with an
amplitude greater than the
individual waves but less than the
maximum possible)
• H: Partial Destructive
(The waves are partially out of
phase, resulting in a resultant wave
with a reduced amplitude)
• I: Partial Destructive
(The waves are partially out of
phase, resulting in a resultant wave
with a reduced amplitude)

• J: Partial Constructive
amplitude greater than the
individual waves but less than the
maximum possible)
• K: Partial Destructive
(Similar to 'I', the waves are partially
out of phase, leading to partial
cancellation)
• L: Partial Destructive
(Similar to 'I', the waves are partially
out of phase, leading to partial
cancellation)

• M: Partial Constructive
amplitude greater than the individual
waves but less than the maximum
possible)
• N: Partial Constructive
amplitude greater than the individual
waves but less than the maximum
possible)
• O: Partial Destructive
(Similar to 'I', the waves are partially out
of phase, leading to partial cancellation)

STANDING WAVES (STATIONARY WAVES)
• A combination of two waves
moving in opposite directions,
each having the same
amplitude and frequency.
• Result of interference
• Standing wave pattern - points
that appear to be standing still
• Only created within the
medium at specific
frequencies (harmonics) of
vibration

STANDING WAVES (STATIONARY WAVES)
A standing wave emerges
from two identical waves
traveling oppositely. Nodes
(zero displacement) form at
fixed points; antinodes
(maximum displacement) lie
midway. Only discrete
frequencies (harmonics)
satisfy boundary constraints
on a string of length 𝐿 fixed at
both ends:

INSTRUCTIONS: Give what is asked.

WAVES IN ONE DIMENSION
One‐dimensional waves—
such as pulses on a
rope—confine oscillations
along a single axis.
Analysis tracks
displacement vs. position
and time, ideal for
illustrating boundary
reflections and standing
modes.

REFLECTION IN ONE DIMENSION
• When a wave
encounters a barrier,
like with 1-D waves,
it is reflected.
• When a straight wave
encounters a straight
barrier head on, it is
reflected back on the
original path.

REFLECTION IN TWO DIMENSION
• The wave pulse moves
toward a rigid barrier that
reflects the wave: the
incident wave moves
forward, and the reflected
wave moves to the right.
• The law of reflection states
that the angle of incidence is
equal to the angle of
reflection.

WAVES IN TWO DIMENSION
Surface waves on water spread in
two axes. Visualizing these
requires:
• Wave Front: Curve connecting
points oscillating in phase,
often drawn as concentric
circles around a source.
• Ray: Line perpendicular to wave
fronts indicating energy
propagation direction.

WAVES IN TWO DIMENSION
• Example: Ripples on the surface
of water.
• Behavior: The wave spreads out
in a plane—both x and y
directions—forming circular
wavefronts.
• Real-world analogy: Dropping a
pebble into a pond and
watching the ripples expand
outward.

• When a straight wave
front meets a straight
barrier: Incident and
reflected rays make
equal angles with the
normal (perpendicular
to barrier). Law of
Reflection: 𝜃 𝑖 = 𝜃 𝑟 .

• The incident wave is represented by an
arrow pointing upward. The ray of the
reflected wave points to the right. The
barrier is represented by a line. The line
which drawn at a right angle, or
perpendicular, to the barrier is called
the NORMAL LINE.
• The Angle between the normal line and
the reflected ray is called the angle of
reflection. The law of reflection states
that “THE ANGLE OF INCIDENCE IS
EQUAL TO THE ANGLE OF
REFLECTION”

REFRACTION IN TWO DIMENSION
• When the waves move from deep to
shallow water, their speed decreases,
and the direction of the waves changes.
• Because the waves in the deep water
generate the waves in the shallow
water, their frequency is not changed.
• Based on the decrease f in the
speed of the waves means that the
wavelength is shorter in the shallower
water.

REFLECTION AND REFRACTION FACTS
• Do you know that echoes are
caused by the reflection of
sound off hard surfaces, like
the walls of a large warehouse
or a distant rock face?
• RAINBOWS are created when
white light passes through a
raindrop; refraction separates
the light into its colors.

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