Oscillations and waves form one of the most formula-dense yet predictable units in the NEET syllabus — questions on SHM energy, time periods of springs and pendulums, standing waves in pipes, beats and the Doppler effect appear almost every year. The good news is that a single idea (a restoring influence proportional to displacement) generates almost the whole unit, so once the logic is clear the formulas stop feeling arbitrary.
Periodic Motion and the Logic of Simple Harmonic Motion
Any motion that repeats itself after a fixed interval is periodic; if the repetition happens on either side of a fixed equilibrium position it is oscillatory. The smallest interval after which the motion repeats is the time period $T$, its reciprocal is the frequency $\nu = 1/T$ (unit hertz), and $\omega = 2\pi\nu$ is the angular frequency.
The simplest and most important oscillation is simple harmonic motion (SHM), defined by the condition that the restoring force is directed towards the mean position and is proportional to the displacement from it:
$$F = -kx \quad \Rightarrow \quad a = -\omega^2 x, \qquad \omega = \sqrt{k/m}$$
The minus sign is the whole physics of the topic — force and acceleration always point opposite to displacement, i.e. towards the centre. Solving this gives the displacement equation
$$x(t) = A\sin(\omega t + \phi)$$
where $A$ is the amplitude (maximum displacement), $(\omega t + \phi)$ is the phase, and $\phi$ is the initial phase or epoch. A cosine form describes the same motion, only with a different choice of starting instant.
Differentiating gives the two other standard results:
- Velocity: $v = A\omega\cos(\omega t + \phi)$, so $|v| = \omega\sqrt{A^2 - x^2}$; maximum $v_{max} = A\omega$ at the mean position, zero at the extremes.
- Acceleration: $a = -A\omega^2\sin(\omega t + \phi)$, so $|a|_{max} = A\omega^2$ at the extremes, zero at the mean position.
Note the phase relationships: velocity leads displacement by $\pi/2$, and acceleration is exactly out of phase (by $\pi$) with displacement. A useful geometric picture: SHM is the projection, on a diameter, of a particle moving uniformly on a circle of radius $A$ with angular speed $\omega$ (the "reference circle").
Energy in SHM and Standard Oscillating Systems
Because the force is $-kx$, the potential energy stored is
$$U = \tfrac{1}{2}kx^2 = \tfrac{1}{2}m\omega^2A^2\sin^2(\omega t+\phi)$$
and the kinetic energy is
$$K = \tfrac{1}{2}mv^2 = \tfrac{1}{2}m\omega^2A^2\cos^2(\omega t+\phi)$$
Their sum is constant: $E = \tfrac{1}{2}m\omega^2A^2 = \tfrac{1}{2}kA^2$. Note that $E \propto A^2$ — doubling the amplitude quadruples the energy. Both $K$ and $U$ vary with frequency $2\nu$, i.e. twice the frequency of the oscillation itself, because each completes two cycles per oscillation. Kinetic and potential energies are equal when $x = \pm A/\sqrt{2}$, and each averages to $E/2$ over a full cycle.
The time period of any SHM can be written as $T = 2\pi\sqrt{\text{inertia factor}/\text{restoring factor}}$. Standard results:
- Spring–block system: $T = 2\pi\sqrt{m/k}$. Independent of gravity, so the period is the same whether the spring is horizontal or vertical.
- Springs in series: $1/k_{eff} = 1/k_1 + 1/k_2$ (softer, longer period). In parallel: $k_{eff} = k_1 + k_2$ (stiffer, shorter period).
- Simple pendulum: $T = 2\pi\sqrt{L/g}$, valid only for small angular amplitude (typically under about 5°, where $\sin\theta \approx \theta$). Independent of the mass of the bob.
- Liquid in a U-tube of column length $h$ on each side: $T = 2\pi\sqrt{h/g}$.
For a pendulum, if the lift accelerates upward with $a$, replace $g$ by $g+a$; downward, by $g-a$; in free fall the pendulum stops oscillating ($T \to \infty$).
Damped and Forced Oscillations, Resonance
Real oscillators lose energy to friction and air drag. Modelling the damping force as $-bv$ (proportional to speed) gives a solution whose amplitude decays exponentially:
$$x(t) = A e^{-bt/2m}\cos(\omega' t + \phi), \qquad \omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$$
So a damped oscillator is slightly slower than the undamped one, and its mechanical energy also decays exponentially, $E(t) \approx \tfrac{1}{2}kA^2e^{-bt/m}$.
If an external periodic force of angular frequency $\omega_d$ drives the oscillator, after the transient dies out the system oscillates at the driver's frequency, not its own. The steady-state amplitude depends on how close $\omega_d$ is to the natural frequency $\omega_0$. When $\omega_d \approx \omega_0$, the amplitude becomes very large — this is resonance. With light damping the resonance peak is tall and sharp; heavy damping flattens and broadens it and shifts the peak slightly. Classic examples: pushing a swing in rhythm, a radio circuit tuned to a station, marching soldiers breaking step on a bridge.
Wave Motion and the Travelling Wave Equation
A wave transports energy and momentum through a medium without bulk transport of matter — each particle merely oscillates about its own mean position and hands energy to its neighbour. Waves needing a material medium are mechanical waves (sound, water waves, waves on a string); electromagnetic waves need none.
- Transverse wave: particle displacement is perpendicular to the direction of propagation (wave on a stretched string). Possible only in media with shear rigidity, hence not in ideal gases or liquids' interiors.
- Longitudinal wave: particle displacement is along the propagation direction, producing compressions and rarefactions (sound in air).
A sinusoidal travelling wave moving along $+x$ is written
$$y(x,t) = A\sin(kx - \omega t + \phi)$$
where $k = 2\pi/\lambda$ is the angular wave number (unit rad m⁻¹). The wave speed is
$$v = \frac{\omega}{k} = \nu\lambda = \frac{\lambda}{T}$$
Distinguish clearly between wave speed $v$ (constant, fixed by the medium) and particle speed $\partial y/\partial t = -A\omega\cos(kx-\omega t)$, whose maximum is $A\omega$. Speeds in media:
- Transverse wave on a string: $v = \sqrt{T/\mu}$, with $T$ the tension and $\mu$ the linear mass density.
- Longitudinal wave in a solid rod: $v = \sqrt{Y/\rho}$; in a fluid: $v = \sqrt{B/\rho}$.
- Sound in a gas (Laplace's adiabatic correction): $v = \sqrt{\gamma P/\rho} = \sqrt{\gamma RT/M}$. Hence $v \propto \sqrt{T}$ in kelvin, and $v$ is independent of pressure (since $P/\rho$ is fixed at constant temperature). Sound travels faster in humid air because moist air is less dense.
Superposition: Interference, Standing Waves and Beats
When two or more waves overlap, the resultant displacement at each point is the algebraic sum of the individual displacements — the principle of superposition. Two waves of the same frequency and amplitude with a phase difference $\phi$ give a resultant of amplitude $2A\cos(\phi/2)$: constructive interference for $\phi = 0, 2\pi, 4\pi \ldots$ and destructive for $\phi = \pi, 3\pi \ldots$
Reflection: at a rigid boundary (denser medium) the reflected wave suffers a phase change of $\pi$ — a crest returns as a trough. At a free/open boundary there is no phase change.
Standing (stationary) waves arise when two identical waves travel in opposite directions, typically the incident and reflected waves in a bounded medium:
$$y = 2A\sin kx \cos\omega t$$
The wave no longer travels; every particle does SHM but with a position-dependent amplitude $2A\sin kx$. Points of zero amplitude are nodes, points of maximum amplitude are antinodes; adjacent nodes are $\lambda/2$ apart, and node-to-nearest-antinode is $\lambda/4$. Energy is not transported past a node.
Resonant frequencies:
- String fixed at both ends (length $L$): $\nu_n = \dfrac{n}{2L}\sqrt{\dfrac{T}{\mu}}$, $n = 1,2,3,\ldots$ — all harmonics present.
- Open pipe (both ends open): $\nu_n = \dfrac{nv}{2L}$, all harmonics present; fundamental $= v/2L$.
- Closed pipe (one end closed): $\nu_n = \dfrac{(2n-1)v}{4L}$ — only odd harmonics; fundamental $= v/4L$, which is half that of an open pipe of the same length.
Beats occur when two waves of slightly different frequencies $\nu_1$ and $\nu_2$ superpose at a point: the intensity waxes and wanes at the beat frequency $|\nu_1 - \nu_2|$. The ear can resolve beats only up to about 10 per second. Loading a tuning fork with wax lowers its frequency, which is the standard trick used to decide which of the two possible frequencies a fork actually has.
The Doppler Effect in Sound
When there is relative motion between a source and an observer, the observed frequency differs from the emitted one, because wavefronts arrive at a changed rate. For sound (which needs a medium), motion of the source and of the observer are not equivalent. The general result, with all velocities measured along the line joining source and observer and taken positive in the direction from source to observer:
$$\nu' = \nu_0 \left(\frac{v + v_o}{v + v_s}\right)$$
In practice, use the physical rule: relative approach raises the pitch, relative recession lowers it. Common cases:
- Source approaching a stationary observer: $\nu' = \nu_0 \dfrac{v}{v - v_s}$
- Source receding: $\nu' = \nu_0 \dfrac{v}{v + v_s}$
- Observer approaching a stationary source: $\nu' = \nu_0 \dfrac{v + v_o}{v}$
- Observer receding: $\nu' = \nu_0 \dfrac{v - v_o}{v}$
Important points: the wavelength in front of a moving source is genuinely compressed (a real physical decrease in wavelength, not just an illusion of perception), whereas for a moving observer with a stationary source the wavelength in the medium is unchanged — only the rate at which wavefronts are encountered changes. This distinction is why the Doppler formula treats source motion and observer motion asymmetrically even though both produce a shift in the same direction for approach or recession. Also note that the Doppler effect for sound depends on velocities relative to the medium (the air), so a wind blowing from source to observer effectively adds to $v$ in both the numerator and denominator, and must be included explicitly if the problem provides it — a detail rarely tested but occasionally offered as a distractor.
Common Mistakes and Exam Traps
- Assuming the period of a pendulum is independent of amplitude at all angles. $T = 2\pi\sqrt{L/g}$ holds only for small angular amplitude; for larger swings the real period is slightly longer, and a question that explicitly gives a large amplitude is testing whether you notice the approximation breaks down.
- Forgetting that $K$ and $U$ oscillate at twice the frequency of the SHM itself. Since both go as $\sin^2$ or $\cos^2$, they repeat every $T/2$, not every $T$ — a frequent error is to equate the frequency of the energy graphs with the frequency of the oscillation.
- Mixing up the harmonics allowed in open versus closed pipes. An open pipe supports every harmonic ($\nu, 2\nu, 3\nu,\ldots$), but a closed pipe supports only odd harmonics ($\nu, 3\nu, 5\nu,\ldots$) — quoting the second harmonic for a closed pipe, or assuming both pipe types share the same fundamental for equal length, are common slips.
- Applying the wrong Doppler formula, or ignoring the sign convention, when both source and observer move. All velocities must be measured along the line joining source and observer and taken positive in the direction from source to observer; plugging in speeds without fixing this sign convention first is the most frequent source of an inverted (too high or too low) answer.