Refraction

🎯 Learning Objectives

By the end of this chapter, you should be able to:

Describe the physical mechanism of refraction and its relation to changes in acoustic impedance and wave speed.
Derive Snell’s law for acoustic waves at plane and curved interfaces.
Analyze the effects of refraction at boundaries between different media, including air, water, and solid materials.
Calculate transmitted and reflected wave amplitudes using impedance matching principles.
Understand angle-dependent refraction and critical angles, including total internal reflection scenarios.
Model sound propagation through stratified media, gradients, and layered materials.
Apply refraction concepts to architectural acoustics, underwater acoustics, and urban noise propagation.
Integrate theoretical formulations with interactive simulations to visualize wave bending and energy transmission.

Introduction:

Refraction is the bending of a wave as it passes from one medium to another in which the wave speed changes. Although commonly demonstrated with light, refraction occurs for all wave types: sound, water, and seismic waves.

Refraction fundamentally arises from one principle:

A wave changes direction when its speed changes across a boundary.

Why Refraction Happens:

When a wavefront crosses into a new medium at an angle, one part of the wavefront slows down or speeds up first. This causes the wavefront to rotate.
This directional change is refraction.

Snell’s Law:

The relationship between incident and refracted angles is given by Snell’s Law:

n_1 \sin\theta_1 = n_2 \sin\theta_2

Where:

$n_1, n_2$ are refractive indices
$\theta_1, \theta_2$ are angles to the normal

The refractive index is:

n = \frac{c}{v}

where $c$ is the speed of light in vacuum and
$v$ is the wave speed in the medium.

Python Example: Snell's Law Visualization

Change θ1 and see how the refracted angle changes.
Change c1 and c2 to simulate air-water, air-glass, etc.

import numpy as np
import matplotlib.pyplot as plt

# Parameters
theta1_deg = 10          # incidence angle in degrees
c1 = 340.0               # speed of sound in medium 1 [m/s]
c2 = 1500.0              # speed of sound in medium 2 [m/s]
# Convert to radians
theta1 = np.radians(theta1_deg)

# Geometry
src = np.array([0, 0])          # source position
boundary_x = 5.0                # x-position of plane boundary
ray_length = 8.0
# Incident ray: from source to boundary
x_incident = [src[0], boundary_x]
y_incident = [src[1], src[1] + np.tan(theta1) * (boundary_x - src[0])]
# Compute refraction using Snell's law
sin_theta2 = (c2 / c1) * np.sin(theta1)
# Handle total internal reflection case
if sin_theta2 > 1.0:
  theta2 = np.pi/2  # 90 degrees
else:
  theta2 = np.arcsin(sin_theta2)
# Refracted ray: from boundary onward
x_refracted = [boundary_x, boundary_x + ray_length*np.cos(theta2)]
y_refracted = [y_incident[1], y_incident[1] + ray_length*np.sin(theta2)]

# Plotting
plt.figure(figsize=(8,5))
plt.plot(x_incident, y_incident, 'b-', linewidth=2, label="Incident ray")
plt.plot(x_refracted, y_refracted, 'r-', linewidth=2, label="Refracted ray")
# Plot boundary
plt.axvline(x=boundary_x, color='k', linestyle='--', linewidth=2, label="Boundary")
# Labels
plt.xlabel("X [m]")
plt.ylabel("Y [m]")
plt.title(f"Refraction of Sound: θ1={theta1_deg}°, c1={c1}, c2={c2}")
plt.legend()
plt.grid(True)
plt.axis('equal')
plt.show()

Press Run Code: Output will appear here.

Python Example: Frequency Effect on Wavelength in Medium

How frequency and medium speed affect the wavelength, which is key in refraction phenomena

Press Run Code: Output will appear here.

Python Example: Ray Bending Across Layered Media

Shows ray bending at an interface for layered media, helps visualize sound path changes in multi-layer environments

import numpy as np
import matplotlib.pyplot as plt

# Parameters
layers = [
  {"c": 340.0, "thickness": 2.0},   # Layer 1: Air, speed 340 m/s
  {"c": 1500.0, "thickness": 2.0},  # Layer 2: Water, speed 1500 m/s
  {"c": 2000.0, "thickness": 2.0}   # Layer 3: Denser medium, speed 2000 m/s
]

incident_angle_deg = 5  # Angle in degrees relative to normal
incident_angle = np.radians(incident_angle_deg)

# Compute ray path
x_points = [0]
y_points = [0]
theta = incident_angle
x = 0
y = 0
for layer in layers:
  thickness = layer["thickness"]
  c = layer["c"]

# horizontal displacement using tan(theta)
  dx = thickness * np.tan(theta)
  x += dx
  y += thickness
  x_points.append(x)
  y_points.append(y)

# Snell's law: n1*sin(theta1) = n2*sin(theta2), using c1*sin(theta1) = c2*sin(theta2)
  if layer != layers[-1]:
      c_next = layers[layers.index(layer)+1]["c"]
      theta = np.arcsin((c_next/c)*np.sin(theta))

# Plotting
plt.figure(figsize=(8,5))
plt.plot(x_points, y_points, 'o-', color='red', linewidth=2, markersize=6)
plt.xlabel("Horizontal distance [m]")
plt.ylabel("Depth / Layer thickness [m]")
plt.title(f"Ray Bending Across Layered Media (Incident angle = {incident_angle_deg}°)")
plt.grid(True)
# Draw layer boundaries
y_boundary = 0
for layer in layers:
  y_boundary += layer["thickness"]
  plt.axhline(y=y_boundary, color='blue', linestyle='--', linewidth=1)

plt.show()

Press Run Code: Output will appear here.

Wavefront Explanation:

Case 1 - Wave slows down ( $v_2 < v_1$ )

Bends toward the normal
$\theta_2 < \theta_1$

Case 2 - Wave speeds up ( $v_2 > v_1$ )

Bends away from the normal
$\theta_2 > \theta_1$

Interactive Refraction Demonstration

Index of Refraction:

Material	n
Air	1.0003
Water	1.33
Glass	1.5–1.9
Diamond	2.42

Consequences of Refraction:

Apparent Depth

Objects underwater appear closer to the surface because rays bend away from the normal as they leave water.

d_{app} = \frac{d_{real}}{n}

Dispersion

Refractive index depends slightly on wavelength.
From prism spectra to rainbows, dispersion separates colors because shorter wavelengths slow down more.

Total Internal Reflection (TIR)

Occurs when a ray travels from a medium with higher refractive index to lower refractive index:

\sin\theta_c = \frac{n_2}{n_1}

If $\theta > \theta_c$ , all light is reflected.

Applications:

Fiber optic cables
Endoscopes
Mirage formation

Refraction of Sound:

Sound refracts toward the region where its speed is lower, which depends on temperature.

Examples:

At night, cool air near ground → sound bends downward
During daytime, warm ground layer → sound bends upward

This explains why traffic noise carries farther at night.

Refraction of Water Waves:

Water wave speed depends on depth.
Shallow regions slow down the wave, causing refraction.

Effects

Wave focusing
Wave wrapping around obstacles
Coastal refraction patterns

Fermat’s Principle:

Refraction can be derived from:

Light follows the path of least time

The varying speeds in each medium lead directly to Snell’s law.

Lenses and Image Formation

Converging (Convex) Lens

Focuses rays
Forms real or virtual images

Diverging (Concave) Lens

Spreads rays
Always forms virtual images

Gradient-Index (GRIN) Refraction:

A continuous gradient $n(x)$ causes curved ray paths.

Examples

Mirage over hot surfaces
Graded-index optical fibers
Atmospheric bending of starlight

Example Problems

Problem 1: Basic Refraction

A beam enters water from air at $30^\circ$ .
Find the refracted angle.

1.00\sin 30^\circ = 1.33\sin\theta_2

\theta_2 = 22^\circ

Problem 2: Critical Angle for Glass–Air

\sin\theta_c = \frac{1.0}{1.5} = 0.666

\theta_c = 41.8^\circ

Problem 3: Apparent Depth

A coin lies at 1 m depth in water.

d_{app} = \frac{1}{1.33} = 0.75\text{ m}

Applications:

Cameras
Eyeglasses
Microscopes
Telescopes
Fiber-optic communication
Underwater imaging
Mirage formation

📝 Key Takeaways

Refraction occurs when an acoustic wave passes from one medium to another with a different wave speed or acoustic impedance, causing the wave to bend.
Snell’s law governs the refraction angle: $\frac{\sin\theta_1}{c_1} = \frac{\sin\theta_2}{c_2}$ , where $c_1, c_2$ are wave speeds in the two media.
Acoustic impedance mismatch determines the proportion of reflected and transmitted energy at the interface.
Critical angle and total internal reflection occur when a wave moves from a slower medium to a faster medium and the incident angle exceeds a certain threshold.
Angle-dependent transmission affects sound propagation in layered media, underwater acoustics, and stratified atmospheres.
Refraction effects can produce focusing, shadow zones, and bending around obstacles, significantly affecting room acoustics and environmental noise propagation.
Practical modeling of refraction combines analytical formulas, impedance-based calculations, and numerical simulations for complex geometries.

🧠 Quick Quiz

1) Snell’s law for sound waves across two media with different wave speeds is expressed as:

A) $c_1 \sin\theta_1 = c_2 \sin\theta_2$
B) $\frac{\sin\theta_1}{c_1} = \frac{\sin\theta_2}{c_2}$
C) $c_1 \cos\theta_1 = c_2 \cos\theta_2$
D) $\frac{c_1}{\sin\theta_1} = \frac{c_2}{\sin\theta_2}$

2) Total internal reflection occurs when:

A) A wave passes from a faster to a slower medium at any angle
B) A wave passes from a slower to a faster medium and the incident angle exceeds the critical angle
C) Wave frequency is higher than the medium’s resonance frequency
D) The impedance is perfectly matched

3) The acoustic impedance mismatch affects:

A) The wavelength only
B) The proportion of reflected and transmitted energy
C) The speed of sound in the first medium
D) The direction of the incident wave only

4) Which effect can result from refraction in the atmosphere or ocean?

A) Shadow zones
B) Focusing of sound energy
C) Bending around obstacles
D) All of the above

5) In practical acoustics, refraction modeling requires:

A) Only analytical Snell’s law calculations
B) Only measuring wave speeds in air
C) A combination of analytical formulas, impedance calculations, and numerical simulations
D) Ignoring the medium properties entirely

Chapter - 11

Key Words: Refraction, Snell's law, Acoustic impedance, Sound propagation, Wave transmission

Refraction

🎯 Learning Objectives

Introduction:

Why Refraction Happens:

Snell’s Law:

Python Example: Snell's Law Visualization

Python Example: Frequency Effect on Wavelength in Medium

Python Example: Ray Bending Across Layered Media

Wavefront Explanation:

Case 1 - Wave slows down ( $v_2 < v_1$ )

Case 2 - Wave speeds up ( $v_2 > v_1$ )

Interactive Refraction Demonstration

Index of Refraction:

Consequences of Refraction:

Apparent Depth

Dispersion

Total Internal Reflection (TIR)

Applications:

Refraction of Sound:

Examples:

Refraction of Water Waves:

Effects

Fermat’s Principle:

Lenses and Image Formation

Converging (Convex) Lens

Diverging (Concave) Lens

Gradient-Index (GRIN) Refraction:

Examples

Example Problems

Problem 1: Basic Refraction

Problem 2: Critical Angle for Glass–Air

Problem 3: Apparent Depth

Applications:

📝 Key Takeaways

🧠 Quick Quiz

Chapter - 11

Key Words: Refraction, Snell's law, Acoustic impedance, Sound propagation, Wave transmission

Refraction

🎯 Learning Objectives

Introduction:

Why Refraction Happens:

Snell’s Law:

Python Example: Snell's Law Visualization

Python Example: Frequency Effect on Wavelength in Medium

Python Example: Ray Bending Across Layered Media

Wavefront Explanation:

Case 1 - Wave slows down (v2<v1v_2 < v_1v2​<v1​)

Case 2 - Wave speeds up (v2>v1v_2 > v_1v2​>v1​)

Interactive Refraction Demonstration

Index of Refraction:

Consequences of Refraction:

Apparent Depth

Dispersion

Total Internal Reflection (TIR)

Applications:

Refraction of Sound:

Examples:

Refraction of Water Waves:

Effects

Fermat’s Principle:

Lenses and Image Formation

Converging (Convex) Lens

Diverging (Concave) Lens

Gradient-Index (GRIN) Refraction:

Examples

Example Problems

Problem 1: Basic Refraction

Problem 2: Critical Angle for Glass–Air

Problem 3: Apparent Depth

Applications:

📝 Key Takeaways

🧠 Quick Quiz

Case 1 - Wave slows down ( $v_2 < v_1$ )

Case 2 - Wave speeds up ( $v_2 > v_1$ )