Wave Properties and the Wave Equation

Do the stadium wave: you stand up and sit down, yet "the wave" races around the stadium. Nothing material travels — what moves is a pattern, carrying energy and information while the medium merely wiggles in place. That decoupling of stuff from signal is the essence of every wave.

A wave is a disturbance that transfers energy without transferring matter.

The wave equation:

$$\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}$$

Travelling wave solution:

$$y(x,t) = A\sin(kx - \omega t + \phi)$$

Types of waves

• Transverse: displacement $\perp$ propagation (light, string waves).
• Longitudinal: displacement $\parallel$ propagation (sound).

Energy in a wave — The intensity (power per unit area) of a wave is:

$$I = \frac{1}{2}\rho v \omega^2 A^2$$

Intensity drops as $1/r^{2}$ for a point source (inverse-square law).

Physics link: The wave equation appears everywhere — strings, sound, light, quantum mechanics. The solutions always have the same structure: oscillation in space and time.

Common pitfall: Wave speed is a property of the medium (tension, density, temperature), not of the source. Shaking a rope harder makes taller waves, not faster ones — only changing the rope itself changes $v$.