gen-define-standing-waves-on-a-string-and-give-one-short-worked-1083

Define standing waves on a string and give one short worked example.

# Standing Waves on a String ## Definition **Standing waves** are wave patterns that remain stationary in space, formed by the interference of two waves with the same frequency and amplitude traveling in opposite directions along a string. They appear as fixed patterns of **nodes** (points of zero displacement) and **antinodes** (points of maximum displacement). ### Key characteristics: - **Nodes**: Fixed points with no oscillation - **Antinodes**: Points of maximum oscillation - The distance between adjacent nodes = λ/2 - Only certain wavelengths (frequencies) can form standing waves on a fixed-length string ## Worked Example **Problem**: A guitar string of length L = 0.65 m is fixed at both ends. What is the wavelength of the fundamental (first harmonic) standing wave? **Solution**: For a string fixed at both ends, both ends must be nodes. The fundamental mode has the simplest pattern: just two nodes (at the ends) and one antinode (in the middle). This means half a wavelength fits on the string: $$\frac{\lambda_1}{2} = L$$ $$\lambda_1 = 2L = 2(0.65 \text{ m})$$ $$\boxed{\lambda_1 = 1.3 \text{ m}}$$ *Note: The next harmonics would be λ₂ = L, λ₃ = 2L/3, following the pattern λₙ = 2L/n where n = 1, 2, 3...*