4. Oscillations & Waves

1. 4. Oscillations & Waves

2. Topic Outline

3. 4.4 Wave Characteristics

4. “The impetus is much quicker than water, for it often happens that the wave flees the place of its creation, while the water does not; like waves made in a field of grain by the wind, where we see the waves running across the field while the grain remains in place.”Leonardo da Vinci

5. Wave Terms • A pulse is a single disturbance that transfers energy over a distance • An oscillation is a periodic (regularly repeating) disturbance • A wave is a periodic disturbance that transfers energy over a distance without any net transfer of the medium • Mechanical waves (e.g. sound, water waves) travel through a medium (e.g. air, water) • Electromagnetic waves (e.g. light, X-rays, UV rays) do not require a medium to propagate

6. Transverse Waves • In transverse waves, the particles oscillate at right angles to the direction of travel • Examples of transverse waves include: water waves, electromagnetic waves • The point of maximum amplitude is called a crest, the point of minimum amplitude is called a trough • Although mechanical transverse waves can travel along the surface of a liquid, they do not travel through liquids or gasses

7. Longitudinal Waves • In longitudinal waves, the particles oscillate along the direction of travel • Examples of longitudinal waves include: sound waves, the primary waves of an earthquake • The point where the particles are closest together is called a compression, the point where the particles are farthest apart is called a rarefaction

8. Representing Waves • Waves can be represented by: • an arrow (ray) showing the direction that the wave is propagating in • lines showing the wave crests (wavefronts) looking down on the waves from above

9. Wave Terms • The displacement (x, m) of a particle is how far it has moved from its rest position • The wavelength (l, m) is the distance between two consecutive particles that have the same displacement, i.e. to consecutive crests/compressions or troughs/rarefactions • The velocity (v, m.s-1) of a wave is the speed and direction in which it is travelling • The frequency (f, s-1 or Hz) of a wave is the number of wavelengths that pass a given point every second • The period (T, s) of a wave is the amount of time taken for one wavelength to pass a given point • The amplitude (A, m) is the maximum displacement of a particle from its rest position • The intensity (I, J.m-2.s-1) is the energy that a wave transports per unit time across a unit area of the medium. The energy transported is proportional to the square of the amplitude, I aA2

10. Frequency and Period • Frequency is the inverse of period: f = 1/T • Period is the inverse of frequency: T = 1/f f = frequency, Hz (or s-1) T = period, s

11. Wave Velocity • The velocity of a wave in a given medium is calculated by: v = fl v = velocity, m.s-1 f = frequency, Hz l= wavelength, m Worksheet: Wave Velocity Questions

12. Displacement-Position Graph • A displacement-position graph shows a ‘snapshot’ of a wave at a given time

13. Displacement-Time Graph • A displacement-time graph shows how the displacement of one particle in the medium varies over time • The following graph shows how the displacement of point P (previous graph) varies with time

14. Electromagnetic Waves • Electromagnetic waves consist of a varying electric field and a varying magnetic field • These two fields travel in the same direction and are at right angles to each other • Electromagnetic waves are transverse waves and they can propagate through a vacuum, i.e. they do not require a medium to propagate • The speed of electromagnetic radiation (EMR) in a vacuum (c) is 3.00 x 108 m.s-1 • Electromagnetic waves are produced when a charge is accelerated

15. Electromagnetic Waves

16. Electromagnetic Spectrum • High energy EMR has a high frequency (and short wavelength) • Low energy EMR has a low frequency (and long wavelength)

17. Electromagnetic Spectrum

18. Electromagnetic Spectrum

19. 4.5 Wave Properties

20. Reflection of Pulses String with a fixed end • If a pulse travels along a string that is fixed to a rigid support, the pulse is reflected with a phase change of 180º • The shape of the pulse stays the same, except that it is inverted and travelling in the opposite direction • The amplitude of the pulse is slightly less as some energy is absorbed at the fixed end • When the (upward) pulse reaches the fixed end, it exerts an upward force on the support, the support then exerts and equal and opposite downward force on the string (reaction force), causing the inverted pulse to travel back along the string • http://rt210.sl.psu.edu/phys_anim/waves/indexer_waves.html

21. Reflection of Pulses String with a free end • If a pulse travels along a string that is tethered to a pole but free to move, the pulse is reflected with no phase change • The shape of the pulse stays the same, except that it is travelling in the opposite direction

22. Reflection of Wavefronts • Reflection is when waves bounce off a surface • The first law of reflection is that the angle of incidence equals the angle of reflection qi = qr • The second law of reflection is that the incident ray, normal line and reflected ray all lie on the same plane

23. Reflection at a Plane Boundary

24. Refraction • Refraction is when waves bend as they travel from one medium to another • When a wave travels into a different medium: • the wave speed changes • the wavelength changes • the frequency stays the same • If the wave hits the new medium at an angle, the wave direction will change

28. Refraction • Refraction is when waves bend as they travel from one medium into another • For example, light waves travelling from air to water; ocean waves travelling from deep water to shallow water • Refraction occurs because the speed of the wave changes: • when waves speed up, they bend away from the normal • when waves slow down, they bend towards the normal • Water waves travel faster in deep water, light rays travel faster in less dense media

29. Refraction

30. Refractive Index • The refractive index (n) of a medium relates to how fast waves travel through that medium • Refractive index is a relative scale and does not have units •  As light travels into a denser medium: • refractive index increases • wave velocity decreases • wavelength decreases • frequency stays the same

31. Refractive Index • Some common refractive indices are: nvacuum= 1 nwater= 1.33 nglass= 1.5 nrock salt = 1.5 nruby= 1.76 ndiamond= 2.4 n1 = refractive index of medium 1 n2 = refractive index of medium 2 v = velocity, m.s-1 l = wavelength, m

32. Snell’s Law • Snell’s Law relates the angle of incidence, the angle of refraction and the refractive indices of two media n1sinq1 = n2sinq2 n1 = refractive index of medium 1 n2 = refractive index of medium 2 q1 = angle of incidence q2 = angle of refraction

33. Huygen’s Principle • Huygens’ Principle:Every point on a wave front can be considered as a source of tiny wavelets. These wavelets spread out in the forward direction at the speed of the wave. The new wave front is given by the tangent of all of these wavelets.

34. Diffraction • Diffraction is when waves bend slightly when travelling past an obstacle or through an aperture • Diffraction is noticeable when the size of the obstacle is similar to the size of the wave • For example, ocean waves bending around a headland, shortwave radio bending around the corner of a building, AM radio waves bending over a hill

35. Superposition of Waves • When two waves of the same nature travel past each other, the displacement of the resultant wave is the sum of the displacements of the individual waves at that point

36. Constructive Interference • Constructive interference is when displacements of the individual waves are in the same direction, and the resultant wave has a greater amplitude

37. Destructive Interference • Destructive interference is when the displacements of the individual waves are in opposite directions, and the resultant wave has a lesser amplitude

38. Two-Source Interference • If two point sources produce coherent waves (i.e. same wavelength, same phase), a pattern of constructive and destructive interference occurs around them

40. Two-Source Interference • Where two crests (or two troughs) meet, constructive interference occurs resulting in a greater displacement; this is called an antinode • Where a crest and a trough meet, destructive interference occurs, resulting in no displacement; this is called a node • When viewed from above, lines of antinodes and nodes radiate outward from the point between the sources • If the sources are in phase, there will always be a central antinode (i.e. maxima or line of constructive interference)

42. Path Difference & Phase Difference • We can predict where nodes and anti-nodes will occur in two-source interference • This is done by measuring the path length from one source (S1) to a given point (A), and the path length from the other source (S2) to that same point (A) • If the path difference (S1A – S2A) is a whole number of wavelengths, the waves from the two sources will arrive in phase and constructive interference will occur; this will cause an antinode path difference = nl for an anti-node • If the path difference is a half wavelength greater, the waves from the two sources will arrive 180º out of phase and destructive interference will occur; this will cause a node path difference = (n + ½)l for a node

43. Interference of Light • When monochromatic (one coloured) light shines through a thin double slit (creating two sources of the same light), an interference pattern is observed • This experiment demonstrates the wave properties of light and was first performed by English scientist Thomas Young in 1801

44. Beat Frequency • If two waves of the same nature but slightly different frequency are allowed to interfere, the waves will drift in phase (causing constructive interference and greater amplitude) and then out of phase again (causing destructive interference and lesser amplitude) • This periodic oscillation in the amplitude of the combined waveform is called the beat frequency • The most obvious example is when two musical notes that are slightly out of tune are played together, the resulting sound has a regular increase and decrease in volume

45. Beat Frequency

46. 4.1 Simple Harmonic Motion

47. Simple Harmonic Motion

48. Simple Harmonic Motion • An object is moving with Simple Harmonic Motion (SHM) if: • the object experiences a net force (and therefore an acceleration) towards the equilibrium position • the magnitude of the net force (and therefore acceleration) is proportional to the distance of the object from the equilibrium position • it is moving with a regular, repeating motion with constant period and (theoretically) constant amplitude • the motion of the object can be modelled by a ‘reference circle’ • Examples of SHM include: a pendulum swinging, a mass bouncing on a spring, a guitar string that has been plucked, the tides

49. Terms in SHM • The displacement (x, m or q, rad) of an object moving in SHM is the distance of the object from its equilibrium position • The amplitude (x0 or q0) is the maximum displacement of the object • The period (T, s) is the time taken for one complete oscillation • The frequency (f, Hz) is the number of oscillations per second; f = 1/T • The angular frequency (w, rad.s-1) is equivalent to the angular speed of an object moving in uniform circular motion w= angular frequency, rad.s-1 f = frequency, Hz T = period, s

50. The Reference Circle • The reference circle is used to understand SHM • We consider the motion of an object moving with uniform circular motion, and project this motion into one dimension • The amplitude, x0, of the SHM is given by the radius of the circle • The angular frequency, w, of SHM is given by the angular velocity of the object in the reference circle