Unit 9: Oscillations and Waves

This is a 3–4 MCQ unit. SHM and waves are tightly knit through the formulas for period and energy. Doppler effect (for sound) is a near-certain NEET question. Beats and standing waves on strings/pipes also recur. Expect plenty of "find $T$ or $f$" numericals.

Concept Map

• Periodic and oscillatory motion
• Simple Harmonic Motion
  • Kinematics (x, v, a)
  • Energy
  • Phase
• Spring oscillator (series, parallel combinations)
• Simple pendulum (small-angle derivation)
• Damped and forced oscillations (qualitative)
• Waves — transverse vs longitudinal
• Wave equation, speed
• Speed of sound — Laplace correction
• Superposition, beats, standing waves
• Resonance in pipes
• Doppler effect for sound

Topic 1: Simple Harmonic Motion (SHM)

Sub-topic A: Definition

SHM is a periodic motion where the restoring force is proportional to displacement and directed toward the equilibrium:

with $\omega = \sqrt{k/m}$.

Sub-topic B: Kinematics

General solution: $x(t) = A\sin(\omega t + \phi)$ or $x(t) = A\cos(\omega t + \phi)$.

• Amplitude: $A$ (max displacement).
• Angular frequency: $\omega$ (rad/s).
• Time period: $T = 2\pi/\omega$.
• Frequency: $f = 1/T$.
• Phase: $(\omega t + \phi)$.

Velocity and acceleration:

$\vert v_\text{max}\vert = A\omega$, $\vert a_\text{max}\vert = A\omega^2$.

Useful relation: $v = \omega\sqrt{A^2 - x^2}$. Hence $v^2 + \omega^2 x^2 = \omega^2 A^2$.

Sub-topic C: Energy in SHM

So total energy is constant and proportional to $A^2$. KE is maximum at $x = 0$; PE is maximum at $x = \pm A$.

Average over a full cycle: $\langle K \rangle = \langle U \rangle = \tfrac{1}{2} E = \tfrac{1}{4} m\omega^2 A^2$.

Sub-topic D: Spring Oscillator

Series combination: $1/k_\text{eq} = 1/k_1 + 1/k_2$. Effective stiffness decreases ⟹ period increases.

Parallel combination: $k_\text{eq} = k_1 + k_2$. Stiffness adds ⟹ period decreases.

Sub-topic E: Simple Pendulum

For small angles ($\theta < 10°$), torque $= -mg\ell\sin\theta \approx -mg\ell\theta$. So angular SHM with

Independent of mass and amplitude (to first order). On Earth's surface a pendulum's period varies with $g$ (so changes with altitude, depth, latitude).

For a pendulum on a freely falling lift, $g_\text{eff} = 0$ — it does not oscillate.

For a lift accelerating up with $a$: $g_\text{eff} = g + a$, period decreases. Down: $g_\text{eff} = g - a$.

For a pendulum in a horizontally accelerating vehicle (acceleration $a$): $g_\text{eff} = \sqrt{g^2 + a^2}$.

Sub-topic F: Damped and Forced Oscillations

Damped: amplitude decays as $A(t) = A_0 e^{-bt/(2m)}$ for weak damping. Energy decays as $E_0 e^{-bt/m}$.

Forced oscillation: external periodic driving at frequency $\omega_d$. Amplitude is maximum at resonance when $\omega_d \approx \omega_0$ (natural frequency). Resonance amplitude is limited by damping.

Topic 2: Wave Motion

Sub-topic A: Types

• Mechanical waves: need a medium (sound, water).
  • Transverse: oscillation $\perp$ wave direction (string).
  • Longitudinal: oscillation $\parallel$ wave direction (sound).
• Electromagnetic waves: no medium needed.

Sub-topic B: Wave Parameters

A plane wave moving in $+x$ direction:

with $k = 2\pi/\lambda$ (wave number), $\omega = 2\pi f$ (angular frequency), wave speed $v = \omega/k = f\lambda$.

Sub-topic C: Wave Speeds

• On a stretched string of tension $T$ and linear mass density $\mu$:

• In a solid rod (longitudinal): $v = \sqrt{Y/\rho}$.
• In a fluid: $v = \sqrt{K/\rho}$.
• In a gas (sound): $v = \sqrt{\gamma P/\rho}$ (Laplace), where $\gamma = C_p/C_v$. The earlier Newton formula $v = \sqrt{P/\rho}$ used isothermal $K = P$ and underestimated by ~16%.

At 0 °C in air: $v \approx 331\ \text{m/s}$; at 20 °C, $v \approx 343\ \text{m/s}$. Temperature dependence: $v \propto \sqrt{T}$ (kelvin).

Sub-topic D: Energy and Intensity

Energy density of a wave $\propto A^2 \omega^2$. Intensity (power per unit area) $\propto A^2$. For point source: $I \propto 1/r^2$.

Topic 3: Superposition and Standing Waves

Sub-topic A: Principle of Superposition

When two waves overlap, the net displacement is the algebraic sum of individual displacements.

Sub-topic B: Interference

Two coherent waves with same amplitude and frequency: $y_1 = A\sin(\omega t - kx)$, $y_2 = A\sin(\omega t - kx + \phi)$. Result:

Resultant amplitude $A_R = 2A\vert \cos(\phi/2)\vert$:

• $\phi = 0$: constructive, $A_R = 2A$.
• $\phi = \pi$: destructive, $A_R = 0$.

Sub-topic C: Standing Waves on a String (both ends fixed)

A string of length $L$ supports modes with wavelengths $\lambda_n = 2L/n$, $n = 1, 2, 3, \dots$. Frequencies:

• $n = 1$: fundamental (1st harmonic).
• $n = 2$: 2nd harmonic (or 1st overtone).
• $n = 3$: 3rd harmonic (or 2nd overtone).

All integer harmonics are present.

Sub-topic D: Standing Waves in a Pipe

Open at both ends (open organ pipe): all harmonics

Closed at one end (closed organ pipe): only odd harmonics

Hence for the same $L$, the closed pipe has half the fundamental of the open pipe.

End correction (open end): add $0.6 r$ to $L$ for a tube of radius $r$.

Sub-topic E: Beats

When two sound waves of nearly equal frequencies $f_1, f_2$ interfere, the result has a slow envelope:

with beat frequency

Human ear can perceive beats for $\vert f_1 - f_2\vert \lesssim 10\ \text{Hz}$.

Topic 4: Doppler Effect for Sound

For a source $S$ and observer $O$ moving along the line joining them, with speed of sound $v$, source speed $v_s$, observer speed $v_o$:

Sign convention (take direction from observer to source as positive):

• $v_o > 0$ if observer moves toward source.
• $v_s > 0$ if source moves toward observer.

Special cases:

• Source approaching, observer stationary: $f' = f\cdot v/(v - v_s)$, $f'$ increases.
• Source receding: $f' = f\cdot v/(v + v_s)$, $f'$ decreases.
• Observer approaching stationary source: $f' = f(v + v_o)/v$.
• Observer receding: $f' = f(v - v_o)/v$.

Wind: if wind blows with speed $w$ from source to observer, replace $v$ by $v + w$ in numerator and denominator.

Doppler effect for sound is not symmetric in source vs observer motion (unlike light Doppler, which is symmetric to first order). This is because sound has a preferred frame (medium).

NEET Pattern MCQ Tips

• SHM period: spring $T = 2\pi\sqrt{m/k}$, pendulum $T = 2\pi\sqrt{\ell/g}$.
• SHM energy: $E = \tfrac{1}{2} m\omega^2 A^2$.
• Spring combinations: series vs parallel.
• Standing waves: open vs closed pipes; harmonics present.
• Beats: $f_b = \vert f_1 - f_2\vert$; loading with wax/dust to identify the higher frequency.
• Doppler effect: source/observer moving — apply formula with correct sign.
• Wave on string: $v = \sqrt{T/\mu}$.
• Laplace's correction: $v = \sqrt{\gamma P/\rho}$.

Common Confusions and Traps

• SHM amplitude does not appear in the period.
• A simple pendulum's period is independent of mass.
• For SHM, $a = -\omega^2 x$ — the negative sign is essential (restoring).
• Closed organ pipe contains only odd harmonics; open pipe contains all.
• Doppler effect for sound depends on whether the medium moves — wind alters the result.
• Beat frequency is the absolute difference of frequencies — independent of which is larger.
• Wave on a string carries transverse displacement, but propagates longitudinally along the string.
• The Laplace correction multiplies Newton's formula by $\sqrt{\gamma}$, fixing the ~16% discrepancy.

Quick Revision Card

• $\ddot x + \omega^2 x = 0$; $T = 2\pi/\omega$.
• Spring: $T = 2\pi\sqrt{m/k}$.
• Pendulum: $T = 2\pi\sqrt{\ell/g}$.
• SHM energy: $E = \tfrac{1}{2} m\omega^2 A^2$.
• $v_\text{max} = A\omega$; $a_\text{max} = A\omega^2$.
• Series springs: $1/k_\text{eq} = 1/k_1 + 1/k_2$.
• Parallel: $k_\text{eq} = k_1 + k_2$.
• Wave: $v = f\lambda = \omega/k$.
• String: $v = \sqrt{T/\mu}$.
• Sound in gas: $v = \sqrt{\gamma P/\rho}$, $v \propto \sqrt{T}$.
• Open pipe: all harmonics; closed pipe: odd harmonics.
• Doppler (sound): $f' = f(v + v_o)/(v - v_s)$.
• Beats: $f_b = \vert f_1 - f_2\vert$.

Worked NEET Examples

Example 1: Maximum Velocity and Acceleration

A particle in SHM with amplitude 4 cm and period 0.5 s. Angular frequency: $\omega = 2\pi/T = 4\pi$ rad/s. Maximum velocity: $v_\text{max} = A\omega = 0.04 \times 4\pi \approx 0.5$ m/s. Maximum acceleration: $a_\text{max} = A\omega^2 = 0.04 \times 16\pi^2 \approx 6.3$ m/s².

Example 2: Pendulum Period on Moon

On Earth: $T_E = 2\pi\sqrt{L/9.8}$. On Moon: $T_M = 2\pi\sqrt{L/1.62}$. Ratio: $T_M/T_E = \sqrt{9.8/1.62} = \sqrt{6.04} \approx 2.46$. So Moon pendulum has ~2.5× longer period.

Example 3: Standing Wave on String

A string of length 1 m, mass per unit length 0.01 kg/m, tension 100 N. Wave speed: $v = \sqrt{T/\mu} = \sqrt{10000} = 100$ m/s. Fundamental: $f_1 = v/(2L) = 50$ Hz. Second harmonic: 100 Hz, third: 150 Hz, etc.

Example 4: Beat Frequency

Two tuning forks 250 Hz and 256 Hz. Beats per second: 6.

Example 5: Doppler — Train Approaching

Train horn at 600 Hz approaching observer at 30 m/s; speed of sound 340 m/s. Observer hears:

After passing, $f' = 600 \times 340/(340 + 30) \approx 552$ Hz. Drop in pitch: $658 \to 552$ Hz.

Derivations

SHM from F = −kx

Newton's law: $m\ddot x = -kx$. Define $\omega^2 = k/m$. Then $\ddot x + \omega^2 x = 0$. Solution: $x(t) = A\cos(\omega t + \phi)$. Period: $T = 2\pi/\omega = 2\pi\sqrt{m/k}$.

Pendulum Period (Small Angle)

For a pendulum of length $L$, displacement angle $\theta$ (small), restoring force tangent to circle: $F_t = -mg\sin\theta \approx -mg\theta$. Tangential distance: $s = L\theta$, so $\ddot s = L\ddot\theta$, and Newton's law gives $m L\ddot\theta = -mg\theta$, so $\ddot\theta + (g/L)\theta = 0$, period $T = 2\pi\sqrt{L/g}$.

Wave Equation

For a string, transverse displacement $y(x, t)$ satisfies

A traveling wave solution: $y = A\sin(\omega t - kx)$ with $v = \omega/k$.

For string, $v = \sqrt{T/\mu}$ (derive by Newton's law on a small string element).

Speed of Sound in Gas

Newton's formula: $v = \sqrt{P/\rho}$ (assumed isothermal). Predicted ~280 m/s in air at 0 °C — 15% too low.

Laplace's correction: Process is actually adiabatic (fast oscillations, no heat flow). For adiabatic, $\Delta P/\Delta V = -\gamma P/V$, so bulk modulus is $\gamma P$. Hence

For diatomic ($\gamma = 7/5$) at 273 K: $v = \sqrt{1.4 \times 8.314 \times 273/0.029} \approx 331$ m/s — matches experiment.

Doppler Effect — Source Approaches Stationary Observer

Source emits waves of frequency $f$. In a time $\Delta t = 1/f$, source moves $v_s/f$ closer. So consecutive wavefronts are separated by $\lambda' = (v - v_s)/f$ (where $v$ is sound speed). Observer hears

Higher pitch as source approaches.

Energy in SHM

Total energy:

KE: $K = \tfrac{1}{2} m v^2 = \tfrac{1}{2} m \omega^2 (A^2 - x^2)$.

PE: $U = \tfrac{1}{2} k x^2 = \tfrac{1}{2} m \omega^2 x^2$.

Both K and U oscillate with frequency $2\omega$ (double the SHM frequency).

Average over a cycle: $\langle K \rangle = \langle U \rangle = E/2$.

Pendulum Variants

Compound Pendulum

A rigid body pivoted at point P, distance $\ell$ from CM, MI about pivot $I_P$. Period for small oscillations:

Torsion Pendulum

A disc suspended by a wire with torsion constant $C$. Period:

Liquid in U-Tube

A liquid column of length $\ell$ in a uniform U-tube. Oscillates with period

Wave on a Stretched String — Modes

For a string of length $L$ fixed at both ends, modes have wavelengths $\lambda_n = 2L/n$, so frequencies

For a string fixed at one end, free at other: $\lambda_n = 4L/(2n-1)$ — only odd harmonics, similar to closed pipe.

Doppler Effect — Source and Observer Both Moving

General formula:

with sign convention: positive when moving toward each other.

If wind has speed $w$ blowing from source to observer, replace $v$ by $v + w$.

EM Wave Doppler vs Sound Doppler

For sound (with medium), source and observer are not symmetric. For EM (no medium), Doppler effect depends only on relative velocity:

For non-relativistic speeds: $\Delta f/f \approx \pm v/c$. Sign + for approach, − for recede.

Resonance Tube Experiments

For determining sound speed: a resonance tube partially filled with water, vibrating tuning fork held above open top. As water level drops, resonance occurs at length $L_1, L_2, \dots$ where $L = (n - 1/4)\lambda$. Difference $L_2 - L_1 = \lambda/2$. Hence $v = 2f(L_2 - L_1)$.

End correction: $0.3 \cdot d$ (d = inner diameter).

Formula Sheet