(II) A standing wave on a 1.64 -m-long horizontal string...

(II) A standing wave on a 1.64 -m-long horizontal string displays three loops when the string vibrates at 120 $\mathrm{Hz}$ . The maximum swing of the string (top to bottom) at the center of each loop is 8.00 $\mathrm{cm} .$ (a) What is the function describing the standing wave? (b) What are the functions describing the two equal-amplitude waves traveling in opposite directions that make up the standing wave?

Key Concepts

Wave Functions and Superposition Principle

Wave functions mathematically describe the displacement of points on a medium as a function of space and time, often using sinusoidal functions. The superposition principle states that the net displacement at any point is the sum of the displacements due to individual waves. Thus, a standing wave can be expressed as the sum of two traveling waves moving in opposite directions, each described by its own wave function.

Relationship between Wavelength, Length, and Harmonic Modes

The wavelength of a standing wave on a string is related to the length of the string and the harmonic mode. In each mode, the string accommodates a certain number of half wavelengths based on the number of nodes. Understanding this relationship allows one to determine the wavelength and frequency of the wave based on the physical dimensions of the medium and the number of loops or segments observed.

Standing Waves

A standing wave is formed when two waves of identical frequency and amplitude travel in opposite directions and interfere with each other. This interference creates a pattern with fixed nodes (points of zero displacement) and antinodes (points of maximum displacement), even though energy is continuously oscillating between kinetic and potential forms in the medium.

Harmonics

Harmonics refer to the natural modes of vibration of a medium, such as a string fixed at both ends. Each harmonic corresponds to a specific pattern of nodes and antinodes, and the number of loops or segments in the standing wave is determined by the harmonic order. The fundamental or first harmonic has one antinode in the center, while higher harmonics have additional loops.

Nodes and Antinodes

Nodes are the points along the medium where the displacement remains zero during the vibration, while antinodes are the points where the displacement is at a maximum. In a standing wave, the positions of these nodes and antinodes are fixed in space, and their distribution is key to understanding the behavior and characteristics of the wave mode.

Transcript

Please give Ace some feedback