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# 7 superposition and standing waves

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### 7 superposition and standing waves

1. 1. 17 Superposition andstanding waves1
2. 2. IntroductionWhen two or more waves meet, the result is found bythe principle of superposition.At any instant, the resultant displacement is simply thesum of the displacements of the individual waves.Constructive and destructive interference are obviousexamples of this idea.It also explains the formation of standing waves.Melde’s experiment to showstanding waves2
3. 3. 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 stationarywave) pattern of loops is formed.At certain point – nodes – thetwo waves interferedestructively.There is no vibration. Thereare nodes at the ends of thestringnodeantinode3
4. 4. Half-way between the nodes are antinodes. The stringvibrates with a large amplitude.When the vibration has its maximum amplitude, the twowaves are interfering constructivelyChanging the frequency slightly causes the standingwaves to disappear.Changing the length, tension or thickness of the stringcauses the standing waves to appear at differentfrequencies.The wavelength of the wave is twice the distance fromone node to the next.4
5. 5. Conditions for a standing waveTwo identical but oppositely travelling waves interferewith each other to form a standing wave.Often, one wave is a reflection of the other.5Incident wave in blue, reflected wave in redUsing the principle of superpositionDiagram on slide 6 show two waves which make a standingwave. They are shown at two instants in time.
6. 6. 6Two waves inphaseOut of phase(phase difference = ½ λ)The waves are progressivewaves travelling in oppositedirectionsAbove them are theresultant waves – workedout by adding thedisplacements of the twoprogressive waves.Air columnsWhen the frequency of a loudspeaker is changed, apoint is reached where the noise becomes much louder.Sound waves are reflected by the closed end of thecolumn, forming a standing wave in the air column insidethe cylinder.
7. 7. 7λ/4 3λ/4There is a node at the footof the air column and ananti-node at the topThe lowest frequency atwhich this occurs, thelength of the air column isone quarter of thewavelength of sound.A standing wave is formed again at three times thisfrequency, with three-quarters of the wave fitting inthe column.
8. 8. Questions1. A string of length 1.2 m is stretched and vibrated sothat a standing wave consisting of 2 loops is formed.Sketch this, and calculate the wavelength of thewaves on the string.2. Microwaves are directed at a sheet of steel. Adetector is used to investigate the intensity of thewaves between the source and the plate. A patternof high and low intensity regions is found; theseparation of adjacent high intensity regions is 1.5cm. what is the wavelength of the microwaves?3. Explain why nodes occur in standing waves.4. In a vibrating air column experiment, the air columnis 20 cm long. The lowest frequency which produces astanding wave is 400 Hz. Calculate the wavelengthand speed of the sound wave.8