27.2 Huygens's Principle: Diffraction

Waves are fascinating phenomena that transfer energy through matter. From ripples on a pond to light bending around corners, wave propagation and Huygens's principle explain how waves move and interact with their environment.

Understanding wave behavior is crucial for grasping many physical phenomena. We'll explore transverse waves, Huygens's principle, and diffraction, uncovering how these concepts apply to everyday experiences like guitar strings and rainbows.

Wave Propagation and Huygens's Principle

Propagation of transverse waves

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• Transverse waves oscillate perpendicular to the direction of wave propagation
  • Light waves, electromagnetic waves, and waves on a string (guitar strings, jump ropes)
• Energy transfers through the medium without transferring matter
  • Particles in the medium oscillate about their equilibrium position (water molecules in ocean waves)
  • Adjacent particles interact, causing the wave to propagate (dominoes falling)
• Wave speed depends on the properties of the medium
  • Determined by the medium's elasticity and inertia (sound waves in air vs. water)
  • For a stretched string, wave speed is given by $v = \sqrt{\frac{T}{\mu}}$
    • $T$ is tension (tightness of the string)
    • $\mu$ is linear mass density (mass per unit length)

Huygens's principle in wave behavior

• Every point on a wavefront acts as a source of secondary wavelets (ripples from a pebble dropped in a pond)
  • Secondary wavelets spread out in all directions with the same speed as the original wave
  • The new wavefront is the envelope of these secondary wavelets (outer edge of the ripples)
• Huygens's principle explains various wave behaviors
  • Reflection: Secondary wavelets from the boundary points create the reflected wavefront (light reflecting off a mirror)
  • Refraction: Change in wave speed causes the wavefront to change direction (light bending through a prism)
  • Diffraction: Secondary wavelets from the edges create a new wavefront that bends around the obstacle or spreads out from the aperture (sound waves bending around a corner)
• The Huygens-Fresnel principle extends Huygens's principle by incorporating the amplitude and phase of the secondary wavelets

Diffraction of light around obstacles

• Diffraction occurs when a wave encounters an obstacle or aperture comparable in size to its wavelength
  • Light has a very small wavelength, so diffraction effects are more noticeable with small obstacles or apertures (a single strand of hair)
• Single-slit diffraction: Light passes through a narrow slit
  1. The slit acts as a source of secondary wavelets, which interfere constructively and destructively
  2. The diffraction pattern consists of a central bright fringe and alternating dark and bright fringes on either side
  3. The angular width of the central bright fringe is given by $\theta = \frac{\lambda}{a}$
     • $\lambda$ is the wavelength
     • $a$ is the slit width
• Double-slit diffraction: Light passes through two narrow slits
  1. The slits act as coherent sources of secondary wavelets, which interfere constructively and destructively
  2. The diffraction pattern consists of a series of bright fringes separated by dark fringes
  3. The angular separation between bright fringes is given by $\theta = \frac{\lambda}{d}$
     • $d$ is the slit separation
• Diffraction gratings: Many equally spaced slits
  • Produce a series of sharp, intense bright fringes called spectral orders (rainbow effect from a CD)
  • The angular position of the $m$-th order bright fringe is given by $\sin \theta = \frac{m \lambda}{d}$
    • $m$ is an integer (1, 2, 3, etc.)

Types of Diffraction

• Fresnel diffraction: Occurs when the light source or observation point is close to the diffracting object
• Fraunhofer diffraction: Occurs when the light source and observation point are far from the diffracting object
• Wave optics, which includes diffraction, helps explain the wave-particle duality of light