Wave patterns Knowledge-Based Study

Study notes

1. What Is a Wave Pattern?

   A wave pattern represents the spatial distribution of regular oscillations travelling through a medium at a given instant. In mechanical waves, as a disturbance propagates, particles of the medium oscillate around their equilibrium positions, transferring energy without the net transport of matter.

   Wavefronts and Rays

   To visualize and analyze wave patterns effectively, two geometric tools are commonly used:

   • Wavefront: A continuous line or surface joining points in the wave that share the identical phase of oscillation, such as adjacent crests or troughs.
   • Ray: A directed line perpendicular to the wavefront that indicates the direction of energy propagation.

   Depending on the geometry of the vibration source, common propagating patterns include:

   • Circular (or Spherical) Waves: Generated by point-source disturbances, resulting in concentric circular wavefronts expanding outward.
   • Plane (or Linear) Waves: Produced by extended linear sources, characterized by parallel, evenly spaced straight wavefronts moving in a uniform direction.

   Boundary Interactions

   When propagating wave patterns encounter a change in medium or an obstacle, their geometry changes according to fundamental physical laws.

   Reflection Patterns

   When a plane wave reflects from a flat rigid surface, the incident angle equals the reflected angle. However, curved barriers transform wave patterns:

   • A concave parabolic barrier reflects incident plane waves into converging circular waves focused at a focal point.
   • A point source placed at the focal point of a concave reflector produces outgoing parallel plane waves.

   Refraction Patterns

   When periodic waves cross into a medium with a different wave speed (such as water waves travelling from deep to shallow regions):

   • The frequency $f$ remains constant because it is determined solely by the source.
   • The wave speed $v$ changes, causing a proportional change in wavelength $\lambda$, according to the wave relationship $v = f\lambda$.
   • This compression or expansion of wavefront spacing bends non-normal wavefronts toward or away from the normal line.

   Two-Source Interference

   When two coherent wave sources vibrate in phase with the same frequency and amplitude, their waves overlap across the medium. According to the principle of superposition, the resultant displacement at any point is the algebraic sum of the individual displacements.

   Constructive vs. Destructive Interference

   The observable pattern depends on the path difference ($\Delta d = |d_1 - d_2|$) from the two sources to a given point:

   • Constructive Interference (Antinodes): Occurs where waves arrive in phase, resulting in maximum displacement amplitude. This requires an integer multiple of wavelengths:
     $\Delta d = n\lambda$ (where $n = 0, 1, 2, \dots$).
   • Destructive Interference (Nodes): Occurs where waves arrive completely out of phase ($180^\circ$ or $\pi\text{ rad}$), cancelling each other out to yield zero net displacement. This requires an odd half-integer multiple of wavelengths:
     $\Delta d = \left(n + \frac{1}{2}\right)\lambda$ (where $n = 0, 1, 2, \dots$).

   In two dimensions (such as a ripple tank), this forms a stable pattern of alternating nodal lines and antinodal lines radiating outward from the sources.

   Formation of Standing Waves

   When two identical periodic waves of the same frequency, wavelength, and amplitude travel in opposite directions along the same medium, they interfere to form a standing wave (stationary wave pattern). Unlike travelling waves, standing waves do not transport net energy along the medium.

   Key Structural Features

   • Nodes: Fixed spatial positions where destructive interference is permanent, resulting in zero amplitude at all times.
   • Antinodes: Positions midway between adjacent nodes where constructive interference creates oscillations of maximum amplitude.

   Spatial Relationships

   In any standing wave pattern:

   • The distance between any two consecutive nodes is equal to half a wavelength: $\frac{\lambda}{2}$.
   • The distance between an adjacent node and antinode is equal to a quarter of a wavelength: $\frac{\lambda}{4}$.

   For a string of length $L$ fixed at both ends, stable resonance patterns occur only for discrete wavelengths that satisfy the boundary conditions: $\lambda_n = \frac{2L}{n}$, where $n = 1, 2, 3, \dots$ corresponds to the harmonic number.