Chapter Review
Oscillations and Waves
Simple Harmonic Motion · Wave Motion, Speed and Classification · Superposition, Interference and Stationary Waves
Defining SHM
SHM is oscillatory motion where the restoring force is proportional to displacement from equilibrium and always directed toward it, giving the signature equation $a = -\omega^2 x$.
Key Points
- •The negative sign ensures acceleration always opposes displacement — directed back to equilibrium
- •If $a \propto -x$, the motion is SHM; amplitude does not affect period or frequency
- •At equilibrium ($x = 0$): zero acceleration, maximum velocity
- •At extremes ($x = \pm A$): maximum acceleration ($\omega^2 A$), zero velocity
- •SHM is the projection of uniform circular motion onto a diameter
Formula
$$a = -\omega^2 x$$
SHM Quantities and Phase
Angular frequency $\omega$, period $T$, and frequency $f$ are interrelated by $\omega = 2\pi/T = 2\pi f$. The phase angle $\theta = \omega t + \phi$ determines displacement and velocity direction at any instant.
Key Points
- •$f = 1/T$ and $\omega = 2\pi f$ — frequency (Hz) and angular frequency (rad/s) differ by $2\pi$
- •Displacement: $x = A\sin(\omega t + \phi)$; velocity: $v = A\omega\cos(\omega t + \phi)$
- •$\phi = 0$ starts from mean position; $\phi = \pi/2$ starts from positive extreme
- •Two oscillators with phase difference $\pi$ are always anti-phase (opposite sides of equilibrium)
Mass-Spring System
A mass-spring oscillator obeys Hooke's Law ($F = -kx$), producing SHM with period determined by the ratio of inertia to stiffness.
Key Points
- •$T \propto \sqrt{m}$ and $T \propto 1/\sqrt{k}$ — square-root relationships, not linear
- •Velocity at any point: $v = \sqrt{k/m(A^2 - x^2)}$; maximum at center ($v_{max} = A\omega$)
- •Vertical spring has the same $T$ — gravity only shifts equilibrium downward
- •Find $k$ statically: hang a known mass, measure extension, then $k = mg/x$
Formula
$$T = 2\pi \sqrt{\frac{m}{k}}$$
Simple Pendulum
A pendulum approximates SHM for small angles ($\theta < 15°$) where the restoring force is $-mg\sin\theta \approx -mg\theta$. The period depends only on length and gravity, not mass.
Key Points
- •Small angle approximation: $\sin\theta \approx \theta$ in radians ensures linear restoring force
- •Mass independence: both inertia and weight scale with mass, cancelling out
- •$T \propto \sqrt{L}$ (quadrupling $L$ doubles $T$); $T \propto 1/\sqrt{g}$
- •Length is measured to the center of the bob, not its top
- •Derivation: $F = -mg(x/L)$, so $a = -(g/L)x$, confirming $\omega = \sqrt{g/L}$
Formula
$$T = 2\pi \sqrt{\frac{L}{g}}$$
Energy in SHM
SHM systems continuously swap energy between kinetic and potential forms. Total mechanical energy $E = \frac{1}{2}kA^2$ is conserved and proportional to the square of amplitude.
Key Points
- •At extremes: all energy is potential (PE = $\frac{1}{2}kA^2$, KE = 0)
- •At equilibrium: all energy is kinetic (KE = $\frac{1}{2}mv_{max}^2$, PE = 0)
- •At $x = A/2$: PE = $E/4$ and KE = $3E/4$ — PE is proportional to $x^2$, not $x$
- •KE = PE occurs at $x = A/\sqrt{2} \approx 0.707A$
- •Doubling amplitude quadruples total energy ($E \propto A^2$)
Formula
$$E_{total} = \frac{1}{2}kA^2 = \frac{1}{2}mv_{max}^2$$
Progressive Waves and Classification
A progressive wave transfers energy without permanently displacing matter. Waves are classified as transverse (displacement perpendicular to propagation) or longitudinal (displacement parallel to propagation).
Key Points
- •Transverse: rope waves, light — particles move perpendicular to wave direction
- •Longitudinal: sound in air — particles oscillate parallel, creating compressions and rarefactions
- •Solids support both wave types; fluids support only longitudinal waves
- •The wave equation $v = f\lambda$ links speed, frequency, and wavelength
- •When a wave enters a new medium, frequency stays constant (set by source); $v$ and $\lambda$ change
Formula
$$v = f\lambda$$
Speed of Sound in Gases
Sound speed depends on the elastic properties and density of the medium. Newton's isothermal formula gave 280 m/s (16% too low); Laplace's adiabatic correction with $\gamma$ gave the accurate ~332 m/s.
Key Points
- •Newton (isothermal): $v = \sqrt{P/\rho} \approx 280$ m/s — incorrect, compressions are too rapid for isothermal conditions
- •Laplace (adiabatic): $v = \sqrt{\gamma P/\rho} \approx 332$ m/s — correct, no time for heat exchange
- •Pressure change at constant $T$ does not change $v$ (density tracks pressure proportionally)
- •Lighter gas = faster sound: $v \propto 1/\sqrt{\rho}$; sound is 4× faster in H₂ than O₂
- •Temperature effect: $v \propto \sqrt{T}$ (Kelvin), or $v_t \approx 332 + 0.61t$ m/s for small changes
- •$v_{solid} > v_{liquid} > v_{gas}$ — elasticity increase dominates over density increase
Formula
$$v = \sqrt{\frac{\gamma P}{\rho}}$$
Superposition, Beats and Echo
The superposition principle states that overlapping waves add algebraically. This leads to interference phenomena including beats (periodic loudness fluctuations from close frequencies) and echoes (reflected sound from distant surfaces).
Key Points
- •Beat frequency: $f_{beat} = |f_1 - f_2|$ — the number of loudness maxima per second
- •Beats > 10 Hz are too rapid to distinguish as pulsation; $f_1 = f_2$ gives zero beats (tuning goal)
- •Loading a tuning fork with wax lowers its frequency — use this to resolve beat ambiguity
- •Echo distance: $d = vt/2$ — divide round-trip by 2 for one-way distance
- •Reverberation: when round-trip time < 0.1 s, reflected sound blends with the original
- •Phase difference: $\phi = 2\pi x/\lambda$; points $\lambda$ apart are in phase, $\lambda/2$ apart are out of phase
Formula
$$f_{beat} = |f_1 - f_2|$$
Stationary Wave Formation
Stationary waves form when two identical progressive waves travel in opposite directions and superpose. The result is a wave where energy is trapped between nodes rather than transferred.
Key Points
- •Nodes: points of permanent zero displacement, spaced $\lambda/2$ apart
- •Antinodes: maximum amplitude ($2A$), midway between nodes ($\lambda/4$ from nearest node)
- •All particles within one loop are in phase; adjacent loops are $180°$ out of phase
- •All particles (except nodes) cross equilibrium simultaneously — energy alternates between KE and PE
- •Stationary wave equation: $y = [2A\sin(kx)]\cos(\omega t)$ — space and time factors separate
Harmonics in Strings and Air Columns
Vibrating strings and air columns support discrete resonant frequencies (harmonics) determined by boundary conditions. Strings fixed at both ends support all harmonics; closed pipes support only odd harmonics.
Key Points
- •String frequency: $f_n = \frac{n}{2L}\sqrt{T/\mu}$; harmonic series $f_1, 2f_1, 3f_1...$
- •$f \propto \sqrt{T}$ (quadrupling tension doubles frequency); $f \propto 1/L$; $f \propto 1/\sqrt{\mu}$
- •Open pipe: both ends antinodes, all harmonics ($n = 1, 2, 3...$), $\lambda_1 = 2L$
- •Closed pipe: node at closed end, antinode at open end, odd harmonics only ($n = 1, 3, 5...$), $\lambda_1 = 4L$
- •Same-length pipes: $f_{1,\text{closed}} = \frac{1}{2}f_{1,\text{open}}$ — closed pipe sounds one octave lower
- •Wave speed on string: $v = \sqrt{T/\mu}$, different from sound speed in air
Formula
$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$$
Formulas
SHM Acceleration
The defining equation of SHM: acceleration proportional to displacement, opposite in direction.
Formula
$$a = -\omega^2 x$$
Mass-Spring Period
Inertia on top, stiffness on bottom.
Formula
$$T = 2\pi \sqrt{\frac{m}{k}}$$
Pendulum Period
Length on top, gravity on bottom. Mass-independent.
Formula
$$T = 2\pi \sqrt{\frac{L}{g}}$$
SHM Velocity
Maximum at center, zero at extremes.
Formula
$$v = \omega\sqrt{A^2 - x^2}$$
Energy in SHM
Total energy proportional to amplitude squared.
Formula
$$E = \frac{1}{2}kA^2$$
Universal Wave Equation
Speed equals frequency times wavelength.
Formula
$$v = f\lambda$$
Laplace's Sound Speed
Adiabatic correction with $\gamma = C_p/C_v$.
Formula
$$v = \sqrt{\frac{\gamma P}{\rho}}$$
Beat Frequency
Beats per second equals the difference of two frequencies.
Formula
$$f_{beat} = |f_1 - f_2|$$
String Harmonics
All integer multiples for strings fixed at both ends.
Formula
$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$$
Closed Pipe Harmonics
Odd harmonics only for pipes closed at one end.
Formula
$$f_n = \frac{(2n-1)v}{4L}$$