Oscillations & Waves – Comprehensive MIT‑Level Study Notes
(Physics, Advanced Undergraduate / First‑Year Graduate)
Audience – Students preparing for the MHT CET (or similar competitive exams) who wish to master the subject at a rigorous, research‑ready level.
Prerequisites – Calculus (including multivariable), Linear Algebra, Differential Equations, Elementary Mechanics.
1. Introduction
| Concept | Definition | Key Feature |
|---|---|---|
| Oscillation | Periodic motion of a system about an equilibrium point | Restoring force ∝ displacement (linear regime) |
| Wave | Propagation of a disturbance through a medium | Linear superposition, dispersion relation |
| Simple Harmonic Oscillator (SHO) | Ideal system with linear restoring force | Exact analytic solution |
| Damped Oscillator | SHO with friction/momentum loss | Decay of amplitude; energy dissipation |
| Driven Oscillator | Forced SHO with external periodic force | Resonance, steady‑state response |
| Wave Equation | Partial differential equation linking temporal and spatial derivatives | Linear partial differential equation, hyperbolic |
The goal is to derive each phenomenon from first principles, develop a rigorous mathematical framework, and understand edge cases that often appear in exam problems.
2. Lagrangian & Hamiltonian Formalism for Oscillators
2.1. From Newton to Lagrange
Consider a particle of mass (m) on a 1‑D coordinate (x(t)) with potential (U(x)). The Lagrangian:
[ \mathcal{L}(x,\dot x,t)=T-U=\frac12 m\dot x^2-U(x). ]
Euler–Lagrange equation:
[ \frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot x}\right)-\frac{\partial\mathcal{L}}{\partial x}=0 ;;\Longrightarrow;; m\ddot{x}+U'(x)=0. ]
For a linear restoring force (U(x)=\frac12 kx^2), we recover the SHO equation.
2.2. Hamiltonian & Canonical Equations
Define the conjugate momentum (p=m\dot x). The Hamiltonian
[ H(x,p)=p\dot x-\mathcal{L}= \frac{p^2}{2m}+U(x). ]
Canonical equations:
[ \dot x=\frac{\partial H}{\partial p}=\frac{p}{m},\qquad \dot p=-\frac{\partial H}{\partial x}=-U'(x). ]
These first‑order equations are often more convenient for multi‑degree‑of‑freedom problems.
3. Simple Harmonic Motion (SHM)
3.1. Derivation
For (U(x)=\frac12 kx^2),
[ m\ddot x + kx = 0 \quad \Rightarrow \quad \ddot x + \omega^2 x = 0,\quad \omega=\sqrt{\frac{k}{m}}. ]
General solution:
[ x(t)=A\cos(\omega t)+B\sin(\omega t)=C\cos(\omega t-\delta). ]
3.2. Energy Conservation
Total energy:
[ E = \frac12 m\dot x^2 + \frac12 kx^2 = \text{constant}. ]
The energy partitions sinusoidally between kinetic and potential.
3.3. Normal Modes & Coupled Oscillators
For (N) masses (m_i) connected by springs (k_{ij}), the equations of motion become
[ m_i\ddot x_i + \sum_j k_{ij}(x_i-x_j)=0. ]
In matrix form:
[ \mathbf{M}\ddot{\mathbf{x}} + \mathbf{K}\mathbf{x}=0, \quad \mathbf{M}=\text{diag}(m_i),; \mathbf{K}\text{ symmetric}. ]
Assume (x_i(t)=a_i e^{i\omega t}). Then
[ (\mathbf{K} - \omega^2 \mathbf{M})\mathbf{a}=0. ]
Non‑trivial solutions exist when
[ \det(\mathbf{K}-\omega^2 \mathbf{M})=0. ]
Each root (\omega_n) gives a normal mode vector (\mathbf{a}_n). Orthogonality follows from the symmetry of (\mathbf{M}) and (\mathbf{K}).
Edge Cases
- Singular mass matrix ((\det\mathbf{M}=0)): implies a free coordinate (e.g., a massless spring). Leads to constraints and reduced‑dimensional dynamics.
- Negative stiffness ((k<0)): Unstable equilibrium; solutions grow exponentially.
4. Damped Oscillators
4.1. Equation of Motion
Add a linear damping force (-b\dot x):
[ m\ddot x + b\dot x + kx =0. ]
Characteristic equation:
[ m r^2 + b r + k = 0 \quad \Longrightarrow \quad r = \frac{-b \pm \sqrt{b^2-4mk}}{2m}. ]
Define damping ratio (\zeta = \frac{b}{2\sqrt{mk}}).
| Regime | Solution |
|---|---|
| (\zeta>1) (over‑damped) | (x(t)=C_1 e^{-\lambda_1 t}+C_2 e^{-\ |