Oscillations and Waves

SHM is periodic motion in which restoring force is proportional to displacement and directed toward equilibrium: $F = -kx$, giving $a = -\omega^2 x$ where $\omega = \sqrt{k/m}$.

Solution: $x(t) = A\sin(\omega t + \phi)$, where $A$ = amplitude, $\omega$ = angular frequency, $\phi$ = phase constant.

Key Quantities:

Energy in SHM:

Energy is constant; $K$ and $U$ oscillate at twice the SHM frequency.

Phase relation: Velocity leads displacement by $\pi/2$; acceleration leads displacement by $\pi$.

Simple Pendulum: Mass $m$ on light inextensible string of length $L$. For small amplitudes: $$T = 2\pi\sqrt{L/g}$$ Independent of mass and amplitude (isochronous).

Physical Pendulum: Rigid body suspended from pivot at distance $d$ from CM, moment of inertia $I$ about pivot: $$T = 2\pi\sqrt{\frac{I}{mgd}}$$

Torsional Pendulum: $T = 2\pi\sqrt{I/C}$, where $C$ = torsional constant.

Spring-Mass System: $T = 2\pi\sqrt{m/k}$.

Spring Combinations:

Liquid Column in U-Tube: $T = 2\pi\sqrt{L/(2g)}$, where $L$ = total length of liquid column.

Pendulum in accelerated frame: Effective $g$ replaces $g$. E.g., in lift with upward acceleration $a$: $g_{\text{eff}} = g+a$, period decreases.

Damped SHM: $x = A_0 e^{-bt/2m}\cos(\omega' t + \phi)$, $\omega' = \sqrt{\omega_0^2 - (b/2m)^2}$. Energy decays as $e^{-bt/m}$.

Forced/Resonance: At driving frequency = natural frequency, amplitude is maximum (sharp peak for low damping).

Wave = disturbance propagating through a medium (mechanical waves need a medium; EM waves do not). Two main types:

• Transverse: Particle motion perpendicular to wave direction (e.g., on a string, EM waves)
• Longitudinal: Particle motion along wave direction (e.g., sound in fluids)

Wave Quantities:

• Wavelength $\lambda$, frequency $f$, period $T = 1/f$
• Wave speed: $v = f\lambda = \omega/k$
• Angular frequency $\omega = 2\pi f$
• Wave number $k = 2\pi/\lambda$

Travelling wave: $y(x,t) = A\sin(\omega t - kx + \phi)$ (moving in $+x$).

Wave equation (general): $\dfrac{\partial^2 y}{\partial t^2} = v^2\dfrac{\partial^2 y}{\partial x^2}$.

Wave Speeds:

Sound speed depends on temperature: $v \propto \sqrt{T}$ (in Kelvin).

Intensity: Power per unit area, $I = \dfrac{1}{2}\rho v\omega^2 A^2$.

For point source: $I \propto 1/r^2$.

Sound Level (decibel): $\beta = 10\log_{10}(I/I_0)$ dB, $I_0 = 10^{-12}$ W/m².

Superposition Principle: If two waves overlap, resultant displacement = vector sum. Leads to interference, beats, and standing waves.

Standing Wave: Superposition of two identical waves travelling in opposite directions: $$y = 2A\sin(kx)\cos(\omega t)$$ Nodes at $kx = n\pi$ (zero amplitude); antinodes at $kx = (2n+1)\pi/2$ (max). Distance between consecutive nodes (or antinodes) = $\lambda/2$.

Vibrating String (both ends fixed): Allowed wavelengths $\lambda_n = 2L/n$, frequencies $f_n = nv/(2L)$, $n = 1, 2, 3, \ldots$ Fundamental: $f_1 = v/(2L)$.

Air Column:

End correction for resonance tube: $L_{\text{eff}} = L + 0.6r$ (single end open) or $L + 1.2r$ (both ends open).

Beats: Two slightly different frequency waves give resultant whose amplitude varies. Beat frequency: $$f_{\text{beat}} = |f_1 - f_2|$$ Only audible if difference $\leq 10$ Hz.

Doppler Effect (sound, source frequency $f_0$, speed of sound $v$, source velocity $v_s$ toward observer, observer velocity $v_o$ toward source): $$f' = f_0 \cdot \frac{v + v_o}{v - v_s}$$

Sign convention: velocities are positive toward each other. Examples:

• Source toward stationary observer: $f' = f_0 \cdot v/(v - v_s)$ — higher frequency
• Observer toward stationary source: $f' = f_0 \cdot (v + v_o)/v$ — higher frequency
• Both moving apart: $f' = f_0 \cdot (v - v_o)/(v + v_s)$ — lower frequency

For light (EM): relativistic Doppler $f' = f_0\sqrt{(c+v)/(c-v)}$ approaching.