WAVE MOTION | PHYSICS DECODED

6.1 Progressive Wave: Definition & Understanding

A progressive wave (also called traveling wave) is a disturbance that travels through a medium, transferring energy from one point to another without permanently displacing the medium itself.

What is a Progressive Wave?

Progressive Wave Definition:

"A progressive wave is a wave that continuously moves or propagates through space, carrying energy from one location to another without net transport of matter."

Key Characteristics:

Energy and momentum are transferred
No permanent displacement of medium particles
Wave pattern moves through space
All particles oscillate with same amplitude (in ideal case)
Phase changes continuously with position and time

Types of Progressive Waves

Classification Based on Particle Motion:

1. Transverse Waves:

Particle displacement perpendicular to wave propagation
Can be polarized
Examples: Light waves, electromagnetic waves, waves on strings, water surface waves
Require medium with shear elasticity (except EM waves)

2. Longitudinal Waves:

Particle displacement parallel to wave propagation
Cannot be polarized
Consist of compressions and rarefactions
Examples: Sound waves, pressure waves in fluids, seismic P-waves
Can travel through any elastic medium

Transverse vs Longitudinal Waves

Important Wave Parameters

Parameter	Symbol	Definition	Unit
Wavelength	λ (lambda)	Distance between two consecutive identical points (e.g., crest to crest)	meter (m)
Amplitude	A	Maximum displacement from equilibrium position	meter (m)
Frequency	f or ν	Number of complete oscillations per second	Hertz (Hz) or s^-1
Time Period	T	Time for one complete oscillation	second (s)
Wave Velocity	v	Speed at which wave propagates through medium	m/s
Angular Frequency	ω (omega)	Rate of change of phase, ω = 2πf	rad/s
Wave Number	k	Number of wavelengths per 2π distance, k = 2π/λ	rad/m or m^-1
Phase	φ (phi)	State of oscillation at given time and position	radian (rad)

Fundamental Wave Relationships

KEY WAVE FORMULAS:

1. Wave velocity:
v = fλ = λ/T

2. Frequency-Period relation:
f = 1/T or T = 1/f

3. Angular frequency:
ω = 2πf = 2π/T

4. Wave number:
k = 2π/λ

5. Wave velocity (alternative):
v = ω/k

Important for NEB Exams:

Progressive wave: Travels through medium, transfers energy, no net matter transport
Transverse wave: Particle motion ⊥ to wave direction (light, string waves)
Longitudinal wave: Particle motion ∥ to wave direction (sound waves)
Wavelength λ: Distance between consecutive crests/troughs
Wave equation: v = fλ (most important formula!)
All particles in progressive wave oscillate with same amplitude
Phase changes continuously with position and time

6.2 Mathematical Form of Progressive Wave

The mathematical representation of a progressive wave describes how displacement varies with both position and time. This equation is fundamental to understanding wave behavior.

General Wave Equation

Standard Progressive Wave Equation:

y(x,t) = A sin(kx - ωt + φ)

Or equivalently:
y(x,t) = A sin(2π(x/λ - t/T) + φ)

Where:
• y = displacement at position x and time t
• A = amplitude (maximum displacement)
• k = wave number = 2π/λ
• ω = angular frequency = 2π/T = 2πf
• φ = initial phase constant
• x = position along wave direction
• t = time

Deriving the Wave Equation

Step-by-Step Derivation:

Step 1: Simple Harmonic Motion at Origin

Consider a particle at x = 0 executing SHM:

y(0,t) = A sin(ωt) = A sin(2πt/T)

Step 2: Wave Propagation

If wave travels with velocity v in +x direction, disturbance at x = 0 at time t will reach position x at time (t + x/v).

Or equivalently: disturbance at position x at time t was at origin at earlier time (t - x/v).

Step 3: Displacement at Position x

y(x,t) = A sin(ω(t - x/v))

y(x,t) = A sin(ωt - ωx/v)

Step 4: Substituting Wave Parameters

Since v = fλ = ω/k, we have ω/v = k

Therefore: y(x,t) = A sin(ωt - kx)

Step 5: Standard Form (with conventional sign)

y(x,t) = A sin(kx - ωt)

(Equivalent to previous, just reordered)

Step 6: Including Initial Phase

For general case where oscillation doesn't start from equilibrium:

y(x,t) = A sin(kx - ωt + φ)

Alternative Forms of Wave Equation

Different Representations:

1. Using Sine Function (traveling right):

y = A sin(kx - ωt + φ)

or: y = A sin(2π(x/λ - t/T) + φ)

2. Using Cosine Function (traveling right):

y = A cos(kx - ωt + φ)

or: y = A cos(2π(x/λ - t/T) + φ)

3. Wave Traveling Left (negative x direction):

y = A sin(kx + ωt + φ)

or: y = A sin(ωt + kx + φ)

Note: Plus sign indicates leftward propagation

4. Complex Exponential Form:

y = A e^{i(kx - ωt + φ)}

(Used in advanced physics, take real part)

Understanding the Wave Equation Terms

What Does Each Term Mean?

kx term (Spatial Variation):

Describes how wave pattern varies in space
k = 2π/λ determines spatial frequency
Larger k → shorter wavelength → more oscillations per meter

ωt term (Temporal Variation):

Describes how wave oscillates in time at any point
ω = 2πf determines temporal frequency
Larger ω → faster oscillation

(kx - ωt) term (Wave Propagation):

Minus sign indicates wave traveling in +x direction
For constant phase: kx - ωt = constant
Differentiating: k dx - ω dt = 0
Therefore: dx/dt = ω/k = v (wave velocity)

φ term (Initial Phase):

Determines starting condition at t = 0, x = 0
φ = 0: wave starts from equilibrium
φ = π/2: wave starts from maximum (equivalent to cosine)
φ = π: wave starts from equilibrium but opposite direction

Wave Equation at Specific Conditions

SPECIAL CASES:

1. At a fixed position (x = constant, say x = 0):
y(0,t) = A sin(-ωt + φ) = A sin(φ - ωt)
This is SHM with time period T = 2π/ω

2. At a fixed time (t = constant, say t = 0):
y(x,0) = A sin(kx + φ)
This shows spatial wave pattern (snapshot)

3. Wave traveling right with no initial phase:
y = A sin(kx - ωt)

4. Wave traveling left with no initial phase:
y = A sin(kx + ωt)

5. At equilibrium positions (y = 0):
kx - ωt + φ = nπ, where n = 0, ±1, ±2, ...

6. At maximum displacement (y = ±A):
kx - ωt + φ = (2n+1)π/2, where n = 0, ±1, ±2, ...

Phase and Phase Difference

Understanding Phase:

Phase of Wave:

Phase Φ = kx - ωt + φ

The argument of sine/cosine function

Phase Difference between Two Points:

For two points x₁ and x₂ at same time t:

ΔΦ = k(x₂ - x₁) = (2π/λ)(x₂ - x₁)

Phase difference = (2π/λ) × Path difference

If path difference = λ → Phase difference = 2π (360°)
If path difference = λ/2 → Phase difference = π (180°)
If path difference = λ/4 → Phase difference = π/2 (90°)

Significance of Phase Difference:

ΔΦ = 0, 2π, 4π, ... (even multiples of π) → Points oscillate in phase
ΔΦ = π, 3π, 5π, ... (odd multiples of π) → Points oscillate in opposite phase
ΔΦ = π/2 → 90° out of phase (quarter wavelength apart)

Particle Velocity and Acceleration

Kinematics of Wave Particles:

Displacement:

y = A sin(kx - ωt)

Particle Velocity (∂y/∂t at fixed x):

v_particle = ∂y/∂t = -Aω cos(kx - ωt)

Maximum particle velocity = Aω

Particle Acceleration (∂²y/∂t² at fixed x):

a_particle = ∂²y/∂t² = -Aω² sin(kx - ωt)

a_particle = -ω²y (SHM relation)

Maximum particle acceleration = Aω²

Important Distinction:

Wave velocity (v): Speed of wave pattern propagation = ω/k
Particle velocity: Speed of individual particle oscillation = -Aω cos(kx - ωt)
These are NOT the same!

Progressive Wave: Displacement vs Position & Time

Important for NEB Exams:

Standard form: y = A sin(kx - ωt + φ) for wave traveling right
Wave traveling left: y = A sin(kx + ωt + φ) (note plus sign)
Wave number: k = 2π/λ (spatial frequency)
Angular frequency: ω = 2πf = 2π/T (temporal frequency)
Wave velocity: v = ω/k = fλ (from wave equation)
Phase difference: ΔΦ = (2π/λ) × path difference
Particle velocity: ∂y/∂t = -Aω cos(kx - ωt) ≠ wave velocity
Can use sine or cosine (differ by phase of π/2)

6.3 Conditions for Formation of Stationary Waves

A stationary wave (or standing wave) is formed by the superposition of two progressive waves of equal amplitude and frequency traveling in opposite directions through the same medium.

What is a Stationary Wave?

Stationary Wave Definition:

"A stationary wave is a wave pattern that remains fixed in space, formed by the interference of two identical progressive waves traveling in opposite directions."

Key Characteristics:

Wave pattern does not move (appears "stationary")
Has fixed points of zero displacement (nodes)
Has fixed points of maximum displacement (antinodes)
No net energy transfer (energy confined to regions)
Amplitude varies with position
All particles between two nodes oscillate in phase

Essential Conditions for Stationary Wave Formation

Necessary Conditions:

1. Two Waves Must Travel in Opposite Directions:

One wave traveling forward (+x direction)
Another wave traveling backward (-x direction)
Usually formed by reflection of original wave

2. Equal Amplitude:

Both waves must have same amplitude A
If amplitudes differ, standing wave pattern is imperfect
Results in some progressive wave component

3. Same Frequency (and wavelength):

Both waves must have identical frequency f
Therefore same wavelength λ (since v = fλ)
Same angular frequency ω = 2πf
Same wave number k = 2π/λ

4. Travel in Same Medium:

Both waves must propagate through same medium
Ensures same wave velocity v
Necessary for interference pattern to form

5. Same Type of Wave:

Both must be transverse OR both longitudinal
Cannot form from one transverse + one longitudinal

How Stationary Waves are Formed

Formation Mechanisms:

Method 1: Reflection from Fixed End

Progressive wave travels toward fixed boundary
Reflects back with 180° phase change (inverted)
Incident and reflected waves superpose
Example: String fixed at both ends, sound in closed tube

Method 2: Reflection from Free End

Progressive wave reflects without phase change
Reflected wave maintains same phase
Example: Open end of organ pipe

Method 3: Two Sources

Two identical sources at opposite ends
Generate waves traveling toward each other
Less common in practice

Formation of Stationary Wave by Superposition

Nodes and Antinodes

Key Features of Stationary Waves:

NODES (N):

Definition: Points of zero displacement at all times
Condition: sin(kx) = 0, so kx = nπ → x = nλ/2 (n = 0, 1, 2, ...)
Spacing: Distance between consecutive nodes = λ/2
Particle motion: Particles remain at rest
Energy: Minimum energy (no kinetic or potential energy)
Amplitude: Zero

ANTINODES (AN):

Definition: Points of maximum displacement
Condition: sin(kx) = ±1, so kx = (2n+1)π/2 → x = (2n+1)λ/4
Spacing: Distance between consecutive antinodes = λ/2
Particle motion: Maximum amplitude oscillation
Energy: Maximum energy exchange (KE ↔ PE)
Amplitude: 2A (twice the amplitude of component waves)

Node to Antinode Distance: λ/4

Differences: Progressive vs Stationary Waves

Property	Progressive Wave	Stationary Wave
Wave Pattern	Moves through medium	Fixed in space (appears stationary)
Energy Transfer	Energy propagates continuously	No net energy transfer (confined)
Amplitude	Same for all particles	Varies with position (0 at nodes, max at antinodes)
Nodes & Antinodes	No fixed nodes or antinodes	Fixed nodes and antinodes
Phase	Changes continuously with position	All particles between two nodes in same phase
Phase Change	2π over one wavelength	π (180°) across each node
Formation	Single wave propagating	Superposition of two identical waves in opposite directions
Wavelength	Distance between consecutive crests	Twice the distance between consecutive nodes (2×λ/2 = λ)
Example	Sound traveling through air, light from bulb	Vibrating string, organ pipes, resonance tubes

Important for NEB Exams:

Stationary wave: Formed by superposition of two identical waves traveling in opposite directions
Essential conditions: Same amplitude, same frequency, opposite directions, same medium
Nodes: Points of zero displacement (spacing = λ/2)
Antinodes: Points of maximum displacement (spacing = λ/2)
Node to antinode distance: λ/4
No energy propagation in stationary waves
Amplitude varies with position: 0 at nodes, 2A at antinodes
All particles between two nodes oscillate in phase
Phase change of π across each node

6.4 Mathematical Form of Stationary Wave

We can derive the mathematical equation for a stationary wave by superposing two progressive waves of equal amplitude and frequency traveling in opposite directions.

Deriving Stationary Wave Equation

Complete Derivation:

Given: Two progressive waves

Wave 1 (traveling right):

y₁ = A sin(kx - ωt)

Wave 2 (traveling left, same amplitude & frequency):

y₂ = A sin(kx + ωt)

Resultant displacement (Principle of Superposition):

y = y₁ + y₂

y = A sin(kx - ωt) + A sin(kx + ωt)

Using trigonometric identity:

sin C + sin D = 2 sin[(C+D)/2] cos[(C-D)/2]

Let C = (kx - ωt) and D = (kx + ωt)

C + D = kx - ωt + kx + ωt = 2kx

C - D = kx - ωt - kx - ωt = -2ωt

Therefore:

y = 2A sin(kx) cos(-ωt)

Since cos(-θ) = cos(θ):

y = 2A sin(kx) cos(ωt)

This is the standard equation for a stationary wave

Standard Forms of Stationary Wave Equation

Different Representations:

1. Using k and ω:

y(x,t) = 2A sin(kx) cos(ωt)

Where:
• k = 2π/λ (wave number)
• ω = 2πf = 2π/T (angular frequency)
• 2A = amplitude at antinodes

2. Using λ and T:

y(x,t) = 2A sin(2πx/λ) cos(2πt/T)

3. With phase constant:

y(x,t) = 2A sin(kx + φ) cos(ωt)

φ determines node/antinode positions

4. Alternative form (using sine for time):

y(x,t) = 2A sin(kx) sin(ωt)

(Differs by π/2 phase in time)

Understanding the Equation Components

Analyzing y = 2A sin(kx) cos(ωt):

Amplitude Factor: 2A sin(kx)

This term depends ONLY on position x
Determines amplitude at each point
Maximum value = 2A (at antinodes)
Minimum value = 0 (at nodes)
Shows wave pattern is "standing" (position-dependent amplitude)

Time Factor: cos(ωt)

This term depends ONLY on time t
ALL particles oscillate with this same time dependence
Period of oscillation = T = 2π/ω
Frequency = f = ω/(2π)
Shows all particles between nodes move in phase

Key Insight:

The equation factors into spatial and temporal parts, unlike progressive waves where position and time are coupled in (kx - ωt).

Finding Nodes and Antinodes

Mathematical Conditions:

For NODES (y = 0 for all t):

Amplitude factor must be zero:

2A sin(kx) = 0

sin(kx) = 0

kx = nπ, where n = 0, 1, 2, 3, ...

x = nπ/k = nπ/(2π/λ) = nλ/2

Position of n-th node:
x_n = nλ/2, where n = 0, 1, 2, 3, ...

First node (n=0): x = 0
Second node (n=1): x = λ/2
Third node (n=2): x = λ
etc.

Distance between consecutive nodes = λ/2

For ANTINODES (maximum amplitude = 2A):

sin(kx) = ±1 (maximum value)

kx = (2n + 1)π/2, where n = 0, 1, 2, 3, ...

x = (2n + 1)π/(2k) = (2n + 1)λ/4

Position of n-th antinode:
x_n = (2n + 1)λ/4, where n = 0, 1, 2, 3, ...

First antinode (n=0): x = λ/4
Second antinode (n=1): x = 3λ/4
Third antinode (n=2): x = 5λ/4
etc.

Distance between consecutive antinodes = λ/2

Amplitude Distribution

Amplitude Variation with Position:

From y = 2A sin(kx) cos(ωt), the amplitude at position x is:

Amplitude(x) = |2A sin(kx)| = |2A sin(2πx/λ)|

Position x	sin(kx)	Amplitude	Type
0	0	0	Node
λ/4	1	2A	Antinode
λ/2	0	0	Node
3λ/4	-1	2A	Antinode
λ	0	0	Node

Energy in Stationary Waves

Energy Distribution:

Particle Velocity:

v_particle = ∂y/∂t = -2Aω sin(kx) sin(ωt)

Key Points about Energy:

At Nodes: Particles don't move, zero kinetic and potential energy
At Antinodes: Maximum energy exchange between kinetic and potential
Energy confined: Doesn't propagate along the wave
Energy oscillates: Between kinetic and potential in each segment

When cos(ωt) = 1 (maximum displacement):

All energy is potential (elastic/gravitational)
Particle velocity = 0 everywhere

When cos(ωt) = 0 (zero displacement):

All energy is kinetic
Maximum particle velocity

Stationary Wave at Different Time Instants

Important for NEB Exams:

Stationary wave equation: y = 2A sin(kx) cos(ωt)
Derived by superposing y₁ = A sin(kx - ωt) and y₂ = A sin(kx + ωt)
Equation factors into spatial [2A sin(kx)] and temporal [cos(ωt)] parts
Nodes: x = nλ/2 (n = 0, 1, 2, ...), where sin(kx) = 0
Antinodes: x = (2n+1)λ/4, where sin(kx) = ±1
Amplitude at x: |2A sin(kx)| (varies from 0 to 2A)
Node-to-node distance = λ/2
Antinode-to-antinode distance = λ/2
Node-to-antinode distance = λ/4
All particles between two nodes oscillate in phase

6.5 Calculations: Frequency, Amplitude, Velocity, Time Period

This section covers practical problem-solving techniques for calculating various wave parameters in both progressive and stationary waves.

Problem 1: Basic Wave Calculations

Problem:

A progressive wave has wavelength 0.5 m and frequency 200 Hz. Calculate:

(a) Wave velocity

(b) Time period

(c) Angular frequency

(d) Wave number

Solution:

Given:

Wavelength: λ = 0.5 m
Frequency: f = 200 Hz

(a) Wave velocity:

v = fλ

v = 200 × 0.5

v = 100 m/s

(b) Time period:

T = 1/f

T = 1/200

T = 0.005 s = 5 ms

(c) Angular frequency:

ω = 2πf

ω = 2π × 200

ω = 400π

ω = 1257 rad/s

(d) Wave number:

k = 2π/λ

k = 2π/0.5

k = 4π

k = 12.57 rad/m

ANSWERS: (a) v = 100 m/s, (b) T = 5 ms, (c) ω = 1257 rad/s, (d) k = 12.57 rad/m

Problem 2: Wave Equation Given

Problem:

A wave is described by the equation y = 0.02 sin(4πx - 120πt), where x and y are in meters and t in seconds. Find:

(a) Amplitude

(b) Wavelength

(c) Frequency

(d) Wave velocity

(e) Direction of wave propagation

Solution:

Given equation: y = 0.02 sin(4πx - 120πt)

Comparing with standard form: y = A sin(kx - ωt)

(a) Amplitude:

A = 0.02 m = 2 cm

(b) Wavelength:

From equation: k = 4π

We know: k = 2π/λ

Therefore: 4π = 2π/λ

λ = 2π/(4π) = 1/2

λ = 0.5 m = 50 cm

(c) Frequency:

From equation: ω = 120π

We know: ω = 2πf

Therefore: 120π = 2πf

f = 120π/(2π) = 60

f = 60 Hz

(d) Wave velocity:

Method 1: v = fλ = 60 × 0.5 = 30 m/s

Method 2: v = ω/k = 120π/(4π) = 30 m/s ✓

(e) Direction:

Since equation is (kx - ωt) with minus sign,

Wave travels in positive x-direction (right) →

ANSWERS: (a) A = 2 cm, (b) λ = 50 cm, (c) f = 60 Hz, (d) v = 30 m/s, (e) +x direction

Problem 3: Phase Difference

Problem:

Two points on a progressive wave are separated by a distance of 15 cm. If the wavelength is 60 cm, calculate:

(a) Phase difference between the two points

(b) If one point is at maximum displacement, what is the displacement of the other point? (Given amplitude A = 5 cm)

Solution:

Given:

Path difference: Δx = 15 cm = 0.15 m
Wavelength: λ = 60 cm = 0.6 m
Amplitude: A = 5 cm = 0.05 m

(a) Phase difference:

ΔΦ = (2π/λ) × Δx

ΔΦ = (2π/0.6) × 0.15

ΔΦ = 2π × (0.15/0.6)

ΔΦ = 2π × (1/4)

ΔΦ = π/2 rad

ΔΦ = 90° or π/2 rad

(b) Displacement of second point:

If point 1 is at maximum: y₁ = A sin(φ₁) = A

This means: sin(φ₁) = 1, so φ₁ = π/2

Point 2 phase: φ₂ = φ₁ + ΔΦ = π/2 + π/2 = π

Displacement: y₂ = A sin(π) = A × 0 = 0

Alternative method:

Since phase difference = 90°, when one is at max, other is at equilibrium

ANSWERS: (a) ΔΦ = π/2 rad = 90°, (b) y₂ = 0 (at equilibrium)

Problem 4: Stationary Wave Calculations

Problem:

A stationary wave is formed in a string of length 120 cm fixed at both ends. The string vibrates in 3 segments (3 loops). If the frequency is 150 Hz, calculate:

(a) Wavelength of the wave

(b) Wave velocity

(c) Positions of nodes from one end

Solution:

Given:

Length: L = 120 cm = 1.2 m
Number of segments (loops): n = 3
Frequency: f = 150 Hz

(a) Wavelength:

For a string fixed at both ends with n loops:

L = nλ/2

λ = 2L/n

λ = (2 × 1.2)/3

λ = 2.4/3

λ = 0.8 m = 80 cm

(b) Wave velocity:

v = fλ

v = 150 × 0.8

v = 120 m/s

(c) Positions of nodes:

Node positions: x = nλ/2, where n = 0, 1, 2, 3, ...

With λ = 80 cm:

Node 0 (n=0): x = 0 cm (fixed end)
Node 1 (n=1): x = 40 cm
Node 2 (n=2): x = 80 cm
Node 3 (n=3): x = 120 cm (fixed end)

Nodes at: 0, 40, 80, 120 cm from one end

ANSWERS: (a) λ = 80 cm, (b) v = 120 m/s, (c) Nodes at 0, 40, 80, 120 cm

Problem 5: Stationary Wave Equation

Problem:

A stationary wave is given by y = 0.04 sin(5πx) cos(200πt), where x and y are in meters and t in seconds. Calculate:

(a) Amplitude of component waves

(b) Amplitude at antinode

(c) Wavelength

(d) Frequency

(e) Position of first three nodes

Solution:

Given: y = 0.04 sin(5πx) cos(200πt)

Standard form: y = 2A sin(kx) cos(ωt)

(a) Amplitude of component waves:

2A = 0.04

A = 0.04/2

A = 0.02 m = 2 cm

(b) Amplitude at antinode:

Maximum amplitude = 2A = 0.04 m = 4 cm

(c) Wavelength:

k = 5π

k = 2π/λ

5π = 2π/λ

λ = 2π/(5π) = 2/5

λ = 0.4 m = 40 cm

(d) Frequency:

ω = 200π

ω = 2πf

200π = 2πf

f = 200π/(2π) = 100

f = 100 Hz

(e) Position of first three nodes:

Nodes occur where sin(kx) = 0

kx = nπ, so x = nπ/k = nπ/(5π) = n/5

Node 0 (n=0): x = 0 m
Node 1 (n=1): x = 0.2 m = 20 cm
Node 2 (n=2): x = 0.4 m = 40 cm

First three nodes: 0, 20, 40 cm

ANSWERS: (a) A = 2 cm, (b) Max amplitude = 4 cm, (c) λ = 40 cm, (d) f = 100 Hz, (e) Nodes at 0, 20, 40 cm

Problem 6: Particle Velocity

Problem:

A progressive wave is given by y = 0.05 sin(10x - 300t) where x and y are in meters and t in seconds. Find:

(a) Maximum particle velocity

(b) Particle velocity at x = 0.1 m and t = 0.01 s

(c) Compare with wave velocity

Solution:

Given: y = 0.05 sin(10x - 300t)

A = 0.05 m, k = 10 rad/m, ω = 300 rad/s

(a) Maximum particle velocity:

Particle velocity: v_p = ∂y/∂t = -Aω cos(kx - ωt)

Maximum when cos(kx - ωt) = ±1:

v_p,max = Aω

v_p,max = 0.05 × 300

v_p,max = 15 m/s

(b) Particle velocity at x = 0.1 m, t = 0.01 s:

v_p = -0.05 × 300 × cos(10×0.1 - 300×0.01)

v_p = -15 × cos(1 - 3)

v_p = -15 × cos(-2)

v_p = -15 × 0.416

v_p ≈ -6.24 m/s

(c) Wave velocity:

v_wave = ω/k = 300/10 = 30 m/s

Comparison:

Wave velocity = 30 m/s (constant, speed of pattern)
Max particle velocity = 15 m/s (< wave velocity)
These are different physical quantities!

ANSWERS: (a) v_p,max = 15 m/s, (b) v_p ≈ -6.24 m/s, (c) v_wave = 30 m/s

Key Formulas Summary:

Progressive Waves:

v = fλ = λ/T = ω/k
f = 1/T, ω = 2πf, k = 2π/λ
Phase difference: ΔΦ = (2π/λ)Δx
Max particle velocity = Aω

Stationary Waves:

y = 2A sin(kx) cos(ωt)
Nodes: x = nλ/2
Antinodes: x = (2n+1)λ/4
For string with n loops: L = nλ/2, λ = 2L/n

Common Mistakes to Avoid:

Confusing wave velocity with particle velocity
Forgetting to convert to SI units
Mixing up node/antinode formulas
Wrong sign for wave direction ((kx - ωt) vs (kx + ωt))

Important for NEB Exams:

Always identify: A, λ, f, k, ω from given equation first
v = fλ is the most important formula (memorize!)
Phase difference = (2π/λ) × path difference
For stationary waves: compare with y = 2A sin(kx) cos(ωt)
Component wave amplitude A = (antinode amplitude)/2
Nodes at x = nλ/2, antinodes at x = (2n+1)λ/4
Wave velocity ≠ particle velocity (different concepts)