Travelling Waves and Wave Speed: Quick Revision

A brief note on Travelling Waves and Wave Speed. Read it before a test. Try each question at the end before you open its answer.

Kinds of wave

• A wave carries energy and a pattern. Each particle of the medium only moves about its own place.
• Transverse: the particles move across the direction of travel. Crests and troughs.
• Longitudinal: the particles move along the direction of travel. Compressions and rarefactions.
• Mechanical waves need a medium. Gases and liquids carry only longitudinal waves in their bulk.

The equation of a wave

• Towards $+x$: $y = a\sin(kx - \omega t + \phi)$. Towards $-x$: $kx + \omega t$.
• $k = \dfrac{2\pi}{\lambda}$, in rad/m. $\omega = \dfrac{2\pi}{T} = 2\pi f$, in rad/s.
• Wave speed: $v = \dfrac{\omega}{k} = f\lambda$.
• Particle velocity: $u = -a\omega\cos(kx - \omega t + \phi)$, at most $a\omega$.
• Phase difference between two points: $\Delta\phi = k\Delta x$.

What sets the speed

• String: $v = \sqrt{\dfrac{T}{\mu}}$, with $\mu$ in kg/m.
• Fluid: $v = \sqrt{\dfrac{B}{\rho}}$. Solid rod: $v = \sqrt{\dfrac{Y}{\rho}}$.
• Gas: $v = \sqrt{\dfrac{\gamma p}{\rho}} = \sqrt{\dfrac{\gamma R T}{M}}$ (Laplace). Newton's $\sqrt{p/\rho}$ is too low.
• $v \propto \sqrt{T}$, in kelvin. Pressure alone does not change it. Damp air carries sound a little faster.

A worked example

A wave is $y = 0.02\sin(4x - 48t)$, in SI units, on a rope with $\mu = 0.25\ \text{kg/m}$. Find its wavelength, speed and the tension in the rope.

1. Wavelength. $\lambda = \dfrac{2\pi}{4\ \text{rad/m}} = 1.57\ \text{m}$.
2. Speed. $v = \dfrac{48\ \text{rad/s}}{4\ \text{rad/m}} = 12\ \text{m/s}$.
3. Tension. $v^2 = T/\mu$, and we want $T$, so $T = \mu v^2 = 0.25\ \text{kg/m} \times (12\ \text{m/s})^2 = 36\ \text{N}$.

Practice with new numbers

A program to try

The program works through the example above.

Mistakes to avoid

• Wave speed and particle speed. They are different. One is $\omega/k$, the other at most $a\omega$.
• $k$ and $\lambda$. $k = 2\pi/\lambda$. A larger $k$ is a shorter wave.
• Grams in $\mu$. Use kg/m.
• Celsius in $\sqrt{T}$. Use kelvin.

Check yourself

1. Is a wave on a plucked string transverse or longitudinal?

2. A wave has $k = 2\pi\ \text{rad/m}$ and $\omega = 10\pi\ \text{rad/s}$. Find its speed.

3. The tension in a string is made nine times as large. What happens to the wave speed?

4. Why does pressure alone not change the speed of sound in a gas?