Oscillations & Waves

Simple Harmonic Motion (SHM) is periodic motion where the restoring force is proportional to displacement and directed toward the mean position: a = -$\omega$² x, where $\omega$ is the angular frequency (rad/s).
The displacement is x = A sin($\omega$ t + $\phi$) [$M^{0} \; L^{1} \; T^{0}$] (m), where A is the amplitude (m) and $\phi$ is the initial phase (rad).
Velocity v = A $\omega$ cos($\omega$ t + $\phi$) = $\omega \; \sqrt{A^{2} - x^{2}}$ [$M^{0} \; L^{1} \; T^{-1}$] (m/s) — maximum at the mean position ($v_{max}$ = A $\omega$), zero at the extremes.
Acceleration a = -A $\omega$² sin($\omega$ t + $\phi$) = -$\omega$² x [$M^{0} \; L^{1} \; T^{-2}$] (m/$s^{2}$) — maximum at the extremes ($a_{max}$ = A $\omega$²), zero at the mean position.
Time period T = $\frac{2\pi}{\omega}$ [$M^{0} \; L^{0} \; T^{1}$] (s) and frequency f = 1/T = $\omega$/($2\pi$) [$M^{0} \; L^{0} \; T^{-1}$] (Hz).

SHM energy oscillates between kinetic and potential forms.
KE = $\frac{1}{2}$ m $\omega$² ($A^{2}$ - $x^{2}$) [$M^{1} \; L^{2} \; T^{-2}$] (J), maximum at x = 0.
PE = $\frac{1}{2}$ m $\omega$² $x^{2}$, maximum at x = +/-A.
Total energy E = $\frac{1}{2}$ m $\omega$² $A^{2}$ = constant.
KE equals PE at x = A/$\sqrt{2}$ (not A/2 — this is a frequent NEET trap).

A spring-mass system oscillates with T = $2\pi \; \sqrt{m/k}$ [k: spring constant, [$M^{1} \; L^{0} \; T^{-2}$] (N/m)], independent of amplitude and gravity.
Springs in series: 1/$k_{eff}$ = 1/$k_{1}$ + 1/$k_{2}$ (weaker, longer T).
Springs in parallel: $k_{eff}$ = $k_{1}$ + $k_{2}$ (stiffer, shorter T).
A simple pendulum has T = $2\pi \; \sqrt{L/g}$, valid for small angles ($\theta$ < 15 degrees), independent of mass.
In a lift: accelerating up gives $g_{eff}$ = g + a (T decreases), accelerating down gives $g_{eff}$ = g - a (T increases), free fall gives $g_{eff}$ = 0 (T = infinity, no oscillation).

Wave motion transfers energy without transporting matter.
A progressive wave is described by y = A sin(kx - $\omega$ t), where k = $\frac{2\pi}{\lambda}$ is the wave number (rad/m) and $\lambda$ is the wavelength (m).
Wave speed v = f $\lambda$ = $\omega$/k [$M^{0} \; L^{1} \; T^{-1}$] (m/s).
Speed of a transverse wave in a string: v = $\sqrt{T/\mu}$, where T is the tension (N) and $\mu$ is the linear mass density (kg/m) [$M^{1} \; L^{-1} \; T^{0}$].
Dimensional check: $\sqrt{[M L T^{-2}]/[M L^{-1}]}$ = $\sqrt{[L^{2} T^{-2}]}$ = [L $T^{-1}$].
Speed of sound in air: v = $\sqrt{\gamma P/\rho}$ = $\sqrt{\gamma RT/M}$ (Newton-Laplace formula); v is proportional to $\sqrt{T in kelvin}$; $v_{solid}$ > $v_{liquid}$ > $v_{gas}$.

Standing waves form by superposition of two identical waves travelling in opposite directions: y = 2A sin(kx) cos($\omega$ t).
Nodes (zero amplitude) occur at x = n $\frac{\lambda}{2}$; antinodes (maximum amplitude) at x = (2n+1) $\frac{\lambda}{4}$.
Distance between consecutive nodes = $\frac{\lambda}{2}$.

Vibrating strings (both ends fixed): $f_{n}$ = nv/(2L) for n = 1, 2, 3... (all harmonics). Open organ pipe (both ends open): $f_{n}$ = nv/(2L) for n = 1, 2, 3... (all harmonics). Closed organ pipe (one end closed): $f_{n}$ = nv/(4L) for n = 1, 3, 5... (ODD harmonics ONLY). The fundamental of a closed pipe is half that of an open pipe of the same length.

Beats occur when two waves of slightly different frequencies superpose: $f_{beat}$ = |$f_{1}$ - $f_{2}$| [Hz].

The Doppler effect: f' = f x (v +/- $v_{O}$) / (v -/+ $v_{S}$), where v is the speed of sound, $v_{O}$ is the observer's speed, and $v_{S}$ is the source's speed. Convention: use + in the numerator when the observer moves toward the source; use - in the denominator when the source moves toward the observer. Source approaching: f' > f (higher pitch). Source receding: f' < f (lower pitch).

Solved Numerical 1: SHM with A = 10 cm = 0.10 m, T = 2 s.
$\omega$ = $2\pi$/T = $\frac{2\pi}{2}$ = $\pi$ rad/s.
(a) $v_{max}$ = A $\omega$ = 0.10 x $\pi$ = 0.314 m/s = 31.4 cm/s.
(b) $a_{max}$ = A $\omega$² = 0.10 x $\pi$² = 0.987 m/$s^{2}$.
(c) At x = 6 cm = 0.06 m: v = $\omega \; \sqrt{A^{2} - x^{2}}$ = $\pi$ x $\sqrt{0.10^{2} - 0.06^{2}}$ = $\pi$ x $\sqrt{0.0100 - 0.0036}$ = $\pi$ x $\sqrt{0.0064}$ = $\pi$ x 0.08 = 0.251 m/s = 25.1 cm/s.

Solved Numerical 2: Open organ pipe with $f_{1}$ = 300 Hz.
(a) Third harmonic: $f_{3}$ = 3 x 300 = 900 Hz.
(b) If one end is closed (same length L): open $f_{1}$ = v/(2L) = 300 Hz, so v/L = 600.
Closed $f_{1}$ = v/(4L) = (v/L)/4 = $\frac{600}{4}$ = 150 Hz. Fundamental drops to half. Closed pipe harmonics: 150, 450, 750 Hz (odd multiples only).

Solved Numerical 3: Train approaches at 72 km/h = 20 m/s, whistle frequency f = 640 Hz, $v_{sound}$ = 340 m/s, observer stationary ($v_{O}$ = 0).
Source approaching: f' = f(v + $v_{O}$)/(v - $v_{S}$) = 640(340 + 0)/(340 - 20) = 640 x $\frac{340}{320}$ = 640 x 1.0625 = 680 Hz. Apparent frequency is higher as expected.

The key testable concept is the SHM energy equality condition (KE = PE at x = A/$\sqrt{2}$, not A/2) and the distinction between open pipes (all harmonics) and closed pipes (odd harmonics only).