Superposition and Stationary Waves

Superposition

Where two waves meet, the total displacement at a point is equal to the sum of the two individual displacements

When does superposition occur?

When two waves travel through the same medium

Constructive interference

When two waves meet, if their displacements are in the same direction, the displacements combine to give a bigger displacement

Destructive interference

When a wave with a positive displacement meets a wave with a negative displacement, they will cancel each other out

Total destructive interference

If two waves with equal and opposite displacements meet, they cancel each other out completely

Phase difference

How much a wave lags behind another wave

In phase

Two points a wavelength apart oscillate in time with each other

Completely out of phase (antiphase)

Two points half a wavelength apart are always moving in opposite directions

The phase difference between two points depends on what?

Depends on what fraction of a wavelength lies between them

Equation for phase difference for two points at distance d apart along a wave of wavelength λ:

The phase difference (in radians) = 2πd/λ

What is one cycle in degrees and radians?

360° = 2π radians

How are stationary waves formed?

From the superposition of 2 progressive waves, travelling in opposite directions in the same plane, with the same frequency, wavelength and amplitude

Because of the conditions needed, when do stationary waves usually occur?

When a waveform is reflected back on itself

Do stationary waves transfer energy?

No - no energy is transferred by a stationary wave

What is the first harmonic?

The lowest frequency at which a stationary wave forms

Nodes

Areas where the resultant wave have an amplitude of 0

Antinodes

Half way between the nodes, the resultant wave has an amplitude twice the amplitude of its constituent waves

The first harmonic:

• wavelength

• number of nodes

• number of antinodes

• λ = 2l (length)

• 2 nodes

• 1 antinode

The second harmonic:

• wavelength

• number of nodes

• number of antinodes

• λ = l

• 3 nodes

• 2 antinodes

The third harmonic:

• wavelength

• number of nodes

• number of antinodes

• λ = 2/3l

• 4 nodes

• 3 antinodes

First harmonic equation

f = (1/2l)(√(T/μ))

• l = length of string (m)

• T = tension (N)

• μ = mass per unit length (kgm^-1)