Published by Patrick Mutisya · 14 days ago
A progressive (or travelling) wave is a disturbance that moves through a medium, carrying energy from one point to another without permanent displacement of the medium itself. The wave profile at any instant is the same shape, but it is shifted in space as time progresses.
For a sinusoidal wave travelling in the +x direction the displacement \$y\$ of the medium can be written as
\$y(x,t)=A\cos\!\left(kx-\omega t+\phi\right)\$
where
Energy is stored in the medium as both kinetic and potential energy. For a transverse wave on a string of linear mass density \$\mu\$, the instantaneous energy per unit length is
\$\$\mathcal{E}(x,t)=\frac{1}{2}\mu\left(\frac{\partial y}{\partial t}\right)^{2}
+\frac{1}{2}T\left(\frac{\partial y}{\partial x}\right)^{2}\$\$
where \$T\$ is the tension in the string. The two terms represent kinetic and elastic (potential) energy respectively.
For a harmonic wave the time‑averaged energy per unit length \$\langle\mathcal{E}\rangle\$ is
\$\langle\mathcal{E}\rangle=\frac{1}{2}\mu\omega^{2}A^{2}\$
The power transmitted along the string is the product of the wave speed \$v\$ and the average energy density:
\$\langle P\rangle = v\langle\mathcal{E}\rangle = \frac{1}{2}\mu v\omega^{2}A^{2}\$
Intensity \$I\$ (power per unit area) is defined analogously for waves on a surface (e.g., sound or light):
\$I = \frac{\langle P\rangle}{A_{\text{cross}}}\$
| Quantity | Expression | Physical meaning |
|---|---|---|
| Wave speed | \$v = \dfrac{\omega}{k} = f\lambda\$ | Speed at which the wave profile travels |
| Average kinetic energy density | \$\langle K\rangle = \dfrac{1}{4}\mu\omega^{2}A^{2}\$ | Mean kinetic energy per unit length |
| Average potential energy density | \$\langle U\rangle = \dfrac{1}{4}\mu\omega^{2}A^{2}\$ | Mean elastic energy per unit length |
| Average total energy density | \$\langle\mathcal{E}\rangle = \langle K\rangle + \langle U\rangle = \dfrac{1}{2}\mu\omega^{2}A^{2}\$ | Sum of kinetic and potential contributions |
| Average power transmitted | \$\langle P\rangle = v\langle\mathcal{E}\rangle = \dfrac{1}{2}\mu v\omega^{2}A^{2}\$ | Rate at which energy crosses a point |
Consider a string fixed at one end and driven sinusoidally at the other. The disturbance travels along the string, and each element of the string oscillates about its equilibrium position. Although the particles of the string do not travel with the wave, they repeatedly gain and lose kinetic and potential energy. The net effect is a continuous flow of energy from the driver to the far end, where it may be dissipated or reflected.