Wave Study Guide

📖 Core Concepts Wave – A propagating disturbance that moves away from an equilibrium state of one or more physical quantities. Travelling wave – The whole waveform advances through space with speed v. Standing wave – Formed by superposition of two equal‑amplitude, opposite‑direction waves; produces fixed nodes (zero displacement) and antinodes (max displacement). Mechanical vs. Electromagnetic waves – Mechanical waves require a material medium (stress/strain); EM waves are coupled electric‑magnetic fields and can travel in vacuum. Transverse vs. Longitudinal – In transverse waves the field (or particle motion) is ⟂ to propagation; in longitudinal waves it is ‖ to propagation. Polarization – Orientation of the oscillating field of a transverse wave (e.g., linear, circular). Wave function – $u(\mathbf{r},t)$ maps each point in space and time to a field value (displacement, pressure, electric field, etc.). Superposition principle – Fields add algebraically; leads to constructive or destructive interference. --- 📌 Must Remember Wave equation (homogeneous, isotropic, non‑conducting medium) $$\frac{\partial^{2} u}{\partial t^{2}} = c^{2}\nabla^{2} u$$ 1‑D travelling‑wave solution $$u(x,t)=F(x-vt)+G(x+vt)$$ Relationship among speed, wavelength, and period $$vT=\lambda,\qquad f=\frac{1}{T},\qquad \omega=2\pi f$$ Phase velocity $v{p}=\dfrac{\omega}{k}$ ; Group velocity $v{g}= \dfrac{d\omega}{dk}$ String wave speed $v=\sqrt{T/\mu}$ (T = tension, μ = linear mass density) Speed of sound $c=\sqrt{K/\rho}$ (K = adiabatic bulk modulus, ρ = density) Snell’s law (refraction) $\displaystyle n{1}\sin\theta{1}=n{2}\sin\theta{2}$ de Broglie wavelength $\lambda=\dfrac{h}{p}$ --- 🔄 Key Processes Forming a standing wave Launch two identical waves in opposite directions → they interfere. Nodes occur where the two contributions cancel (phase difference = π). Applying the superposition principle Write each contributing wave $ui$; sum: $u{\text{total}}=\sumi ui$. Check relative phase to decide constructive vs. destructive interference. Deriving phase & group velocities From dispersion relation $\omega(k)$, compute $vp=\omega/k$ and $vg=d\omega/dk$. Using Snell’s law for refraction Identify indices of refraction $n=c/v{\text{medium}}$. Apply law to find transmitted angle $\theta2$. --- 🔍 Key Comparisons Transverse vs. Longitudinal Direction of field: ⟂ vs. ‖ propagation. Typical examples: EM waves (transverse) vs. sound waves (longitudinal). Mechanical vs. Electromagnetic waves Medium: required vs. can propagate in vacuum. Energy carrier: particle‑to‑particle stress vs. field energy. Phase velocity vs. Group velocity What moves: a constant‑phase point vs. the envelope/wave packet. Equality: $vp=vg$ only for linear dispersion $\omega=ck$. Constructive vs. Destructive interference Phase difference: $0,2\pi,\dots$ → amplitude adds; $\pi,3\pi,\dots$ → amplitude cancels. --- ⚠️ Common Misunderstandings “Waves transport matter.” – Only energy, momentum, and information travel; the medium’s particles oscillate around equilibrium. “All EM waves are polarized.” – Polarization describes the orientation of the transverse fields; unpolarized light is a random mixture of polarizations. “Group velocity is always the signal speed.” – In anomalously dispersive media, $vg$ can exceed c or become negative; the true information speed is bounded by the front velocity. “Standing waves need two sources.” – A single wave reflecting from a fixed boundary creates the counter‑propagating partner automatically. --- 🧠 Mental Models / Intuition “Wave as a stadium crowd” – Each person (particle) passes the “wave” to its neighbor, illustrating local propagation without bulk transport. “Phase = snapshot” – Imagine freezing a movie at a particular frame; the pattern you see moves with $vp$. “Group = packet of mail” – The envelope carries the overall message (energy, information) and moves with $vg$. “Polarization as a swing direction” – The swing’s plane (horizontal vs. vertical) mirrors the electric field’s orientation. --- 🚩 Exceptions & Edge Cases Shock waves – Propagate faster than the local speed of sound; involve abrupt discontinuities, not described by the linear wave equation. Shear waves – Exist only in solids (or very viscous fluids); cannot travel in fluids lacking shear rigidity. Evanescent fields – Decay exponentially away from an interface (total internal reflection); not true propagating waves. Non‑linear media – Superposition fails; wave speed can depend on amplitude (e.g., high‑intensity sound). --- 📍 When to Use Which Traveling‑wave form $F(x-vt)$ – Use for problems asking for the shape of a pulse moving unchanged. Plane‑wave expression $A e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ – Ideal for scattering, reflection, and polarization calculations in homogeneous media. Standing‑wave condition $L = n\lambda/2$ (string, pipe) – Apply when a wave is confined by fixed or free boundaries. Snell’s law vs. wave‑vector matching – Use Snell for isotropic media; use vector form $k{1}\sin\theta{1}=k{2}\sin\theta{2}$ for anisotropic or dispersive cases. Group‑velocity formula $vg = d\omega/dk$ – Needed when a wave packet or pulse shape evolution is asked. --- 👀 Patterns to Recognize “$\lambda$ appears with $k$ as $k=2\pi/\lambda$” – Whenever a wavenumber shows up, think wavelength conversion. “$vT=\lambda$” – Spot this relation in any speed‑frequency‑wavelength problem. “Node at fixed end, antinode at free end” – Recognize boundary‑condition patterns for standing waves on strings or air columns. “Linear dispersion → $vp=vg$ → unchanged pulse shape” – Look for a simple $\omega = ck$ relation. “Phase difference of $n\pi$ → interference type” – Quick test for constructive (even $n$) vs. destructive (odd $n$) interference. --- 🗂️ Exam Traps Confusing $vp$ with $vg$ – A common distractor gives the phase velocity when the question asks for signal speed (group velocity). Mis‑applying Snell’s law to sound in moving air – The law assumes stationary media; moving media require Doppler‑shifted frequency instead. Assuming all EM waves are transverse – Gravitational waves are transverse‑tensor waves, not EM; also, near‑field components can have longitudinal electric fields. Using $v=\sqrt{T/\mu}$ for a rope with one free end – That formula applies only to a uniform tensioned string with fixed boundaries for standing‑wave modes. Treating a shock wave as a simple sinusoid – Shock fronts are discontinuous; linear wave equations do not describe them. ---