Waves

A physical perturbation carrying energy (not matter) through space and time.

The Wave Function (1D)

$f(x,t)=A\sin \left[2\pi \left(\frac{x}{\lambda }-\frac{t}{T}\right)\right]=A\sin [kx-\omega t]$

Key Relationships:

• Wavenumber: $k=\frac{2\pi }{\lambda }$
• Angular Frequency: $\omega =2\pi \nu =\frac{2\pi }{T}$
• Phase Velocity: $v_{\varphi }=\lambda \nu =\frac{\omega }{k}$

To be a wave, it must satisfy the Wave Equation: ${\nabla }^{2}f=\frac{1}{v^2}\frac{{\partial }^{2}f}{\partial t^2}$

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Wave Packets & Velocities

Real waves are often “packets” (localized), not infinite sinusoids.

Phase vs. Group Velocity

• Phase Velocity ($v_{\varphi }$): Speed of a single peak. $v_{\varphi }=\frac{\omega }{k}$
• Group Velocity ($v_g$): Speed of the “envelope” (information). $v_g=\frac{\partial \omega }{\partial k}$
• Dispersion: If $v_{\varphi }$ depends on frequency, the packet spreads out over time. If all frequencies move at the same speed, it is non-dispersive.

Bandwidth Theorem

You cannot have a wave perfectly localized in both space and frequency. $\Delta x\Delta k\ge 1\text{and}\Delta t\Delta \nu \ge 1$ A tighter packet requires more frequencies to build.

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Standing Waves (Strings)

Waves trapped between two boundaries (fixed ends).

Harmonics

For a string of length $L$ fixed at both ends ($A(0)=A(L)=0$):

• Wavelength: ${\lambda }_n=\frac{2L}{n}$
• Frequency: ${\nu }_n=\frac{v_{\varphi }}{2L}n=n{\nu }_1$ (for $n=1,2,\mathrm{3...}$)
• Nodes: Points of zero amplitude. There are $n-1$ nodes (excluding ends).

String Velocity

The speed of the wave depends on tension ($T$) and linear mass density ($\mu$): $v_{\varphi }=\sqrt{\frac{T}{\mu }}$