Oscillations and Waves revision notes

Periodic and Oscillatory Motions

Periodic motion is a motion that repeats itself identically at regular intervals of time. The smallest interval of time after which the motion is repeated is called its period, denoted by $T$. The reciprocal of the period gives the frequency ($\nu = 1/T$), representing the number of repetitions per unit time, measured in hertz (Hz). When a body in periodic motion moves to and fro about an equilibrium (mean) position—where the net external force is zero—the motion is termed oscillatory or vibratory motion. A restoring force acts on the body to bring it back to equilibrium when displaced.

→ [JEE TIP] Every oscillatory motion is periodic, but every periodic motion need not be oscillatory. For example, uniform circular motion and planetary orbital motion are periodic but not oscillatory. → [JEE TIP] Fourier's Theorem: Any periodic function can be mathematically expressed as a superposition of sine and cosine functions of different time periods. When superposing functions (e.g., $\sin \omega t + \cos 2\omega t$), the fundamental period of the sum is the least common multiple (LCM) of the periods of the constituent terms.

Kinematics and Dynamics of Simple Harmonic Motion (SHM)

Simple harmonic motion (SHM) is the simplest form of oscillatory motion, where the restoring force on a body is directly proportional to its displacement from the mean position and directed toward it. The displacement $x$ of a particle undergoing SHM is expressed as $x(t) = A \cos(\omega t + \phi)$.

• Amplitude ($A$): The magnitude of maximum displacement of the particle from the origin.
• Phase ($\omega t + \phi$): A time-dependent quantity determining the state of motion (position and velocity) at any time $t$.
• Phase Constant ($\phi$): The initial phase angle at $t=0$.

By successive differentiation of the displacement function, we obtain:

• Velocity: $v(t) = -\omega A \sin(\omega t + \phi)$.
• Acceleration: $a(t) = -\omega^2 A \cos(\omega t + \phi) = -\omega^2 x(t)$.
• Force Law: $F(t) = ma = -m\omega^2 x(t) = -kx$, where $k = m\omega^2$ is the force constant.

→ [JEE TIP] Acceleration in SHM is strictly proportional to the negative of displacement. For $x > 0$, $a < 0$ and vice versa. Velocity leads displacement by $\pi/2$, and acceleration leads displacement by exactly $\pi$.

Uniform Circular Motion and Reference Circle Mapping

The projection of uniform circular motion on any diameter of the circle exactly follows simple harmonic motion. If a reference particle moves uniformly on a circle of radius $A$ with angular speed $\omega$, its projection on the x-axis is $x(t) = A \cos(\omega t + \phi)$ and on the y-axis is $y(t) = A \sin(\omega t + \phi)$. → [JEE TIP] Circular Phasor Trick: Evaluating phase differences and time taken to travel between points in SHM is vastly faster using a reference circle instead of solving trigonometric equations. Note, however, that the true linear restoring force in SHM is physically distinct from the centripetal force required for circular motion.

Energy in Simple Harmonic Motion

Both kinetic and potential energies in SHM vary periodically between zero and a maximum value.

• Potential Energy ($U$): $U = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 A^2 \cos^2(\omega t + \phi)$.
• Kinetic Energy ($K$): $K = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2 A^2 \sin^2(\omega t + \phi)$.
• Total Mechanical Energy ($E$): $E = U + K = \frac{1}{2}kA^2$.

The total energy of a harmonic oscillator is independent of time, consistent with motion under conservative forces. → [JEE TIP] Energy Frequencies: While displacement, velocity, and acceleration oscillate with period $T$, both Kinetic and Potential energies peak twice during each period, meaning they oscillate with period $T/2$ (or double the frequency $2\nu$).

The Simple Pendulum

A simple pendulum consists of a small bob of mass $m$ tied to an inextensible, massless string of length $L$. The tangential component of gravity ($mg \sin\theta$) provides the restoring torque about the support. Using Newton's law of rotational motion, the angular acceleration is $\alpha = -(g/L) \sin\theta$. → [JEE TIP] Small Angle Approximation: True SHM only occurs when the displacement $\theta$ is small enough (typically $< 20^\circ$) that $\sin\theta \approx \theta$. The time period becomes $T = 2\pi\sqrt{L/g}$. A "seconds pendulum" is one designed to have a period of exactly 2 seconds, requiring a length of approximately 1 m on Earth.

Types of Waves & Medium Requirements

Waves transport energy and information without the actual physical transfer or flow of matter as a whole.

• Mechanical Waves: Require a material elastic medium for propagation (e.g., sound, water waves).
• Electromagnetic (EM) Waves: Do not require a medium and travel through a vacuum at $c = 299,792,458 \text{ m/s}$ (e.g., light, X-rays).
• Matter Waves: Associated with quantum mechanical constituents of matter like electrons and protons (used in electron microscopes).

Mechanical waves are categorized by particle motion:

• Transverse Waves: Constituents of the medium oscillate perpendicular to the direction of wave propagation. They require the medium to sustain shearing strain (solids, and liquid surfaces via surface tension).
• Longitudinal Waves: Constituents oscillate parallel to the direction of propagation (compressions and rarefactions). They rely on bulk compressive strain (solids, liquids, gases).
• Water Waves (Complex): Capillary waves (ripples < few cm) are restored by surface tension. Gravity waves (large waves) are restored by gravity. Water particle motion is a complex combination of longitudinal and transverse motions (elliptical orbits).

Displacement Relation for Progressive Waves

A one-dimensional progressive harmonic wave is mathematically represented by: $y(x,t) = a \sin(kx - \omega t + \phi)$.

• Amplitude ($a$): Maximum displacement of the constituents.
• Angular Wave Number ($k$): $k = 2\pi/\lambda$ [rad/m].
• Wave Speed ($v$): $v = \omega/k = \lambda\nu$.

→ [JEE TIP] Any valid progressive wave function MUST be expressible as a function of $(ax \pm bt)$. The wave speed is given by the ratio of the coefficients: $v = | \text{coefficient of } t / \text{coefficient of } x | = \omega/k$.

Wave Speed in Different Media

The speed of a mechanical wave is determined purely by the inertial and elastic properties of the medium, NOT by the wave's frequency or wavelength.

• Transverse wave on a string: $v = \sqrt{T/\mu}$, where $T$ is tension and $\mu$ is linear mass density ($m/L$).
• Longitudinal wave in a fluid: $v = \sqrt{B/\rho}$, where $B$ is the bulk modulus.
• Longitudinal wave in a solid bar: $v = \sqrt{Y/\rho}$, where $Y$ is Young's modulus.

Speed of Sound in Ideal Gases:

• Newton's Formula ($v = \sqrt{P/\rho}$): Failed because it assumed pressure variations were isothermal.
• Laplace Correction ($v = \sqrt{\gamma P/\rho}$): Corrected Newton by pointing out that rapid compressions/rarefactions leave no time for heat transfer; the process is adiabatic.

Superposition, Interference & Reflection of Waves

Principle of Superposition: When two or more waves overlap, the net displacement is the algebraic sum of the individual displacements: $y(x,t) = \sum y_i(x,t)$. If two identical waves with a phase difference $\phi$ overlap, the resultant amplitude relies strictly on $\phi$: $A(\phi) = 2a \cos(\phi/2)$.

• Constructive: $\phi = 0, 2\pi \dots$ (Amplitude = $2a$).
• Destructive: $\phi = \pi, 3\pi \dots$ (Amplitude = $0$).

Reflection of Waves & Echo:

• Rigid Boundary (Fixed Wall): The reflected wave undergoes a phase reversal ($\pi$ phase change). The wall exerts a Newton's Third Law reactive force. $y_r = -a \sin(kx + \omega t)$.
• Open/Free Boundary (Loose Ring): The boundary moves freely. The reflected wave has NO phase change. $y_r = a \sin(kx + \omega t)$.
• Refraction: When a wave is incident obliquely on a boundary between two different media, the transmitted wave obeys Snell's Law.

Standing Waves and Normal Modes

When two identical harmonic waves travel in opposite directions in bounded spaces, they form standing waves: $y(x,t) = [2a \sin kx] \cos \omega t$.

• Nodes: Points of permanently zero displacement ($x = n\lambda/2$).
• Antinodes: Points of maximum amplitude ($x = (n+\frac{1}{2})\lambda/2$).

Normal Modes (Natural Frequencies):

1. String fixed at both ends: Nodes at boundaries. Generates ALL harmonics: $\nu = n(v/2L)$ for $n = 1, 2, 3...$.
2. Air column open at both ends: Antinodes at boundaries. Generates ALL harmonics: $\nu = n(v/2L)$.
3. Air column closed at one end: Node at closed end, antinode at open end. Generates ONLY ODD harmonics: $\nu = (2n+1)(v/4L)$ for $n = 0, 1, 2...$.

Beats

When two sound waves of slightly different frequencies ($\nu_1, \nu_2$) are heard simultaneously, their interference produces periodic waxing and waning of sound intensity known as beats. Beat Frequency: $\nu_{beat} = |\nu_1 - \nu_2|$.

Key Concepts & Definitions

Restoring Force:

A force strictly directed toward the equilibrium position, essential for SHM.

Displacement:

In waves/oscillations, this doesn't just mean position. It can be voltage in a capacitor, pressure/density variations in sound, or angle of a pendulum.

Progressive vs Standing Wave:

Progressive waves transfer energy forward. Standing waves transfer zero net energy; particles have different amplitudes but all particles between two nodes oscillate completely in phase.

Conditions & Limitations

• Damped Oscillations Validity: Damped simple harmonic motion is not strictly simple harmonic. It can only be approximated as SHM for time intervals much less than $2m/b$, where $b$ is the damping constant.
• Small Angle Limit: The pendulum formula $T = 2\pi\sqrt{L/g}$ completely fails for large angles (where $\sin\theta \ne \theta$).
• Transverse Waves in Gases: Fluids absolutely lack shear modulus, meaning they cannot support transverse waves internally.

Previous Year JEE Topics

• Finding Period of Composite Functions: Expressing sums of multiple sines and cosines (e.g., $\sin\omega t + \cos 2\omega t$) and determining their net periodicity by finding the LCM of individual periods.
• Energy Graphs in SHM: Identifying the shape of $U(x)$ and $K(x)$ as parabolas, but $U(t)$ and $K(t)$ as strictly positive sinusoidal curves oscillating at frequency $2\nu$.
• Unknown Tuning Fork (Beats): Determining if an unknown frequency $\nu$ is $(\nu_{known} + \nu_{beat})$ or $(\nu_{known} - \nu_{beat})$ by analyzing the change in beat frequency after altering mass (e.g. adding wax lowers $\nu$; filing raises $\nu$).
• Open vs Closed Organ Pipes: Calculating ratios of fundamental frequencies ($v/2L$ vs $v/4L$) and utilizing the fact that closed pipes lack even harmonics to identify valid modes in multiple-choice questions.

Important Graphs & Diagrams

• SHM Kinematic Curves: The graphs of position $x$, velocity $v$, and acceleration $a$ vs time are sinusoids of the same period $T$. Velocity leads position by $\pi/2$, and acceleration is the exact mirror inverse ($\pi$ phase shift) of position.
• Energy vs Displacement/Time: Potential energy is an upward-opening parabola ($\frac{1}{2}kx^2$). Kinetic energy is an inverted parabola peaking at the origin. Plotted against time, both energies are strictly positive sinusoidal curves peaking twice per fundamental cycle. Total energy is a flat horizontal line.
• Standing Wave Patterns: Fixed strings show integer loops (nodes at both ends). Open pipes show integer loops starting/ending with half-loops (antinodes at ends). Closed pipes show a node at the closed end and an antinode at the open end.

Standard Derivations & Step-by-Step Problem Solving

Proving Simple Harmonic Motion & Finding $T$:

1. Assume a small linear ($x$) or angular ($\theta$) displacement from the equilibrium position.
2. Compute the net restoring force ($F_{net}$) or torque ($\tau_{net}$) pushing it back.
3. Simplify forces using small-angle approximations if necessary (e.g., $\sin\theta \approx \theta$).
4. Express the net force linearly: $F_{net} = -k_{eff} x$.
5. Equate to Newton's second law ($ma$) to find $a = -(k_{eff}/m) x$.
6. Compare to the standard SHM definition $a = -\omega^2 x$ to extract $\omega = \sqrt{k_{eff}/m}$, and compute the time period using $T = 2\pi/\omega$.

🎯 TOP 10 JEE MCQ TRAPS

• [JEE TIP] Trap 1 - The Wave Speed vs. Particle Velocity Divide:

  • Misconception: The absolute speed at which a wave profile propagates through a medium is identical to the instantaneous physical velocity of the individual medium particles.
  • Correct Understanding: These velocities belong to completely different kinematic regimes. Wave propagation speed ($v = \frac{\omega}{k} = \nu\lambda$) is a constant property dictated entirely by the elasticity and inertia of the medium. Conversely, the particle velocity ($v_p = \frac{\partial y}{\partial t} = -v \frac{\partial y}{\partial x}$) is a dynamic variable that executes simple harmonic motion, spiking to its maximum at the equilibrium position ($y=0$) and dropping to exactly zero at the extreme amplitudes.

• [JEE TIP] Trap 2 - Longitudinal Displacement-Pressure Inversion:

  • Misconception: A displacement node inside a propagating longitudinal sound wave simultaneously acts as a pressure node.
  • Correct Understanding: Displacement and pressure variations maintain a strict $90^\circ$ phase difference ($\Delta \Phi = \frac{\pi}{2}$). A displacement node is a spatial coordinate where medium particles remain completely stationary, meaning they experience the maximum localized "squeezing and stretching" from incoming neighboring particles. Therefore, a displacement node is a pressure antinode (maximum pressure density variation), whereas a displacement antinode maps precisely to a pressure node.

• [JEE TIP] Trap 3 - Tuning Fork Frequency Alteration:

  • Misconception: Filing down the metal prongs of a tuning fork alters its structural inertia so that its fundamental frequency drops.
  • Correct Understanding: Filing metal off the prongs reduces the effective mass ($m$) near the vibrating tips while keeping the stiffness constant. Because the natural frequency of an oscillator shares an inverse relationship with mass ($\nu \propto \sqrt{\frac{1}{m}}$), filing a tuning fork strictly increases its fundamental frequency. Conversely, loading the prongs with a layer of wax increases the mass, which scales down and decreases the frequency.

• [JEE TIP] Trap 4 - The Humidity-Density Paradox:

  • Misconception: Humid air carries a high concentration of water droplets, making it denser than dry air and causing sound waves to travel slower through it.
  • Correct Understanding: Water vapor molecules ($\text{H}_2\text{O}$, molar mass $\approx 18\text{ g/mol}$) physically displace heavier diatomic Nitrogen and Oxygen molecules (average atmospheric molar mass $\approx 29\text{ g/mol}$). Consequently, humid air is strictly less dense than dry air. Because the speed of sound is inversely proportional to the square root of mass density ($v = \sqrt{\frac{\gamma P}{\rho}}$), sound waves travel faster in high humidity.

• [JEE TIP] Trap 5 - Ideal Gas Pressure Speed Independence:

  • Misconception: Increasing the ambient pressure of an ideal gas container scales up the speed of sound through that medium.
  • Correct Understanding: According to the Laplace correction ($v = \sqrt{\frac{\gamma P}{\rho}}$), the speed of sound is tied to both pressure and density. However, if the temperature is locked constant, Boyle's Law dictates that the mass density ($\rho$) scales up perfectly proportionally with any increase in pressure ($P$). This keeps the ratio $\frac{P}{\rho}$ completely constant, making the speed of sound entirely independent of pressure changes ($\Delta v = 0$).

• [JEE TIP] Trap 6 - Boundary Reflection Phase Demarcation:

  • Misconception: When a wave pulse strikes a boundary and undergoes reflection, it always suffers a mandatory phase reversal of $\pi$ radians ($180^\circ$).
  • Correct Understanding: A phase reversal of $\pi$ occurs if and only if the wave reflects off a denser, rigid/fixed boundary (due to Newton's third law reaction forces). If a wave reflects off a rarer, open/free boundary—such as a loose ring sliding on a frictionless rod, or the open end of an acoustic pipe—the pulse reflects with exactly zero phase change ($\Delta \Phi = 0$).

• [JEE TIP] Trap 7 - SHM Superposition Periodicity Boundary:

  • Misconception: The mathematical superposition of any two simple harmonic motions tracking along the same axis always yields a combined periodic motion.
  • Correct Understanding: The resulting combined waveform is strictly periodic if and only if the ratio of their individual frequencies ($\frac{\nu_1}{\nu_2}$) evaluates to a rational number. This rational condition ensures that the system can establish a common least common multiple (LCM) time period. If the frequency ratio is irrational (e.g., combining $2\text{ Hz}$ and $\sqrt{2}\text{ Hz}$), the wave profile will never perfectly repeat itself, yielding a non-periodic motion.

• [JEE TIP] Trap 8 - Energy Oscillation Frequency Doubling:

  • Misconception: The kinetic and potential energies of a simple harmonic oscillator vibrate back and forth at the exact same fundamental frequency ($\nu$) as the structural displacement.
  • Correct Understanding: Displacement, velocity, and acceleration track the fundamental frequency $\nu$. However, the kinetic and potential energy functions incorporate squared trigonometric identities ($\sin^2\omega t$ and $\cos^2\omega t$). Because these squared functions hit their maximum energy peaks twice during a single directional cycle of displacement, both kinetic and potential energies oscillate at exactly double the fundamental frequency ($2\nu$).

• [JEE TIP] Trap 9 - The Surface Water Wave Cross-Over:

  • Misconception: Mechanical waves traveling across the surface of an ocean or pond are clean examples of pure transverse mechanical waves.
  • Correct Understanding: Surface water waves are a complex, hybrid combination of both longitudinal and transverse motions. Because water is nearly incompressible, individual surface molecules do not merely move up and down; they trace out circular or elliptical spatial paths, moving up-and-down and back-and-forth simultaneously as the energy rolls past.

• [JEE TIP] Trap 10 - Particle Acceleration Axis Orientation:

  • Misconception: The acceleration vector of a medium particle trapped within a propagating wave is always directed opposite to the horizontal direction of the wave's propagation vector.
  • Correct Understanding: Particle acceleration is a localized internal restoration mechanic defined by the simple harmonic rule $a = -\omega^2 y$. This vector is strictly directed opposite to the particle's vertical displacement from its local equilibrium node (the y-axis). It points up or down toward the central equilibrium axis and carries absolutely no dependence on whether the wave profile is travelling to the left or to the right along the horizontal x-axis.