Oscillations and waves

Introduction to mechanical waves and the wave equation

A mechanical wave carries a disturbance and energy through matter; medium particles oscillate about equilibrium.

A pulse travels along a rope while each point moves up and down near its original position. The wave transports the disturbance, not the material itself. Christiaan Huygens developed a wave view in which disturbance advances through wavefronts, providing historical context for this description.

Transverse and longitudinal waves

In a transverse wave, oscillation is perpendicular to propagation (for example, a stretched string); in a longitudinal wave, it is parallel (for example, sound in air). Mechanical waves require a medium, and their speed depends on its properties.

Quick check

For a sinusoidal wave traveling in the positive direction, points separated by $\Delta x$ differ in phase by $\Delta\phi=2\pi\Delta x/\lambda$. At a fixed location the medium oscillates at the source frequency; at a fixed time the spatial profile repeats every wavelength. In one medium, a changed source frequency changes wavelength according to $v=\lambda f$, while the medium sets the propagation speed.

When applying a model, state which quantities are held fixed and choose the appropriate reference frame. Keeping units beside numerical values helps prevent confusion between quantities or between measured and predicted values. After calculating, check dimensions, signs, and limiting cases: the result should fit the assumptions, such as small oscillations, a uniform medium, or negligible resistance. If real conditions differ, explain the discrepancy instead of applying a formula mechanically.

A useful physical check is to substitute special cases into the relation: equilibrium, maximum displacement, one full cycle, or no relative motion. These cases often make a quantity vanish or reach an extreme, exposing sign errors and confusion between period and frequency. When comparing experiments, change one condition at a time so that the cause of a changed result is clear.