Oscillations and Waves

Oscillations (vibrations) are back-and-forth motions about an equilibrium position. Waves are propagating disturbances that carry energy from one place to another without permanently moving the medium itself. Understanding these phenomena is critical for structural engineering (earthquake resistance, wind-induced vibrations) and acoustics.

Oscillations and Simple Harmonic Motion (SHM)

Oscillations and Simple Harmonic Motion (SHM) Concepts

Any motion that repeats itself at regular intervals is called periodic or harmonic motion. The simplest and most important type is Simple Harmonic Motion (SHM).

Simple Harmonic Motion (SHM)

Motion caused by a restoring force that is directly proportional to the displacement from equilibrium and always directed towards that equilibrium position.
Fx=−kx F_x = -kx
(This is Hooke's Law, where kk is the "spring constant" or "stiffness" of the system).

Oscillations and Simple Harmonic Motion (SHM) Concepts

Because Fx=maxF_x = ma_x, the acceleration in SHM is also proportional to displacement: ax=−(k/m)xa_x = -(k/m)x. This differential equation describes a system where the acceleration is always opposite to the position, leading to a sinusoidal oscillation.

Kinematics of SHM

Kinematics of SHM Concepts

The position (xx), velocity (vv), and acceleration (aa) of an object in SHM as a function of time (tt) are described by sinusoidal functions.

SHM Equations

Position:
x(t)=Acos⁡(ωt+ϕ) x(t) = A \cos(\omega t + \phi)
Velocity:
v(t)=−Aωsin⁡(ωt+ϕ) v(t) = -A\omega \sin(\omega t + \phi)
Acceleration:
a(t)=−Aω2cos⁡(ωt+ϕ)=−ω2x(t) a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)
Where:
  • AA is the Amplitude (maximum displacement from equilibrium).
  • ω\omega is the Angular Frequency (rad/s\text{rad/s}).
  • ϕ\phi is the Phase Constant (determined by initial conditions at t=0t=0).

Important

The angular frequency ω\omega is determined entirely by the physical properties of the system (mass and stiffness), not by how the oscillation is started (amplitude).
For a mass-spring system:
ω=km \omega = \sqrt{\frac{k}{m}}
For a simple pendulum (for small angles θ\theta):
ω=gL \omega = \sqrt{\frac{g}{L}}

Energy in SHM

Energy in SHM Concepts

In an ideal SHM system (no friction), the total mechanical energy (E=K+UE = K + U) is conserved. Energy continuously transforms between kinetic and potential forms.
  • Potential Energy (UU): Maximum at the extreme positions (±A\pm A), zero at equilibrium. U=12kx2U = \frac{1}{2}kx^2.
  • Kinetic Energy (KK): Maximum at equilibrium (vmax=Aωv_{max} = A\omega), zero at the extremes. K=12mv2K = \frac{1}{2}mv^2.
  • Total Energy (EE): Constant. E=12kA2E = \frac{1}{2}kA^2.

Damped and Driven Oscillations

Damped and Driven Oscillations Concepts

Real oscillators experience friction (damping), which removes energy and causes the amplitude to decay over time.
If an external periodic force is applied to the system, it is a driven oscillation. Every system has a natural frequency (ω0\omega_0). If the frequency of the driving force matches the natural frequency (ωdriven=ω0\omega_{driven} = \omega_0), the amplitude of the oscillation can grow tremendously. This phenomenon is called Resonance.

The Danger of Resonance

Resonance is often destructive in engineering. The Tacoma Narrows Bridge collapse (1940) and the swaying of skyscrapers in high winds are classic examples of wind-induced resonance that engineers must meticulously design against.

Mechanical Waves

Mechanical Waves Concepts

While an oscillation is a local vibration, a wave is a disturbance that travels through a medium, transferring energy and momentum from point A to point B without transporting the matter of the medium itself.

Types of Waves

Types of Waves Concepts

  • Transverse Waves: The particles of the medium oscillate perpendicular to the direction of wave propagation. Examples: Waves on a string, light (electromagnetic waves).
    • Longitudinal Waves: The particles oscillate parallel to the direction of propagation (compressions and rarefactions). Examples: Sound waves in air or water, P-waves in earthquakes.

Wave Properties

Wave Properties Concepts

A periodic wave has a consistent shape that repeats in both space and time.

Wavelength (λ\lambda)

The spatial distance over which the wave shape repeats itself (e.g., crest to crest). The SI unit is meters (m).

Frequency (ff)

The number of complete wave cycles that pass a fixed point per unit time. The SI unit is Hertz (Hz), where 1 Hz=1 cycle/s1 \text{ Hz} = 1 \text{ cycle/s}.

Period (TT)

The time required for one complete cycle to pass a fixed point. It is the reciprocal of frequency.
T=1f T = \frac{1}{f}

Wave Speed ($v$)

The speed at which the wave disturbance propagates through the medium.
v=fλ=λT v = f \lambda = \frac{\lambda}{T}

Important

The wave speed vv is determined strictly by the physical properties of the medium (its elasticity and inertia), not by the source creating the wave. For example, the speed of sound in air depends on temperature and pressure, not on how loud you yell.

The Mathematical Description of a Wave

The Mathematical Description of a Wave Concepts

A 1D harmonic wave traveling in the positive x-direction can be described by a wave function y(x,t)y(x,t), which gives the transverse displacement yy of a particle at position xx and time tt.
y(x,t)=Asin⁡(kx−ωt) y(x,t) = A \sin(kx - \omega t)

The Mathematical Description of a Wave Concepts

Where:
  • AA is the amplitude.
  • kk is the wave number (k=2π/λk = 2\pi / \lambda).
  • ω\omega is the angular frequency (ω=2πf\omega = 2\pi f).

Energy and Intensity of Waves

Energy and Intensity of Waves Concepts

Waves transport energy. The power (PP) transmitted by a harmonic wave on a string is proportional to the square of its amplitude and the square of its frequency.
For 3D waves (like sound or light), we describe the energy flow using Intensity (II), which is the power transmitted across a unit area perpendicular to the direction of propagation.
I=PA I = \frac{P}{A}

Energy and Intensity of Waves Concepts

For a point source emitting energy equally in all directions (spherical waves), the intensity decreases as the inverse square of the distance (rr) from the source: I∝1/r2I \propto 1/r^2.

Interference and Standing Waves

Interference and Standing Waves Concepts

When two or more waves travel through the same medium simultaneously, they obey the Principle of Superposition: The net displacement of the medium at any point is the algebraic sum of the individual wave displacements at that point.
  • Constructive Interference: Waves arrive "in phase" (crest meets crest), resulting in a larger combined amplitude.
  • Destructive Interference: Waves arrive "out of phase" (crest meets trough), resulting in a smaller or zero combined amplitude.
When two identical waves traveling in opposite directions interfere, they create a Standing Wave. The wave appears to vibrate in place rather than travel.

Nodes and Antinodes

  • Nodes: Points on a standing wave that never move (complete destructive interference).
  • Antinodes: Points that oscillate with maximum amplitude (constructive interference).

Interference and Standing Waves Concepts

Standing waves only form at specific frequencies called harmonics or resonant frequencies, which depend on the boundary conditions of the medium (e.g., a guitar string fixed at both ends).

The Simple and Physical Pendulum

Pendulum Mechanics

A Simple Pendulum consists of a point mass (mm) suspended by a massless, unstretchable string of length LL. For small angular displacements (θ<15∘\theta < 15^\circ), the restoring force (a component of gravity) is approximately proportional to the displacement, resulting in Simple Harmonic Motion.
A Physical Pendulum is any real, rigid object swinging from a pivot point. Its period depends on its Moment of Inertia (II) about the pivot and the distance (dd) from the pivot to its center of gravity.

Period of a Physical Pendulum

Calculates the time required for one complete swing of a rigid body pendulum.

$$ T = 2\pi \sqrt{\frac{I}{mgd}} $$

Sound Waves and the Doppler Effect

The Doppler Effect

Sound waves are longitudinal mechanical waves. When a source of sound and an observer are in relative motion, the observer perceives a frequency different from the one emitted by the source. This is the Doppler Effect.
If the source and observer are moving towards each other, the perceived frequency increases (higher pitch). If they are moving apart, the perceived frequency decreases.

The Doppler Effect Equation

Calculates the observed frequency of a wave due to relative motion.

$$ f' = f \left( \frac{v \pm v_o}{v \mp v_s} \right) $$
Key Takeaways
  • Simple Harmonic Motion (SHM) occurs when a restoring force is proportional to displacement (F=−kxF = -kx). It is described by sinusoidal functions (x(t)=Acos⁡(ωt)x(t) = A\cos(\omega t)).
  • The angular frequency (ω\omega) of SHM depends on the system's mass and stiffness, not amplitude. Resonance occurs when a driving frequency matches this natural frequency.
  • Waves transfer energy through a medium. They are characterized by wavelength (λ\lambda), frequency (ff), and wave speed (v=fλv = f\lambda). Wave speed depends only on the medium.
  • Transverse waves oscillate perpendicular to propagation; Longitudinal waves oscillate parallel.
  • Interference (superposition) leads to phenomena like Standing Waves, characterized by nodes (zero amplitude) and antinodes (max amplitude).
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