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17 Superposition and standing waves

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Introduction When two or more waves meet, the result is found by the principle of superposition.At any instant, the resultant displacement is simply the sum of the displacements of the individual waves.Constructive and destructive interference are obvious examples of this idea.It also explains the formation of standing waves.

Melde’s experiment to show standing waves

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Melde’s experimentThe vibrator sends waves along the string.They reflect at the other end.The outgoing and reflected waves then interfere.At certain frequencies, a standing wave (or stationary wave) pattern of loops is formed.

At certain point – nodes – the two waves interfere destructively.There is no vibration. There are nodes at the ends of the string

node

antinode

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Half-way between the nodes are antinodes. The string vibrates with a large amplitude.

When the vibration has its maximum amplitude, the two waves are interfering constructively

Changing the frequency slightly causes the standing waves to disappear. Changing the length, tension or thickness of the string causes the standing waves to appear at different frequencies.

The wavelength of the wave is twice the distance from one node to the next.

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Conditions for a standing wave

Two identical but oppositely travelling waves interfere with each other to form a standing wave.Often, one wave is a reflection of the other.

Incident wave in blue, reflected wave in red

Using the principle of superpositionDiagram on slide 6 show two waves which make a standing wave. They are shown at two instants in time.

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Two waves in phase

Out of phase(phase difference = ½ λ)

The waves are progressive waves travelling in opposite directions

Above them are the resultant waves – worked out by adding the displacements of the two progressive waves.Air columns

When the frequency of a loudspeaker is changed, a point is reached where the noise becomes much louder.Sound waves are reflected by the closed end of the column, forming a standing wave in the air column inside the cylinder.

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λ/4 3λ/4There is a node at the foot of the air column and an anti-node at the top

The lowest frequency at which this occurs, the length of the air column is one quarter of the wavelength of sound.

A standing wave is formed again at three times this frequency, with three-quarters of the wave fitting in the column.

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Questions 1. A string of length 1.2 m is stretched and vibrated so

that a standing wave consisting of 2 loops is formed. Sketch this, and calculate the wavelength of the waves on the string.

2. Microwaves are directed at a sheet of steel. A detector is used to investigate the intensity of the waves between the source and the plate. A pattern of high and low intensity regions is found; the separation of adjacent high intensity regions is 1.5 cm. what is the wavelength of the microwaves?

3. Explain why nodes occur in standing waves.4. In a vibrating air column experiment, the air column

is 20 cm long. The lowest frequency which produces a standing wave is 400 Hz. Calculate the wavelength and speed of the sound wave.