Wave Motion, Superposition, and Standing Waves
A traveling wave, y(x,t)=A sin(kx∓ωt), is really just Unit 109's kinematics dressed up for a continuous medium. What's new here is superposition: when two waves overlap, their displacements simply add, and the specific case of two identical waves traveling in opposite directions produces something that doesn't travel at all — a standing wave, locked into fixed nodes and antinodes by whatever boundary conditions (fixed or free ends) the medium has. This is the mathematical machinery behind every musical instrument's specific pitches.
What you'll learn
- Write the equation of a traveling wave given its amplitude, wavelength, and frequency (or period).
- Apply the superposition principle to find the resultant of two overlapping waves.
- Derive the standing-wave pattern produced by two identical counter-propagating waves.
- Calculate the harmonic frequencies of a string fixed at both ends.
- Calculate the harmonic frequencies of an air column, for both open-open and closed-open boundary conditions.
- Calculate wave speed on a string from its tension and linear mass density.
1. The Traveling Wave Equation and Wave Speed
A sinusoidal traveling wave is described by y(x,t)=A sin(kx∓ωt), where k=2π/λ and ω=2πf — the minus sign for a wave moving in the +x direction, plus for -x. This is exactly the same mathematics as calc1's introductory treatment, now used as the building block for everything in this unit.
On a stretched string, wave speed depends only on the string's own physical properties: v=√(T/μ), where T is tension and μ is mass per unit length. Notably absent from this formula: the wave's own frequency or amplitude — changing those changes the wavelength (via v=fλ, with v fixed), never the speed.
2. Superposition and Standing Waves
The superposition principle says overlapping waves simply add their displacements at every point. Two identical waves traveling in opposite directions produce a striking special case: y(x,t)=2A sin(kx)cos(ωt) — a pattern whose spatial shape (sin kx) and time-dependence (cos ωt) have separated. Every point still oscillates (at whatever rate cos ωt dictates), but the *shape* of the pattern — where the nodes and antinodes sit — never moves.
Nodes (sin kx=0) are points of permanent zero displacement; antinodes (|sin kx|=1) oscillate with the pattern's full amplitude, 2A. Adjacent nodes are always separated by exactly λ/2.
3. Harmonics of Strings and Air Columns
A string fixed at both ends must have a node at each end, restricting it to wavelengths λ_n=2L/n for n=1,2,3,... — the full harmonic series, with frequencies f_n=nv/(2L). An air column open at both ends obeys the identical mathematics (antinode at each open end plays the same structural role as a node at each fixed end).
A column closed at one end and open at the other is different: the closed end forces a node, the open end an antinode — a combination only odd multiples of the fundamental wavelength can satisfy, giving f_n=nv/(4L) for n=1,3,5,... only. This is why, for the same length, a closed-open pipe's fundamental is exactly half an open-open pipe's — and why clarinets (closed-open, roughly) and flutes (closed-open, roughly, but conceptually treated as open-open) sound so different even at similar physical lengths.
Key equations
- y(x,t) = A sin(kx∓ωt) — The general equation of a sinusoidal traveling wave.
- v = √(T/μ) — Wave speed on a stretched string — depends only on the string's own tension and density, not on the wave's frequency or amplitude.
- y(x,t) = 2A sin(kx)cos(ωt) — The standing-wave pattern from two identical waves of amplitude A traveling in opposite directions — note the spatial part (sin kx) and time part (cos ωt) are now separated, unlike a traveling wave.
- f_n = nv/(2L), n=1,2,3,... — Harmonic frequencies of a string fixed at both ends, or an air column open at both ends — both support the same harmonic series, since both boundary conditions require a node (string) or antinode (open pipe) at each end.
- f_n = nv/(4L), n=1,3,5,... — Harmonic frequencies of an air column closed at one end and open at the other — only odd harmonics are supported, since the closed end must be a node and the open end an antinode.