Standing waves Knowledge-Based Study

Study notes

1. What is a Standing Wave?

   A standing wave (or stationary wave) is a wave pattern that remains in a constant position over time. Unlike traveling waves, which propagate through a medium and transport energy from one location to another, standing waves store energy within fixed regions of the medium.

   A standing wave is formed by the superposition (interference) of two traveling waves of the same frequency, wavelength, and amplitude moving in opposite directions through the same medium.

   Key Features of Standing Waves

   The interference of the two opposing traveling waves creates a distinct spatial pattern defined by two critical types of points:

   • Nodes: Points along the medium that undergo complete destructive interference at all times. The displacement at a node is always zero.
   • Antinodes: Points where constructive interference occurs at maximum amplitude. The particles at an antinode oscillate with the maximum possible displacement of the medium.

   Spatial Relationships

   The distance between any two consecutive nodes, or between any two consecutive antinodes, is equal to half a wavelength:

   Distance between adjacent nodes = $\frac{\lambda}{2}$

   The distance between an adjacent node and antinode is one-quarter of a wavelength:

   Distance between adjacent node and antinode = $\frac{\lambda}{4}$

   Standing Waves on a String with Fixed Ends

   When a string of length $L$ is fixed at both ends, the ends are constrained to be displacement nodes. This boundary condition restricts the allowed wavelengths to discrete values called normal modes or harmonics.

   For a string with fixed ends, an integer number $n$ of half-wavelengths must fit exactly onto the length of the string:

   $$L = n \frac{\lambda_n}{2}, \quad n = 1, 2, 3, \dots$$

   Solving for the allowed wavelengths gives:

   $$\lambda_n = \frac{2L}{n}$$

   Since wave speed is given by $v = f \lambda$, the allowed resonant frequencies are:

   $$f_n = n \frac{v}{2L} = n f_1$$

   • $n = 1$ (Fundamental frequency or First Harmonic): Contains two nodes at the ends and one antinode in the middle.
   • $n = 2$ (Second Harmonic): Contains three nodes and two antinodes.
   • $n = 3$ (Third Harmonic): Contains four nodes and three antinodes.

   Key Differences

   It is essential to distinguish between traveling waves and standing waves:

   • Energy Transport: Traveling waves transmit net energy through space, whereas standing waves trap energy between nodes with no net energy flow along the medium.
   • Phase Relationships: In a traveling wave, adjacent points have varying phases depending on their spatial separation. In a standing wave, all points between two adjacent nodes vibrate in phase, while points on opposite sides of a node vibrate $180^\circ$ ($\pi$ radians) out of phase.
   • Amplitude Variation: In a traveling wave, all particles oscillate with the same amplitude. In a standing wave, particle amplitude depends on position, varying from zero at nodes to maximum at antinodes.