Simple Harmonic Motion (SHM) is the foundation of all oscillatory phenomena tested in NEET. Mathematically, SHM is defined by the condition that the restoring force is proportional to and directed opposite to the displacement from the mean position: F = −kx, leading to the equation of motion a = −ω^{2}x, where ω is the angular frequency in rad/s. The displacement of a particle executing SHM is x = A sin(ωt + φ), where A is the amplitude (maximum displacement in metres), t is time, and φ is the initial phase angle. The velocity at any position is v = ω√($A^{2}$ − $x^{2}$), which reaches its maximum v_max = Aω when x = 0 (mean position) and falls to zero at the extremes x = ±A. The acceleration a = −ω^{2}x is zero at the mean position and reaches its maximum magnitude a_max = Aω^{2} at the extremes. The time period T = 2π/ω and frequency f = 1/T = ω/(2π) are related through ω.

Energy in SHM distributes between kinetic and potential forms while keeping the total constant. The kinetic energy KE = ½mω^{2}($A^{2}$ − $x^{2}$) is maximum at the mean position and zero at the extremes. The potential energy PE = ½mω^{2}$x^{2}$ is zero at the mean position and maximum at the extremes. The total mechanical energy E = KE + PE = ½mω^{2}$A^{2}$ remains constant throughout the motion. A critical NEET point: kinetic energy equals potential energy not at x = A/2, but at x = A/√2 ≈ 0.707A, where each equals E/2 = ¼mω^{2}$A^{2}$.

Two important oscillating systems feature in NEET. For a spring-mass system, T = 2π√(m/k), where k is the spring constant in N/m. This period is completely independent of amplitude and gravitational acceleration — it works identically in zero gravity. Springs connected in series give 1/k_eff = 1/k_{1} + 1/k_{2} (softer effective spring, longer period); springs in parallel give k_eff = k_{1} + k_{2} (stiffer, shorter period). For a simple pendulum, T = 2π√(L/g), valid for small angles (θ < 15°), and is independent of the bob's mass and the amplitude. In a non-inertial frame such as a lift accelerating upward at acceleration a, the effective gravity is g_eff = g + a, reducing the period. In a downward-accelerating lift, g_eff = g − a, increasing the period. In free fall, g_eff = 0, making T infinite — the pendulum does not oscillate.

Wave motion transfers energy through a medium without net transport of matter. A progressive sinusoidal wave travelling in the +x direction is described by y = A sin(kx − ωt), where k = 2π/λ is the wave number (rad/m) and λ is the wavelength. The wave speed v = fλ = ω/k. For transverse waves on a string, v = √(T/μ), where T is tension in N and μ = m/L is the linear mass density in kg/m. Dimensional verification: √([M$LT^{-2}$]/[$ML^{-1}$]) = [$LT^{-1}$] ✓. The speed of sound in a gas is v = √(γP/ρ) = √(γRT/M), where γ = Cp/Cv, R is the gas constant, T is absolute temperature in Kelvin, and M is molar mass. This gives v ∝ √T_K, meaning sound speed increases with temperature. The ordering v_solid > v_liquid > v_gas reflects the decreasing elastic modulus across states of matter.

Standing waves arise from the superposition of two identical waves travelling in opposite directions: y = 2A sin(kx) cos(ωt). The amplitude factor 2A sin(kx) varies with position. Points where sin(kx) = 0 are nodes — permanently at rest, located at x = nλ/2 for integer n. Points where |sin(kx)| = 1 are antinodes — maximum amplitude, located at x = (2n+1)λ/4. Consecutive nodes (or consecutive antinodes) are separated by λ/2.

Vibrating strings (fixed at both ends) and open organ pipes (open at both ends) both support all harmonics: f_n = nv/(2L) for n = 1, 2, 3…. The boundary conditions for strings require nodes at both ends; for open pipes, antinodes at both ends — both lead to the same frequency formula. A closed organ pipe (one end closed, one end open) has a node at the closed end and an antinode at the open end. The shortest allowed standing wave has L = λ/4, giving fundamental f_{1} = v/(4L) — exactly half the fundamental of an open pipe of the same length. The asymmetric boundary condition forbids even harmonics: only odd harmonics n = 1, 3, 5… are present, with f_n = nv/(4L). The first overtone of a closed pipe is therefore the 3rd harmonic (3f_{1}), not the 2nd.

When two waves of slightly different frequencies f_{1} and f_{2} superpose, the resultant amplitude varies periodically at the beat frequency f_beat = |f_{1} − f_{2}|. One beat corresponds to one cycle of loudness variation.

The Doppler effect describes the apparent change in frequency when source and observer are in relative motion. The general formula is f' = f(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. The sign convention: add v_O to the numerator when the observer moves toward the source; subtract v_S from the denominator when the source moves toward the observer. Opposite directions use the opposite signs. When source approaches: f' > f (higher pitch). When source recedes: f' < f (lower pitch). Numerically, a train at 72 km/h = 20 m/s approaching an observer with whistle at 640 Hz and v_sound = 340 m/s gives f' = 640 × 340/(340 − 20) = 680 Hz.

The three most tested NEET traps in this chapter: (1) KE = PE at x = A/√2, not A/2; (2) closed pipes produce only odd harmonics — first overtone is 3rd harmonic; (3) Doppler sign: toward means + in the numerator and − in the denominator.