Reflection on the Basis of Wave Theory

1. Concept Overview (Beginner-Friendly)

When a wave (like light) strikes a smooth surface such as a mirror, it does not disappear. It bounces back. This bouncing back is called reflection.

Huygens’ Principle helps us understand why reflection happens and how the law of reflection

[\text{Angle of incidence}] [= \text{Angle of reflection}]

comes naturally from wave behavior.

In simple words:

• A wavefront is a surface on which all points vibrate in the same phase.
• When it hits a reflecting surface, every point on the surface becomes a source of secondary wavelets.
• The envelope (outer boundary) of these wavelets forms the reflected wavefront.
• Geometry of this construction forces the reflected wavefront to travel such that the laws of reflection are obeyed.

So reflection is not something mirrors “decide” — it’s a natural consequence of how waves behave.

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2. Explanation + Mathematical Derivation

Consider a plane wavefront incident on a reflecting surface AB.

Let:

• [ \theta_i ] = angle of incidence
• [ \theta_r ] = angle of reflection
• [v] = speed of light in the medium
• [t] = time for wavefront propagation

Derivation Using Huygens’ Principle

Let the incident wavefront reach point A on the surface first, and later reach point B.

During time [t]:

• Point A generates secondary wavelets, forming a sphere of radius ([vt]).
• Meanwhile, the incident wavefront travels from point B to C (distance ([BC = vt])).

Construct the reflected wavefront (A’D) as the envelope of these wavelets.

Using geometry:

[
\angle BAC = \theta_i
]

[
\angle D A’ C = \theta_r
]

In the triangles, we have:

[
\dfrac{BC}{AC} = \dfrac{vt}{AC}
]

[
\dfrac{A’D}{AC} = \dfrac{vt}{AC}
]

Thus,

[\sin \theta_i = \sin \theta_r] [\Rightarrow \theta_i = \theta_r]

So, the law of reflection emerges automatically from the wave nature of light.

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3. Dimensions & Units

• Angles are dimensionless.
• Distances: metres ([m])
• Time: seconds ([s])

Wave theory primarily uses geometric reasoning → no dimensional result needed.

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4. Key Features

1. Reflection is fully explained using Huygens’ wavelets.
2. The wavefronts before and after reflection remain coherent.
3. The angle of incidence equals the angle of reflection.
4. The speed and wavelength of light do not change during reflection.
5. Only the direction of propagation changes.

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5. Important Formulas (Table)

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6. Conceptual Questions with Solutions

1. Why does reflection occur according to wave theory?

Because every point on the reflecting surface becomes a source of secondary wavelets, and their envelope forms the reflected wavefront.

2. Does the wavelength of light change after reflection?

No. The speed of light remains the same in the same medium, so wavelength remains unchanged.

3. Why must the angle of incidence equal the angle of reflection?

The geometry of Huygens’ construction forces ([ \sin\theta_i = \sin\theta_r ]), so the angles must be equal.

4. What happens to the wavefront shape after reflection?

Plane wavefront remains plane; spherical remains spherical.

5. Does the frequency of light change after reflection?

No. Frequency is determined by the source.

6. Why does only the direction of light change?

The reflecting surface changes the propagation direction but not speed or wavelength.

7. Is reflection proof that light is a wave?

Reflection can be explained by both wave and ray models. But Huygens’ principle gives a deeper explanation.

8. What would happen if secondary wavelets were not spherical?

Wave theory would fail; Huygens assumed spherical wavelets for consistency with observed phenomena.

9. Do all surfaces reflect light?

Yes—smooth surfaces reflect regularly; rough surfaces diffuse it.

10. Why does reflection preserve phase?

All secondary wavelets originate from points in phase; hence the reflected wavefront is coherent.

11. What happens to amplitude during reflection?

It may reduce depending on the surface, but angle laws remain unchanged.

12. Can Huygens’ principle explain mirror images?

Yes—because it explains the precise direction of reflected rays/wavefronts.

13. Why is reflection reversible?

Because the laws of reflection are symmetric.

14. What happens to the energy of the wavefront?

Part is reflected, part may be absorbed; depends on the material.

15. Why does a plane mirror not distort images?

Because it preserves the linearity of plane wavefronts.

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7. FAQ / Common Misconceptions

1. Light reflects because it “hits” a surface.

Incorrect. Reflection arises from wavelet formation, not “bouncing” like a ball.

2. The mirror changes light’s speed.

False. Speed and wavelength do not change in the same medium.

3. Law of reflection is arbitrary.

No—it is a mathematical consequence of wavefront geometry.

4. Wave theory cannot explain reflection.

Huygens’ principle explains it perfectly.

5. Reflected light loses frequency.

Frequency always remains the same.

6. Reflection and absorption cannot coexist.

They can—some energy is absorbed, some reflected.

7. A rough surface reflects no light.

It reflects diffusely, not regularly.

8. Reflection requires a shiny surface.

Any surface can reflect; shininess only affects clarity.

9. The image is formed on the mirror.

No—image is formed by back-projected rays or wavefronts.

10. Light slows down at reflection.

No change in speed occurs.

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8. Practice Questions (With Step-By-Step Solutions)

1. A plane wavefront strikes a mirror at 30°. What is the angle of reflection?

[\theta_r = \theta_i] [= 30^\circ]

2. A wavefront takes 2×10⁻⁹ s to travel from B to C. Speed of light is [3\times10^8]. Find BC.

[BC = vt] [= 3\times10^8 \times 2\times10^{-9}] [= 0.6\ \text{m}]

3. Does speed change during reflection? Why?

No—because the medium remains the same.

4. A reflected wavefront makes 45° with the surface normal. What is the incidence angle?

[
\theta_i = \theta_r = 45^\circ
]

5. Prove [ \sin\theta_i = \sin\theta_r ] using Huygens’ construction.

From congruent geometric triangles:
[\dfrac{BC}{AC}] [= \dfrac{A’D}{AC}] [\Rightarrow \sin\theta_i = \sin\theta_r]