Describe how waves can undergo: (a) reflection at a plane surface (b) refraction due to a change of speed (c) diffraction through a narrow gap

3 – Waves

3.1 What is a wave?

• Wave: a disturbance that transfers energy from one place to another without the permanent transport of matter.
• Wave‑front: an imaginary line (in 2‑D) or surface (in 3‑D) joining points that are in the same phase of vibration (e.g. all crests).
• Wavelength \( \lambda \): distance between two successive points that are in phase on a wave‑front.
• Frequency \( f \): number of wave‑fronts that pass a given point each second (unit Hz). Frequency does not change when a wave passes from one medium to another.
• Amplitude: maximum displacement of the medium from its rest position; determines the intensity of the wave.
• Speed \( v \): rate at which a wave‑front moves; related to \( f \) and \( \lambda \) by

  \[

  v = f\lambda

  \]

• Transverse vs Longitudinal

  • Transverse – particle motion ⟂ to direction of travel (e.g. light, water‑surface waves).
  • Longitudinal – particle motion ∥ to direction of travel (e.g. sound in air, compression waves in a spring).

3.2 Reflection at a Plane Surface

When a wave meets a smooth, flat (plane) boundary it is sent back into the medium from which it came.

• Law of reflection:

  \[

  \thetai = \thetar

  \]

  where \( \thetai \) is the angle of incidence and \( \thetar \) the angle of reflection, both measured from the normal.

• The incident ray, reflected ray and the normal lie in the same plane.
• Wave characteristics after reflection (provided the medium on the incident side does not change):

  • Speed \( v \) – unchanged
  • Frequency \( f \) – unchanged
  • Wavelength \( \lambda \) – unchanged
  • Amplitude – unchanged (ignoring absorption)

• For electromagnetic waves incident on a good conductor (e.g. metal mirror) the reflected wave undergoes a phase reversal of \(180^{\circ}\) (sign change).

Critical angle and total internal reflection (TIR)

• Occurs when a wave travels from a medium of higher speed (or higher refractive index) to one of lower speed (lower index).
• Critical angle:

  \[

  \thetac = \sin^{-1}\!\left(\frac{v2}{v1}\right)=\sin^{-1}\!\left(\frac{n2}{n1}\right)\qquad (n1>n_2)

  \]

• If the angle of incidence \( \thetai > \thetac \) the wave is completely reflected – no refracted ray is produced.
• Everyday example: light guided down an optical fibre; the light repeatedly undergoes TIR at the core–cladding interface.

3.3 Refraction – Change of Speed

When a wave passes from one medium into another where its speed is different, the direction of the wave‑front changes. This bending is called refraction.

Snell’s law

• Standard form (using refractive indices):

  \[

  n1\sin\theta1 = n2\sin\theta2

  \]

• Speed form (useful for sound or other non‑optical waves):

  \[

  \frac{\sin\theta1}{\sin\theta2}= \frac{v1}{v2}

  \]

Refractive index

• Definition:

  \[

  n = \frac{v{\text{reference}}}{v{\text{medium}}}

  \]

  For light the reference speed is the speed in vacuum, \(c\); thus \( n = c/v \).

• Because \( v = f\lambda \) and the frequency \( f \) remains constant at a boundary, the wavelength changes:

  \[

  \lambda2 = \lambda1\frac{v2}{v1}= \lambda1\frac{n1}{n_2}

  \]

Direction of bending

• If the wave enters a slower medium (higher \( n \)) it bends towards the normal.
• If the wave enters a faster medium (lower \( n \)) it bends away from the normal.

Critical angle revisited

• From Snell’s law, when a wave moves from a higher‑\( n \) medium to a lower‑\( n \) medium, the denominator can become larger than 1, making \(\sin\theta_2 > 1\). The limiting case defines the critical angle:

  \[

  \sin\thetac = \frac{n2}{n1}\qquad (n1>n_2)

  \]

• For \( \thetai > \thetac \) the wave is totally internally reflected (see Section 3.2).

Examples

• Light entering water from air – bends towards the normal ( \( n_{\text{water}}\approx1.33 \) ).
• Sound entering a denser gas (e.g. from air into carbon‑dioxide) – speed decreases, wave bends towards the normal.
• Optical fibre: core \( n\approx1.48 \), cladding \( n\approx1.46 \); light launched at an angle \(>\theta_c\) is trapped by repeated TIR.

3.4 Diffraction – Spreading Through a Narrow Gap

Diffraction is the bending and spreading of a wave when it encounters an obstacle or aperture whose dimensions are comparable to the wavelength.

Single‑slit (or narrow gap) condition

• Noticeable diffraction when the slit width \( a \) satisfies

  \[

  a \lesssim \lambda

  \]

• If \( a \gg \lambda \) the wave emerges almost unchanged; if \( a \le \lambda \) the emerging wave spreads into a fan‑shaped pattern.
• Angular width of the central maximum (small‑angle approximation):

  \[

  \theta \approx \frac{\lambda}{a}

  \]

Diffraction at a sharp edge

• A sharp edge can be treated as a line of point sources (Huygens’ principle). The wave bends around the edge, producing a bright region just beyond the edge and a series of diminishing fringes.

Double‑slit interference (combined diffraction + interference)

• Each slit diffracts the incident wave; the two diffracted wave‑fronts then interfere.
• Condition for bright (constructive) fringes:

  \[

  d\sin\theta = m\lambda \qquad (m = 0, \pm1, \pm2,\dots)

  \]

  where \( d \) is the centre‑to‑centre separation of the slits.

• Dark (destructive) fringes occur when the path‑difference equals \((m+\tfrac12)\lambda\).

Examples

• Sound through a doorway – even when the source is not directly in line of sight, the sound is heard on the other side because the doorway width is comparable to the wavelength of audible sound (≈0.5 m for 680 Hz).
• Radio waves (λ ≈ 1 m) bending around buildings and hills.
• Laser light (λ ≈ 600 nm) passing through a slit of width ≈ 0.5 µm produces a visible diffraction pattern on a screen.

3.5 Summary Table

3.6 Key Equations to Remember

• Wave speed: \( v = f\lambda \)
• Law of reflection: \( \thetai = \thetar \)
• Snell’s law (refractive‑index form): \( n1\sin\theta1 = n2\sin\theta2 \)
• Refractive index: \( n = \dfrac{c}{v} = \dfrac{\sin\theta1}{\sin\theta2} \)
• Critical angle: \( \thetac = \sin^{-1}\!\left(\dfrac{n2}{n1}\right) \) (\( n1>n_2 \))
• Diffraction (single slit, small‑angle): \( \theta \approx \dfrac{\lambda}{a} \)
• Double‑slit constructive interference: \( d\sin\theta = m\lambda \)

Mastering reflection, refraction (including critical angle and total internal reflection) and diffraction gives a solid foundation for many modern technologies – mirrors, lenses, fibre‑optic communication, ultrasound imaging, and radio broadcasting.