# v = fλ PROGRESSIVE WAVES 1 Candidates should be able to :

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1 PROGRESSIVE WAVES 1 Candidates should be able to : Describe and distinguish between progressive longitudinal and transverse waves. With the exception of electromagnetic waves, which do not need a material substance for their transmission, all other wave motions (e.g. sound waves, water waves etc.) are disturbances which pass through a substance. Define and use the terms displacement, amplitude, wavelength, period, phase difference, frequency and speed of a wave. Derive from the definitions of speed, frequency and wavelength, the wave equation : v = fλ A series of waves (similar to those produced on a water surface) can be generated in a long piece of rope which is fixed at one end and moved repeatedly at right angles to its length (as shown in the diagram opposite). hand movement wave motion rope Select and use the wave equation : v = fλ Explain what is meant by reflection, refraction and diffraction of waves such as sound and light. Waves (similar to sound waves) are produced in a spring coil which is fixed at one end while the other end is repeatedly moved as shown in the diagram opposite. Wave motion Spring coil Hand movement A pebble dropped onto a still water surface causes a vibration which produces concentric, circular waves which spread outwards across the water surface. direction of wave mon

2 CLASSIFICATION OF WAVES 2 In each of the wave examples considered, the wave disturbance is initiated at a particular point and this sets the particles of the substance at that point into vibration. This causes neighbouring particles to vibrate in the same way and so the wave progresses through the substance. In the case of a single wave pulse initiated by an up and down movement of the end of a springy coil, the motion of the first ring causes the next ring to move and so on. In this way, the wave travels from one end of the coil to the other as shown in the diagram opposite. Particle vibration direction of wave motion Waves can be : MECHANICAL OR ELECTROMAGNETIC Waves which need a substance for their transmission. (e.g. sound waves can be transmitted Through air, water and steel, but not Through a vacuum). Waves which DO NOT need a substance for their transmission (e.g. light waves travel from stars and galaxies travel through empty space to reach us here on Earth). All the waves considered so far are : PROGRESSIVE WAVES Waves which travel through a material or through a vacuum and transfer energy from one point to another. EXAMPLES Water waves Sound waves Waves along a spring coil or rope Seismic waves EXAMPLES Gamma-rays X-rays Ultra-violet (UV) Visible light Infra-red (IR) Microwaves Radio waves

3 Waves can be : TERMS ASSOCIATED WITH WAVES 3 TRANSVERSE OR LONGITUDINAL crest wavelength (λ) amplitude (A) Waves in which the particles of the medium in which a wave is moving vibrate perpendicular To the direction of wave travel. Waves in which the particles of the medium in which a wave is moving vibrate parallel to the direction of wave travel. displacement (y) trough λ λ particle vibration 90 wave direction Particle vibration compression expansion Wave direction DISPLACEMENT (y) / metre (m) The distance and direction of a vibrating particle in a wave from its undisturbed position. A spring coil can be used as shown above to observe TRANSVERSE wave motion. The coil is fixed at one end and the free end is continuously vibrated in a direction PERPENDICULAR to the axis of the coil. A spring coil can be used as shown above to observe LONGITUDINAL wave motion. The coil is fixed at one end and the free end is continuously vibrated in a direction PARALLEL to the axis of the coil. AMPLITUDE (A) / metre (m) The maximum displacement of any particle in a wave from its undisturbed position. TRANSVERSE WAVE EXAMPLES LONGITUDINAL WAVE EXAMPLES Water waves Secondary seismic waves All electromagnetic waves Sound waves Primary seismic waves WAVELENGTH (λ) / metre (m) The distance between two consecutive points on a wave which are in phase with each other (i.e. the vibrating particles at these points have the same displacement and velocity). It is the distance between two consecutive crests or troughs.

4 4 PHASE DIFFERENCE (Φ) / degrees or radians PERIOD (T) / second (s) The time taken for one complete wave to pass a fixed point. OR The time taken for one complete oscillation of a particle in the wave. The phase difference between two vibrating particles in a wave is the fraction of a cycle between the vibrations of the two particles. Phase difference is measured in DEGREES or RADIANS. 1 cycle = 360 = 2π radians A B C D E FREQUENCY (f) / hertz (Hz) The number of complete waves passing a fixed point per second. OR The number of complete oscillations per second of a particle in the wave. 1 Hz = 1 vibration per second. 1kHz = 10 3 Hz 1 MHz = 10 6 Hz. In the diagram above, particles at points D and E which are one wavelength apart, vibrate in phase with each other. The phase difference between the partcles at these two points is 360 (=2π rads) (which is the same as 0 ). Particles at points A and B which are ½ a wavelength apart, vibrate in antiphase. The phase difference between the particles at these two points is 180 (π rads). What is the phase difference between : (a) Particles at points A and C? (b) Particles at points C and D?

5 WAVE SPEED (v or c) / metre per sec (m s -1 ) This is the speed with which energy is transmitted by a wave. WAVE SPEED EQUATION Consider a wave having wavelength (λ), period (T), frequency (f) and moving with speed (v) In time (T) the wave will travel a distance (λ). wave speed, v = distance travelled = λ time taken T 2 (a) Sound waves travel at a speed of 340 m s -1 in air at 20 C. 5 If the frequency of these waves is 2.5 khz, what is their wavelength? (b) Infra-red electromagnetic waves travel at a speed of 3.0 x 10 8 m s -1. Calculate the frequency of these waves if their wavelength is 1.2 x 10-6 m. 3 A wave of frequency 2.5 Hz travels along a stretched cable at a speed of 20 m s -1. Calculate the phase difference between two points on the cable which are 2.0 m apart. Express your answers in : (a) Degrees, (b) Radians and (c) Fractions of a wavelength. And since, T = 1/f, Therefore : v = λ 1/f v = f λ 4 Calculate the speed of a progressive wave of frequency 350 Hz, if the least distance between two points on the wave having a phase difference of π/6 rad is 0.08 m. (m s -1 ) (Hz) (m) 5 A laser emits a single burst of light which lasts for a time of 0.02 µs. If the wavelength of the light is 500 nm, calculate the number of complete waves emitted by the laser. PRACTICE QUESTIONS 6 Using graph paper, draw two waves, X and Y, such that Y has twice the wavelength and a quarter of the amplitude of X. 1 Determine the amplitude and wavelength of waves a and b shown in the diagram opposite. 7 Use the data given below to calculate the frequency range of light detectable by the human eye. Speed of light in air = 3.0 x 10 8 m s -1. Approximate range of visible light wavelengths is 400 nm (violet) to 700 nm (red).

6 6 REFLECTION, REFRACTION AND DIFFRACTION OF WAVES A ripple tank as shown opposite may be used to demonstrate reflection, refraction and diffraction of water waves. illumination Vibrating bar stroboscope REFRACTION Refraction occurs when waves change their speed. Water waves slow down when they pass from deep into shallower water. The waves, produced by a vibrating, straight-edged bar, can be viewed from above or projected onto a screen. The straight lines observed on the screen show the positions of the wavefronts (i.e. the crests of the wave). Rubber band Wavefronts Projected onto screen motor water RIPPLE TANK Waves slow down in shallower water Obstacles are placed in the path of the waves to see what effects are produced. Flat piece of plastic wave direction reflector REFLECTION A flat, rectangular piece of perspex placed in the tray, makes the water shallower and as the waves pass over it they are slowed down When the water waves are incident at the DEEP-SHALLOW boundary at an angle, they change direction and their wavelength (λ) decreases. This is because v = fλ and since v increases and f stays the same, λ must decrease. When a ray of light is directed at a parallelsided glass block at an angle, it is also refracted. This is because the light waves slow down and so change direction when they pass from air into glass. water The waves are sent at an angle to a straight Reflector. The angle between the Reflected wavefronts and the surface = the angle between the incident wavefronts and the surface. So the direction of the reflected wave is at the same angle to the reflector as the direction of the incident wave. When a ray of light is directed at a plane mirror, the angle between the reflected ray and the mirror = the angle between the incident ray and the mirror. Sound waves are reflected in The same way.

7 7 DIFFRACTION Diffraction is the spreading of waves when they pass through a gap or around an obstacle. The fact that the diffraction effect is more significant when the gap width is comparable to the wavelength of the incident waves enables us to explain why, in everyday life, We can observe diffraction for some types of wave but not for others. Sound waves, for example, diffract as they pass through open doorways because their wavelengths are comparable to the size of the opening. This is why a person speaking in a corridor can be overheard in an adjoining room, in spite of the fact that there is a thick wall in the way. In a ripple tank, plane waves directed at a gapped barrier are seen to spread out as they pass through the gap. By varying the gap width (d) and the wavelength (λ) of the waves, it is seen that the extent of the diffraction effect depends on both (λ) and (d). Diffracted waves Straight waves gap In the aerial photograph shown opposite, sea waves are greatly diffracted as they pass through the gap between two large rocks. Again, the effect is observable because the wavelength is of the same order of magnitude as the gap width. Light wave diffraction, on the other hand, is rarely observable in normal circumstances. This is because visible light wavelengths ( nm) are tiny in comparison to the size of the gaps and objects we normally encounter. λ When the gap width (d) is LARGE compared to the wavelength (λ) of the incident waves, the diffraction effect is SMALL. d As the gap width (d) is decreased (with the wavelength (λ) kept the same), the diffraction effect becomes GREATER. The diffraction effect is GREATEST when the WAVELENGTH (λ) of the waves is EQUAL to the gap WIDTH (d). λ=d When light from a laser is directed at a very narrow slit, it diffracts into the space beyond the slit to give the type of image shown in the photograph opposite. With an adjustable gap, the effect of narrowing the gap can be investigated.

8 HOMEWORK QUESTIONS 1 (a) Describe the differences between transverse and longitudinal waves. (b) The diagram below shows a progressive longitudinal wave formed in a slinky spring by an oscillator connected to a signal generator. 2 (a) State what is meant by the DIFFRACTION of waves. 8 (b) Draw diagrams to illustrate how plane water waves are diffracted when they pass through a gap : (i) About 2 wavelengths wide, (ii) About 10 wavelengths wide. (c) Suggest why the diffraction of light waves cannot usually be observed, except under laboratory conditions. (OCR AS Physics - Module June 2004) To signal generator oscillator Slinky spring 3 (a) All waves are either longitudinal or transverse. State one example of each. (b) Define : (i) The FREQUENCY of a wave. (ii) The PERIOD of a wave. (c) The diagram below shows the variation of displacement with position at a particular instant for a progressive wave travelling in air. (i) Draw arrows to show the direction of the vibrations produced by the oscillator - Label these V. (ii) Label with a C the centre of a compression on the slinky. (iii) Show the wavelength of the wave and label this λ. (c) State and explain the effect on the wavelength of increasing the frequency of the oscillator. (OCR AS Physics - Module June 2005) (i) State the amplitude of the wave shown in the diagram. (ii) Describe the motion of an air particle at position A as one full cycle of the wave passes. (iii) State one way in which the motion of an air particle at position B is similar to, and one way in which it is different from, the motion of an air particle at A as the wave passes. (iv) Use the diagram to determine the wavelength of the sound wave. (v) The speed of the sound wave is 340 m s -1. Calculate the frequency of the sound. (OCR AS Physics - Module January 2003)

9 4 The diagram below shows the displacement-time graph for a particle in a medium as a progressive wave passes through the medium. 5 The diagram below shows, at a given instant, the surface of the 9 water in a ripple tank when plane water waves are travelling from left to right. (a) Determine from the graph : (i) The AMPLITUDE of the wave. (ii) The PERIOD of the wave. (b) (i) What is the FREQUENCY of the wave? (ii) The speed of the wave is 1500 m s -1. Calculate its WAVELENGTH. (iii) Use the grid in the diagram shown below to sketch the displacement-position graph for the wave at a particular instant. Mark the scale on the position axis and draw at least two full cycles. (OCR AS Physics - Module June 2004) (a) Copy the diagram and on your copy show : (i) The AMPLITUDE of the wave - Label this A. (ii) The WAVELENGTH of the wave - Label this λ. (b) On your copy of the diagram : (i) Draw the position of the wave a short time, about 1/10th of a period, later. (ii) Draw arrows to show the directions in which the particles at Q and S are moving during this short time. (c) State the PHASE DIFFERENCE between the movement of particles at P and Q. (d) The frequency of the wave is 25 Hz and the distance between P and Q is 1.8 cm. Calculate : (i) The PERIOD of the wave. (ii) The SPEED of the wave. (e) (i) Suggest how the speed of the waves in the ripple tank could be changed. (ii) The frequency of the wave source is kept constant and the wave speed is halved. State what change occurs to the wavelength. (OCR AS Physics - Module January 2005)

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