Wave Properties and the Wave Equation

Do the stadium wave: you stand up and sit down, yet the pattern races around the stadium. Nothing material travels, what moves is a wave: a disturbance that transfers energy without transferring matter.

Types of waves

The wave equation governing this motion is:

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

Wave Parameters and Energy

A travelling wave is given by $y(x,t) = A\sin(kx - \omega t + \phi)$. Here, Amplitude ($A$) is max displacement, wave number ($k = 2\pi/\lambda$), and angular frequency ($\omega = 2\pi f$).

Speed is $v = f\lambda = \omega/k$. Intensity ($I$) is power per unit area:

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

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

Common pitfall: Wave speed $v$ is a property of the medium (tension, density), not the source. Shaking a rope harder changes $A$, not $v$.