standing wave
Summary
A standing wave is the pattern made when two waves of the same frequency travel in opposite directions, usually a wave and its reflection. It has fixed nodes and antinodes and ideally carries no net energy along the medium. Only wavelengths that fit the boundary conditions occur, giving the harmonics.
Detail
Meaning: A standing wave forms when two waves of equal frequency and similar amplitude travel in opposite directions and superpose, usually an incident wave and its reflection. Nodes never move (destructive interference); antinodes oscillate with maximum amplitude. Adjacent nodes are λ/2 apart. Ideally, no net energy travels along the medium. The boundaries allow only certain wavelengths, called harmonics.
Equation: f = nv/(2L) for a string fixed at both ends or a pipe open at both ends, with n = 1, 2, 3, …; f is frequency (Hz), v is wave speed (m/s), L is length (m), n is the harmonic number (unitless). A pipe closed at one end uses f = nv/(4L) with n = 1, 3, 5, …
On the MCAT: Passages use a vibrating string or an air column. You compare frequencies, count nodes, or predict the effect of changing length, tension, or an open versus closed end. A fixed string end is a node, a closed pipe end is a displacement node, and an open pipe end is a displacement antinode. Higher string tension raises v and therefore f. Shorter length also raises f.
Common confusion: For a string fixed at both ends, n equals the number of antinodes (loops), not the number of nodes, which is n + 1. Also, "standing" does not mean the medium is still: it oscillates in place, and only the pattern fails to travel.
Sources
- Physics
- Fundamentals of Physics
Written for the MCAT and checked before it was published. For study reference only. Spotted an error? Let us know.