Oscillations and Waves

Motion that repeats after a fixed time interval is periodic. Harmonic motion is a more specific sinusoidal form of periodic motion; simple harmonic motion (SHM) occurs when the restoring acceleration is proportional to displacement and directed toward equilibrium.

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

The position ($x$), velocity ($v$), and acceleration ($a$) of an object in SHM as a function of time ($t$) are described by sinusoidal functions.

In an ideal SHM system (no friction), the total mechanical energy ($E = K + U$) is conserved. Energy continuously transforms between kinetic and potential forms.

• Potential Energy ($U$): Maximum at the extreme positions ($\pm A$), zero at equilibrium. $U = \frac{1}{2}kx^2$.
• Kinetic Energy ($K$): Maximum at equilibrium ($v_{max} = A\omega$), zero at the extremes. $K = \frac{1}{2}mv^2$.
• Total Energy ($E$): Constant. $E = \frac{1}{2}kA^2$.

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 ($\omega_0$). If the frequency of the driving force matches the natural frequency ($\omega_{driven} = \omega_0$), the amplitude of the oscillation can grow tremendously. This phenomenon is called Resonance.

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.

• 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.

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

A 1D harmonic wave traveling in the positive x-direction can be described by a wave function $y(x,t)$, which gives the transverse displacement $y$ of a particle at position $x$ and time $t$.

Waves transport energy. The power ($P$) 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 ($I$), which is the power transmitted across a unit area perpendicular to the direction of propagation.

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

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.

Two coherent waves of the same frequency traveling in opposite directions can form a standing-wave pattern. In a bounded system such as a string fixed at both ends, the boundary conditions permit only discrete normal-mode frequencies; these resonant frequencies form the allowed harmonics.

A Simple Pendulum consists of a point mass ($m$) suspended by a massless, unstretchable string of length $L$. For sufficiently small angular displacements, $\sin\theta\approx\theta$ (with $\theta$ in radians), so the restoring torque is approximately proportional to angular displacement and the pendulum behaves as SHM. The approximation becomes progressively less accurate as amplitude increases.

A Physical Pendulum is any real, rigid object swinging from a pivot point. Its period depends on its Moment of Inertia ($I$) about the pivot and the distance ($d$) from the pivot to its center of gravity.

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.

• OpenStax University Physics Volume 1 — Oscillations
• OpenStax University Physics Volume 1 — Traveling Waves
• BIPM — SI Brochure
• NIST — CODATA Values of the Fundamental Physical Constants