Is the “loudness” (intensity level) of the sound measured in decibels (dB)

Recall that for SHM energy

Total Energy = PE_MAX = ½kA²

so energy is proportional to A²

All mechanical waves consist of particles undergoing SHM, and so it can be shown that :

Standing waves

The wave is trapped in a finite region of space (e.g. fixed boundaries) and the energy is stored in the extended oscillations;

Standing waves form through the superposition of identical waves travelling in opposite directions.

• Acoustics (e.g. standing waves on a violin string or in wind instruments);

• Quantum Physics (e.g. 2D electron standing waves in Bohr’s atomic model);

• Electromagnetism (e.g. standing waves in microwave ovens, radio antennas);

• Optics (e.g. optical cavities in lasers, interferometers);

• Engineering (e.g. mechanical waves on bridges);

If two or more travelling waves are moving through a region of space, the resultant value of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves

Two travelling waves can pass through each other without being destroyed or altered

The process and the effect of the combination of separate waves in the same region of space is called interference.

Derive the formula for: Superposition of two waves travelling in opposite directions

Nodes are defined where 2A sin(kx) = 0, and the particle at this x does not oscillate (i.e. y = 0 at any time t).

Antinodes are defined where sin(kx) = ±1, where the particle oscillates from y = 2A to y = -2A .

A Node occurs where the two travelling waves have the same magnitude of displacement, but the displacements are in opposite directions. Net displacement is zero at that point.

An Antinode occurs where the standing wave vibrates at maximum amplitude

• The distance between two nodes is 𝝀/𝟐

• The distance between a node and an antinode is 𝝀/𝟒

Boundary conditions: For strings, the ends must be nodes

The boundary conditions determine a set of modes of vibrations

Each mode of vibration is defined by the frequency of the standing wave

Modes of vibrations of a string

The first mode of vibration:

• is called the fundamental

• has one antinode

• has the lowest frequency, f₁

• has longest wavelength, λ₁

Subsequent modes of vibration:

• are called overtones

• have higher frequencies, fn

• have shorter wavelenghts, λn

The natural frequencies of a string attached to both ends are given by:

Standing waves in pipes

• Standing waves can be set up in pipes (or air columns) as the result of interference between longitudinal sound waves travelling in opposite directions

• A closed end of the pipe is a:

- displacement node (no longitudinal motion of molecules)

- pressure antinode (maximum pressure variations)

• An open end of the pipe is a:

- displacement antinode (molecules of air vibrate with maximum amplitude)

- pressure node (no pressure variation)

Principal of superposition

Waves obey the principle of superposition: the resultant displacement at a point where similar waves from several different sources overlap is the (vector) sum of the individual wave displacements.

When two or more waves overlap, we say that ‘interference’ occurs

When crest meets crest, we get a bigger crest: constructive interference

When crest meets trough, we get no resultant displacement: destructive interference

Conditions for interference

To observe interference of waves, two conditions must be met:

1. The sources of the waves must be coherent, i.e they must maintain a constant phase with respect to each other (constant phase difference).

2. The sources must emit waves of the same frequency and wavelength. This is monochromatic light.

We also need to know the path difference: the difference in distance from each source to the interference point.

Lets consider two sources of wave, A and B

The path difference at P is δ=AP-BP

T otal Constructive interference Total Destructive interference