Table of Contents
- Introduction
- What Are Vibrations?
- The Simple Harmonic Oscillator
- Damped Oscillations
- Driven Oscillations and Resonance
- Coupled Oscillators
- What Are Waves?
- Mechanical Waves: Transverse and Longitudinal
- The Wave Equation
- Wave Parameters: Wavelength, Frequency, Speed, Amplitude
- Superposition Principle and Interference
- Standing Waves and Normal Modes
- Energy Transport in Waves
- Vibrations and Waves in Real Systems
- Conclusion
1. Introduction
Vibrations and waves are among the most fundamental phenomena in classical physics. From the vibration of a tuning fork to the ripples on a pond, from sound in air to light in space — oscillatory behavior lies at the heart of nature.
This article presents a comprehensive overview of classical vibrations and wave phenomena, providing mathematical foundations and physical insights for both simple and complex systems.
2. What Are Vibrations?
Vibrations refer to periodic motion of a body or system about an equilibrium position. The most basic type is simple harmonic motion (SHM), where the restoring force is proportional to displacement.
This type of motion can be found in:
- A mass on a spring
- A pendulum (for small angles)
- Atoms in a solid lattice
3. The Simple Harmonic Oscillator
A system obeys SHM when the force satisfies Hooke’s law:
\[
F = -kx
\]
According to Newton’s second law:
\[
m\frac{d^2x}{dt^2} = -kx
\Rightarrow \frac{d^2x}{dt^2} + \omega^2 x = 0
\]
Where \( \omega = \sqrt{k/m} \) is the angular frequency.
The solution is:
\[
x(t) = A\cos(\omega t + \phi)
\]
- \( A \): amplitude
- \( \phi \): phase constant
- \( \omega \): determines the frequency \( f = \omega / (2\pi) \)
The motion is sinusoidal, periodic, and conservative (energy oscillates between kinetic and potential forms).
4. Damped Oscillations
In real systems, friction or resistance causes the amplitude to decay over time. The motion becomes:
\[
m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0
\]
Where \( b \) is the damping coefficient. Solution depends on damping:
- Underdamped: oscillations decay exponentially
- Critically damped: fastest return to equilibrium, no oscillation
- Overdamped: slower return without oscillation
Underdamped solution:
\[
x(t) = Ae^{-\gamma t} \cos(\omega’ t + \phi)
\]
Where \( \gamma = b/(2m) \), \( \omega’ = \sqrt{\omega^2 – \gamma^2} \)
5. Driven Oscillations and Resonance
Applying a periodic force:
\[
F(t) = F_0 \cos(\omega_d t)
\]
The system responds with oscillations at the driving frequency \( \omega_d \). Resonance occurs when:
\[
\omega_d \approx \omega_0
\]
At resonance, the amplitude is maximum. This phenomenon is central to:
- Tuning radios
- Acoustic cavities
- Mechanical failure (e.g., Tacoma Narrows Bridge)
6. Coupled Oscillators
Two or more oscillators connected together (e.g., pendulums connected by a spring) form coupled systems.
- Exhibits normal modes — patterns where all components oscillate at fixed ratios
- Frequencies are found by solving a system of differential equations
- Important in molecular vibrations, musical instruments, and crystal lattices
7. What Are Waves?
Waves are disturbances that propagate through space and time, carrying energy but not matter.
Each point in a wave medium undergoes oscillatory motion. Unlike pure vibrations (localized), waves transport oscillations over a region.
Types:
- Mechanical (sound, water)
- Electromagnetic (light, radio)
- Matter waves (quantum)
8. Mechanical Waves: Transverse and Longitudinal
- Transverse: Oscillations are perpendicular to propagation (e.g., string vibrations)
- Longitudinal: Oscillations are parallel (e.g., sound waves in air)
Both are governed by the same wave principles but differ in motion and medium.
9. The Wave Equation
For a uniform string under tension:
\[
\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}
\]
Where \( y(x,t) \) is the transverse displacement, and \( v \) is the wave speed.
General solution:
\[
y(x, t) = f(x – vt) + g(x + vt)
\]
Represents right- and left-traveling waves.
10. Wave Parameters: Wavelength, Frequency, Speed, Amplitude
Key quantities:
- Amplitude \( A \): max displacement
- Wavelength \( \lambda \): distance between crests
- Frequency \( f \): oscillations per second
- Wave speed \( v \): speed of propagation
Relationship:
\[
v = f \lambda
\]
These parameters define how fast, how strong, and how regular the wave is.
11. Superposition Principle and Interference
When two or more waves overlap, their displacements add algebraically:
\[
y_{\text{net}}(x, t) = y_1(x, t) + y_2(x, t)
\]
This leads to:
- Constructive interference (amplitudes add)
- Destructive interference (amplitudes cancel)
This is the basis for:
- Standing waves
- Beats
- Interference fringes in optics
12. Standing Waves and Normal Modes
When a wave reflects and overlaps with itself, standing waves form:
\[
y(x, t) = 2A \sin(kx) \cos(\omega t)
\]
Nodes (zero displacement) and antinodes (maximum displacement) appear at fixed locations.
Allowed frequencies:
\[
f_n = \frac{n v}{2L}, \quad n = 1, 2, 3, …
\]
Important in:
- Musical instruments
- Vibrating strings and air columns
- Quantum systems
13. Energy Transport in Waves
For a string:
- Kinetic energy from motion of elements
- Potential energy from tension and curvature
Total power transmitted by a wave:
\[
P = \frac{1}{2} \mu \omega^2 A^2 v
\]
Where \( \mu \) is mass per unit length. Energy increases with square of amplitude and frequency.
14. Vibrations and Waves in Real Systems
Real-world examples combine both vibration and wave phenomena:
- Earthquakes: Seismic vibrations and wavefronts
- Guitar strings: Standing waves and resonance
- Bridges and buildings: Resonant vibrations (engineering importance)
- Sound in air: Longitudinal waves from vibrating sources
15. Conclusion
Vibrations and waves lie at the heart of classical physics. Vibrations describe local, periodic motion, while waves represent the transmission of such motion across space.
Their mathematical treatment reveals deep symmetry and structure — from musical harmonics to electromagnetic theory, and from classical mechanics to quantum fields.
Mastery of classical waves and vibrations is essential not just for physics, but for understanding the world around us — and the universe beyond.
