Waves and Optics

📖 Learn

Waves are disturbances that transfer energy through matter or space without transferring matter. Optics is the study of light behavior, including reflection, refraction, and the properties of lenses and mirrors.

Wave Properties

Property	Symbol	Unit	Definition
Wavelength	λ (lambda)	meters (m)	Distance between consecutive identical points (crest to crest)
Frequency	f	Hertz (Hz)	Number of complete waves passing a point per second
Period	T	seconds (s)	Time for one complete wave cycle (T = 1/f)
Amplitude	A	meters (m)	Maximum displacement from equilibrium
Wave Speed	v	m/s	Speed at which wave energy travels (v = fλ)

The Wave Equation

v = fλ

Wave speed equals frequency times wavelength. This fundamental equation connects all wave properties.

Types of Waves

Type	Description	Examples
Transverse	Particles vibrate perpendicular to wave direction	Light, water surface waves, waves on a string
Longitudinal	Particles vibrate parallel to wave direction	Sound waves, compression waves in a spring
Mechanical	Require a medium to travel through	Sound, water waves, seismic waves
Electromagnetic	Do not require a medium; travel through vacuum	Light, radio waves, X-rays

The Electromagnetic Spectrum

All electromagnetic waves travel at the speed of light (c ≈ 3 × 10⁸ m/s) in a vacuum.

Type	Wavelength Range	Frequency Range
Radio waves	> 1 m	< 10⁸ Hz
Microwaves	1 mm - 1 m	10⁸ - 10¹¹ Hz
Infrared	700 nm - 1 mm	10¹¹ - 10¹⁴ Hz
Visible light	400 - 700 nm	~10¹⁴ Hz
Ultraviolet	10 - 400 nm	10¹⁵ - 10¹⁶ Hz
X-rays	0.01 - 10 nm	10¹⁶ - 10¹⁹ Hz
Gamma rays	< 0.01 nm	> 10¹⁹ Hz

Reflection

Law of Reflection: The angle of incidence equals the angle of reflection.

θᵢ = θᵣ (angles measured from the normal)

Refraction

Refraction is the bending of light when it passes from one medium to another due to a change in speed.

Snell's Law: n₁ sin(θ₁) = n₂ sin(θ₂)

Where n = index of refraction (n = c/v)

Key refraction concepts:

Light bends toward the normal when entering a denser medium (n increases)
Light bends away from the normal when entering a less dense medium (n decreases)
Critical angle: The angle above which total internal reflection occurs
Total internal reflection: All light reflects when going from denser to less dense medium at angles above critical angle

Lenses

Lens Type	Shape	Effect on Light	Image Properties
Convex (Converging)	Thicker in middle	Converges parallel rays to focal point	Can be real/inverted or virtual/upright depending on object distance
Concave (Diverging)	Thinner in middle	Diverges parallel rays	Always virtual, upright, smaller

Lens Equation

1/f = 1/dₒ + 1/dᵢ

Where: f = focal length, dₒ = object distance, dᵢ = image distance

Magnification: M = -dᵢ/dₒ = hᵢ/hₒ

Positive M = upright image; Negative M = inverted image

💡 Examples

Work through these wave and optics problems.

Example 1: Wave Equation

Problem: A wave has a frequency of 500 Hz and a wavelength of 0.68 m. What is the wave speed?

Solution

Given: f = 500 Hz, λ = 0.68 m

Using v = fλ:

v = (500)(0.68) = 340 m/s

(This is approximately the speed of sound in air at room temperature)

Example 2: Electromagnetic Waves

Problem: Red light has a wavelength of 700 nm. What is its frequency? (c = 3 × 10⁸ m/s)

Solution

Given: λ = 700 nm = 700 × 10⁻⁹ m = 7 × 10⁻⁷ m

Using v = fλ → f = v/λ:

f = (3 × 10⁸)/(7 × 10⁻⁷) = 4.3 × 10¹⁴ Hz

Example 3: Snell's Law

Problem: Light travels from air (n = 1.0) into water (n = 1.33) at an angle of 45° from the normal. What is the angle of refraction?

Solution

Given: n₁ = 1.0, n₂ = 1.33, θ₁ = 45°

Using Snell's Law: n₁ sin(θ₁) = n₂ sin(θ₂)

(1.0) sin(45°) = (1.33) sin(θ₂)

0.707 = 1.33 sin(θ₂)

sin(θ₂) = 0.707/1.33 = 0.532

θ₂ = sin⁻¹(0.532) = 32.1°

(Light bends toward the normal when entering denser medium)

Example 4: Lens Equation

Problem: An object is placed 30 cm from a convex lens with focal length 10 cm. Where is the image? Is it real or virtual?

Solution

Given: dₒ = 30 cm, f = 10 cm

Using 1/f = 1/dₒ + 1/dᵢ:

1/10 = 1/30 + 1/dᵢ

1/dᵢ = 1/10 - 1/30 = 3/30 - 1/30 = 2/30 = 1/15

dᵢ = 15 cm

Positive dᵢ means the image is real (on opposite side from object)

M = -dᵢ/dₒ = -15/30 = -0.5 (inverted, half-size)

Example 5: Critical Angle

Problem: What is the critical angle for light traveling from glass (n = 1.5) to air (n = 1.0)?

Solution

At critical angle, θ₂ = 90°:

n₁ sin(θc) = n₂ sin(90°)

1.5 sin(θc) = 1.0 × 1

sin(θc) = 1/1.5 = 0.667

θc = sin⁻¹(0.667) = 41.8°

Light hitting the glass-air boundary at angles > 41.8° will totally internally reflect.

✏️ Practice

Solve these wave and optics problems. Use c = 3 × 10⁸ m/s for light speed.

1. A wave has a period of 0.02 seconds. What is its frequency?

2. Sound travels at 340 m/s in air. If a sound wave has a frequency of 440 Hz (middle A), what is its wavelength?

3. Blue light has a wavelength of 450 nm. What is its frequency?

4. Light hits a mirror at 35° from the normal. At what angle does it reflect?

5. Light travels from air (n = 1.0) into glass (n = 1.5) at 60° from the normal. Find the angle of refraction.

6. An object is placed 20 cm from a convex lens with focal length 15 cm. Find the image distance.

7. For the lens in problem 6, calculate the magnification. Is the image upright or inverted?

8. A concave lens has focal length -20 cm. An object is placed 30 cm from the lens. Where is the image?

9. What is the critical angle for diamond (n = 2.42) to air (n = 1.0)?

10. A radio station broadcasts at 100 MHz (100 × 10⁶ Hz). What is the wavelength of the radio waves?

Answer Key

f = 1/T = 1/0.02 = 50 Hz
λ = v/f = 340/440 = 0.77 m
f = c/λ = (3 × 10⁸)/(450 × 10⁻⁹) = 6.67 × 10¹⁴ Hz
By the law of reflection: 35° (angle of incidence = angle of reflection)
sin(θ₂) = (1.0/1.5)sin(60°) = 0.577; θ₂ = 35.3°
1/dᵢ = 1/15 - 1/20 = 4/60 - 3/60 = 1/60; dᵢ = 60 cm
M = -60/20 = -3 (inverted, magnified 3×)
1/dᵢ = 1/(-20) - 1/30 = -3/60 - 2/60 = -5/60; dᵢ = -12 cm (virtual image, same side as object)
sin(θc) = 1/2.42 = 0.413; θc = 24.4°
λ = c/f = (3 × 10⁸)/(100 × 10⁶) = 3 m

✅ Check Your Understanding

1. Why do all electromagnetic waves travel at the same speed in a vacuum but different speeds in materials?

Show Answer

In a vacuum, there's nothing to interact with, so all EM waves travel at c (the speed of light). In materials, the electric field of the wave interacts with electrons in the material, causing a slight delay in propagation. Different frequencies interact differently with these electrons, which is why different colors of light travel at slightly different speeds in glass (causing dispersion/rainbows through prisms).

2. Why does light bend when it enters a new medium?

Show Answer

Light bends due to a change in speed. When light enters a denser medium at an angle, the part of the wave that enters first slows down while the rest is still traveling at the original speed. This causes the wavefront to pivot, changing the direction of travel. It's like a marching band turning a corner—the inside members take smaller steps (slower) while outside members maintain normal steps.

3. Why can't total internal reflection occur when light travels from a less dense to a denser medium?

Show Answer

Total internal reflection requires the refracted ray to bend away from the normal (increasing the angle). This only happens when light goes from a denser to a less dense medium (higher n to lower n). When going from less dense to denser, light bends toward the normal, so there's always a refracted ray—no angle exists where all light reflects.

4. A virtual image forms 10 cm from a lens on the same side as the object. Is the lens convex or concave?

Show Answer

The lens is concave (diverging). Concave lenses always produce virtual images on the same side as the object. While convex lenses can produce virtual images when the object is between the lens and focal point, the fact that the image is on the same side indicates diverging behavior, characteristic of concave lenses.

🚀 Next Steps

Review any concepts that felt challenging
Move on to the next lesson when ready
Return to practice problems periodically for review