When a wave reaches the end of a string, rope, spring, or air column, it does not simply disappear. Part of the wave energy can reflect back, and the shape of the reflected wave depends on the boundary. Reflection explains echoes, musical instrument vibrations, and patterns on strings.
Understanding it helps connect traveling waves to the stationary patterns called standing waves.
A standing wave forms when two waves of the same frequency and amplitude travel in opposite directions and overlap. Their superposition creates fixed points of no motion called nodes and points of maximum motion called antinodes. A fixed end reflects a wave with phase inversion, while a free end reflects it without inversion.
These boundary rules determine which wavelengths and frequencies can fit on a string or in a pipe.
Key Facts
- Wave speed on a string: v = sqrt(T/mu), where T is tension and mu is mass per unit length.
- Frequency, wavelength, and speed are related by v = f lambda.
- At a fixed end, the reflected wave is inverted, so the reflected pulse has a 180 degree phase change.
- At a free end, the reflected wave is not inverted, so there is no phase change at reflection.
- Standing wave condition for a string fixed at both ends: L = n lambda/2, where n = 1, 2, 3, ...
- Harmonic frequencies for a string fixed at both ends: f_n = n v/(2L).
Vocabulary
- Incident wave
- An incident wave is the wave that travels toward a boundary before reflection occurs.
- Reflected wave
- A reflected wave is the wave that travels away from a boundary after bouncing back.
- Node
- A node is a point in a standing wave that remains still because destructive interference is always occurring there.
- Antinode
- An antinode is a point in a standing wave with the largest oscillation amplitude.
- Phase inversion
- Phase inversion is a 180 degree flip of a reflected wave, such as when a pulse reflects from a fixed end.
Common Mistakes to Avoid
- Assuming every reflected wave is inverted. This is wrong because fixed ends invert waves, but free ends reflect waves without inversion.
- Confusing wave speed with particle speed. The wave speed is how fast the disturbance travels along the medium, while particle speed is how fast a point on the medium moves up and down.
- Placing antinodes at fixed ends. This is wrong because a fixed end cannot move, so it must be a node.
- Using any wavelength for a standing wave on a fixed string. This is wrong because only wavelengths that satisfy L = n lambda/2 fit the boundary conditions.
Practice Questions
- 1 A string fixed at both ends is 1.20 m long and wave speed on the string is 48 m/s. Find the fundamental frequency and the frequency of the third harmonic.
- 2 A wave travels along a rope at 30 m/s with frequency 5.0 Hz. Find its wavelength, then determine the length of a string fixed at both ends that would fit the second harmonic using this wavelength.
- 3 A pulse travels toward the end of a string. Explain how the reflected pulse differs if the end is fixed compared with if the end is free, and state where nodes or antinodes occur at the boundary.