Superposition and Standing Waves: Quick Revision

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

Adding waves

• Superposition: where waves meet, the displacements add, $y = y_1 + y_2$.
• Two equal waves with a phase difference $\phi$: $A = 2a\cos(\phi/2)$.
• Unequal waves: $A^2 = a_1^2 + a_2^2 + 2a_1a_2\cos\phi$.
• In phase ($\phi = 0$): they help each other. Out of phase ($\phi = \pi$): they cancel.
• Path difference to phase difference: $\phi = \dfrac{2\pi}{\lambda}\Delta x$. Intensity $I \propto A^2$.

Reflection and standing waves

• Fixed end: the wave comes back inverted. Free end: it comes back upright.
• Two equal waves going opposite ways add to $y = 2a\sin kx\cos\omega t$.
• Nodes never move. Next nodes are $\lambda/2$ apart, and a node is $\lambda/4$ from the nearest antinode.

Strings and pipes

• String tied at both ends: $f_n = \dfrac{nv}{2L}$, every harmonic, with $v = \sqrt{T/\mu}$.
• Pipe open at both ends: $f_n = \dfrac{nv}{2L}$, every harmonic.
• Pipe closed at one end: $f = \dfrac{(2n - 1)v}{4L}$, odd harmonics only.
• A closed end is a displacement node; an open end is an antinode.

A worked example

A string $0.65\ \text{m}$ long has a tension of $90\ \text{N}$ and $2.5\ \text{g}$ of mass in each metre. Find its fundamental.

1. Speed. $v = \sqrt{\dfrac{90\ \text{N}}{0.0025\ \text{kg/m}}} = 189.7\ \text{m/s}$.
2. Wavelength. In the fundamental, $\lambda = 2L = 1.3\ \text{m}$.
3. Frequency. $f_1 = \dfrac{189.7\ \text{m/s}}{1.3\ \text{m}} = 146\ \text{Hz}$.

Practice with new numbers

A program to try

The program works out the example above, and the first three notes of a closed pipe.

Mistakes to avoid

• Adding amplitudes when the waves are not in phase. Use $2a\cos(\phi/2)$.
• Node spacing of $\lambda$. It is $\lambda/2$.
• Even harmonics from a closed pipe. It has odd ones only.
• $f \propto T$. It is $f \propto \sqrt{T}$.

Check yourself

1. Two equal waves meet with a phase difference of $180^\circ$. What is the amplitude of the sum?

2. Next nodes of a standing wave are $40\ \text{cm}$ apart. What is the wavelength?

3. A string has a fundamental of $200\ \text{Hz}$. What is its third harmonic?

4. Which has the lower fundamental: an open pipe or a closed pipe of the same length?