Standing waves on a 1.0-m-long string that is fixed at both ends are seen at successive frequencies of 36 Hz and 48 Hz. b. Draw the standing-wave pattern when the string oscillates at 48 Hz.

Key Concepts

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This results in a wave pattern that appears to be stationary, characterized by nodes (points of no displacement) and antinodes (points of maximum displacement). In a fixed string, standing waves can only occur at specific frequencies, known as the harmonic frequencies.

Harmonics

Harmonics are the integer multiples of a fundamental frequency at which standing waves can form on a string. For a string fixed at both ends, the fundamental frequency (first harmonic) corresponds to one antinode in the center and two nodes at the ends. Higher harmonics have more nodes and antinodes, with each harmonic frequency being a multiple of the fundamental frequency, allowing for various standing wave patterns.

Frequency and Wavelength Relationship

The frequency of a wave is inversely related to its wavelength, as described by the wave equation: v = fλ, where v is the wave speed, f is the frequency, and λ is the wavelength. For a string fixed at both ends, the speed of the wave remains constant, meaning that as the frequency increases, the wavelength decreases. This relationship is crucial for understanding how different frequencies produce distinct standing wave patterns on the string.