Superposition, Interference and Stationary Waves
Formation and Characteristics
Stationary waves form when two progressive waves of equal amplitude and frequency travel in opposite directions and Superposition occurs.
$$y = [2A \sin(kx)] \cos(\omega t)$$
The displacement is the product of a space-dependent amplitude and a time-dependent oscillation.
$2A \sin(kx)$=Position-dependent amplitude(m)
$k$=Wave number ($2\pi/\lambda$)(rad/m)
$\omega$=Angular frequency(rad/s)
$\sin(kx) = 0$
→Nodes (Zero amplitude)
$\sin(kx) = \pm 1$
→Antinodes (Maximum amplitude)
Nodes: Points of zero displacement that remain permanently at rest. Consecutive nodes are $\lambda/2$ apart.
Antinodes: Points midway between nodes that oscillate with maximum amplitude $2A$. Node-to-antinode distance is $\lambda/4$.
Energy Trapping: Unlike progressive waves, stationary waves do not transfer energy; they localize it between adjacent nodes.
Phase and Energy Localization
Energy alternates between potential and kinetic forms within 'loops', but cannot flow past the Nodes.
Phase Difference: All particles within a single loop are in-phase, while particles in adjacent loops are $180^\circ$ out-of-phase.
Mean Position: All particles (except nodes) cross their equilibrium position simultaneously.
Energy Balance: When amplitude is maximum, energy is purely potential; when passing equilibrium, it is purely kinetic.
Vibrations in Stretched Strings
A string fixed at both ends supports discrete Harmonics where the length $L$ is a multiple of half-wavelengths.
$$f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}}$$
The frequency depends on tension, length, and the linear mass density of the string.
$f_n$=Frequency of $n^{th}$ harmonic(Hz)
$L$=Length of vibrating string(m)
$T$=Tension in string(N)
$\mu$=Linear mass density (mass/length)(kg/m)
$n = 1$
→Fundamental Frequency (First Harmonic)
Harmonic Series: Frequencies are integer multiples ($f_1, 2f_1, 3f_1...$). The $n^{th}$ harmonic has $n$ loops.
Wavelength Rule: $\lambda_n = 2L/n$. Each higher harmonic halves the previous wavelength.
Tension Relation: $f \propto \sqrt{T}$. Quadrupling tension doubles frequency.
Length Relation: $f \propto 1/L$. Halving the vibrating length doubles the frequency — this is how guitarists raise pitch by pressing frets.
Density Relation: $f \propto 1/\sqrt{\mu}$. Thicker strings vibrate at lower frequencies, which is why bass strings are wound with extra wire.
Dimensional Check: In $f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$, units: $\frac{1}{m}\sqrt{\frac{N}{kg/m}} = \frac{1}{m}\sqrt{\frac{kg \cdot m/s^2}{kg/m}} = \frac{1}{s}$ = Hz.
Air Columns (Pipes)
Acoustic resonance depends on boundary conditions: Open ends act as antinodes, while closed ends act as nodes.
$$f_{closed} = \frac{(2n-1)v}{4L}$$
Frequency in a closed pipe involves only odd harmonics due to the node-antinode constraint.
$v$=Speed of sound in air(m/s)
$2n-1$=Odd harmonic multiplier(unitless)
$L$=Length of air column(m)
Open Pipe
→$f_n = \frac{nv}{2L}$ (All harmonics present, $n = 1, 2, 3...$)
$L \to \infty$
→Fundamental frequency approaches zero — longer pipes produce deeper notes
Open Pipe: Both ends are antinodes. Supports all harmonics ($n = 1, 2, 3...$). Fundamental wavelength $\lambda_1 = 2L$.
Closed Pipe: Closed end is a node, open end is an antinode. Only odd harmonics ($n = 1, 3, 5...$). Fundamental wavelength $\lambda_1 = 4L$.
Richness Comparison: An open pipe is 'richer in harmonics' because it supports all integer multiples, while a closed pipe skips every even harmonic.
Proportionality: $f \propto 1/L$ for both pipe types. Doubling the pipe length halves the fundamental frequency.
Fundamental Ratio: For the same length, $f_{1,\text{closed}} = \frac{1}{2} f_{1,\text{open}}$ — a closed pipe sounds one octave lower.