Tuesday 25th March 2014 Mr Spence IB Physics: Waves Tuesday 25th March 2014 Mr Spence Present/copy up to slide 60, SHM. Copy 6 slides per page, double sided = 4 sides of A4 each.

WAVES Wave Characteristics – “what do they look like?” Longitudinal and Transverse 3 Representations Defining the wave The wave equation 2. Wave Properties – “how do they act?” Reflection and transmission Snell’s Law Diffraction 3. Standing Waves 4. The Doppler Effect

Waves Waves can transfer energy and information without a net (total) motion of the medium through which they travel. They involve vibrations (oscillations) in either the x or y directions.

Transverse waves The oscillations are perpendicular to the direction of energy transfer. peak Direction of energy transfer oscillation trough

Transverse waves Water ripples EM Radiation On a rope Earthquake: S waves

Longitudinal waves The oscillations are parallel to the direction of energy transfer. Direction of energy transfer oscillation

Longitudinal waves compression rarefraction

Longitudinal waves Sound Slinky: back and forth Earthquake: P waves The most common example of longitudinal waves. Frequency in Hz, amplitude in Decibels. Slinky: back and forth Earthquake: P waves

Wave fronts Wave fronts highlight the part of a wave that is moving together (in phase). = wavefront Ripples formed by a stone falling in water

Rays Rays highlight the direction of energy transfer.

Wave Characteristics: Displacement - x This measures the change that has taken place as a result of a wave passing a particular point. Zero displacement refers to the average position. = displacement

Amplitude - A The maximum displacement from the mean (equilibrium) position. Often measured in dB or metres. amplitude Examples: 5dB is a whisper, 90dB is painful. Large sea waves may be around 5 metres.

Period - T The time taken (in seconds) for one complete oscillation. It is also the time taken for a complete wave to pass a given point. One complete wave

Frequency - f The number of oscillations in one second. Measured in Hertz. 50 Hz = 50 vibrations/waves/oscillations in one second. Period and frequency are reciprocals of each other f = 1/T T = 1/f

Wavelength - λ The shortest distance between points that are in phase (points moving together or “in step”). wavelength

Wave speed - v The speed at which the wave fronts pass a stationary observer. 330 m.s-1

v = λ/T = fλ The Wave Equation The time taken for one complete oscillation is the period T. In this time, the wave will have moved one wavelength λ. The speed of the wave therefore is distance/time v = λ/T = fλ Example: Purple light has a wavelength of around 6x10-7m and a frequency of 5x1014Hz. What is the speed of purple light? Answer: The same speed as every other colour.

Graphic Representation 1: Displacement-Time graph This looks at the movement of one point of the wave over a period of time Time s -1 -2 0.1 0.2 0.3 0.4 displacement cm 1

Displacement/Time graph This looks at the movement of one point of the wave over a period of time Time s -1 -2 0.1 0.2 0.3 0.4 displacement cm IMPORTANT NOTE: Although the representation here looks transverse it may be a longitudinal wave (eg sound) 1 PERIOD

Graphical Representation 2: Displacement - Distance graph This is a “snapshot” of the wave at a particular moment displacement cm IMPORTANT NOTE: This wave could also be either transverse or longitudnal 1 WAVELENGTH Distance cm 0.4 0.8 1.2 1.6 -1 -2

Wave intensity This is defined as the amount of energy per unit time flowing through a unit area. It is normally measured in W.m-2 For example, imagine a window with an area of 1m2. If one joule of light energy flows through that window every second we say the light intensity is 1 W.m-2.

Intensity at a distance from a light source I = P/4πd2 d is the distance from the light source (in m) and P is the power of the light source (in Watts)

Intensity and amplitude The intensity of a wave is proportional to the square of its amplitude I α a2 (or I = ka2) This means if you double the amplitude of a wave, its intensity quadruples! I = ka2 If amplitude = 2a, new intensity = k(2a)2 new intensity = 4ka2

Electromagnetic spectrum λ ≈ 700 - 420 nm λ ≈ 10-1 - 103 m λ ≈ 10-12 - 10-14 m λ ≈ 10-2 - 10-3 m λ ≈ 10-9 - 10-11 m λ ≈ 10-4 - 10-6 m λ ≈ 10-7 - 10-8 m

What do they all have in common? They can travel in a vacuum They travel at 3 x 108m.s-1 in a vacuum (the speed of light) They are transverse They are electromagnetic waves (electric and magnetic fields at right angles to each oscillating perpendicularly to the direction of energy transfer)

Refraction Occurs when a wave changes speed (normally when entering another medium). It may refract (change direction/bend).

Examples Sound travels faster in warmer air: Light slows down as it goes from air to glass/water:

Snell’s law There is a wonderful relationship between the speed of the wave in the two media and the angles of incidence and refraction. There is a nice geometrical proof as well, if interested (which you should be…) i Derivation: https://www.youtube.com/watch?v=twURUE075y8 r

Snell’s law = speed in substance 1 sinθ1 speed in substance 2 sinθ2

Snell’s law In the case of light only, we usually define a quantity called the index of refraction for a given medium as n = c = sinθ1/sinθ2 cm where c is the speed of light in a vacuum and cm is the speed of light in the medium c vacuum cm

Snell’s law Thus for two different media sinθ1/sinθ2 = c1/c2 = n2/n1

Refraction – a few notes The wavelength changes, the speed changes, but the frequency stays the same

Refraction – a few notes When the wave enters at 90°, no change of direction takes place.

Diffraction Waves spread as they pass an obstacle or through an opening. This can be described using Huygen’s construction.

Diffraction Diffraction is strongest when the opening or obstacle is similar in size to the wavelength of the wave. That’s why we can hear people from around a wall but not see them!

Diffraction of radio waves                                                                                                                                                                                                                                                                     Diffraction of radio waves Aerials in hilly areas can often receive radio waves but not shorter wavelength TV or Microwaves (mobile phone).

The Principle of Superposition “When two or more waves meet, the resultant displacement is the sum of the individual displacements.” Constructive and Destructive Interference: When two waves of the same frequency superimpose, we can get constructive interference or destructive interference. + = = +

Phase difference is the time difference or phase angle by which one wave/oscillation leads or lags another. 180° or π radians

Superposition In general, the displacements of two (or more) waves can be added to produce a resultant wave. (Note, displacements can be negative)

Interference patterns In areas of constructive interference, bright bands occur. Destructive interference = no intensity.

Path difference Whether there is constructive or destructive interference observed at a particular point depends on the path difference of the two waves

Constructive interference if path difference is a whole number of wavelengths

Constructive interference if path difference is a whole number of wavelengths antinode

Destructive interference occurs if the path difference is a half number of wavelengths node

Simple harmonic motion (SHM) periodic motion in which the restoring force is proportional and in the opposite direction to the displacement

Simple harmonic motion (SHM) periodic motion in which the restoring force is in the opposite durection and proportional to the displacement F = -kx

Graph of motion A graph of the motion will have this form displacement Time displacement

Graph of motion A graph of the motion will have this form Amplitude x0 Period T Time displacement

Graph of motion Notice the similarity with a sine curve 2π radians angle π/2 π 3π/2 2π

x = x0sinθ Graph of motion Notice the similarity with a sine curve Amplitude x0 x = x0sinθ 2π radians angle π/2 π 3π/2 2π

Graph of motion Amplitude x0 Period T Time displacement

x = x0sinωt Graph of motion Amplitude x0 Period T x = x0sinωt Time displacement where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

When x = 0 at t = 0 Amplitude x0 Period T x = x0sinωt Time displacement where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

When x = x0 at t = 0 Amplitude x0 Period T x = x0cosωt displacement Time where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

x = x0sinωt v = v0cosωt When x = 0 at t = 0 Amplitude x0 Period T Time displacement where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

x = x0cosωt v = -v0sinωt When x = x0 at t = 0 Amplitude x0 Period T displacement Time where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

To summarise! where ω = 2π/T = 2πf = (angular frequency in rad.s-1) When x = 0 at t = 0 x = x0sinωt and v = v0cosωt When x = x0 at t = 0 x = x0cosωt and v = -v0sinωt It can also be shown that v = ±ω√(x02 – x2) and a = -ω2x where ω = 2π/T = 2πf = (angular frequency in rad.s-1)

Maximum velocity? When x = 0 At this point the acceleration is zero (no resultant force at the equilibrium position).

Maximum acceleration? When x = +/– x0 Here the velocity is zero amax = -ω2x0

Oscillating spring We know that F = -kx and that for SHM, a = -ω2x (so F = -mω2x) So -kx = -mω2x k = mω2 ω = √(k/m) Remembering that ω = 2π/T T = 2π√(m/k)

S.H.M. Where is the kinetic energy maxiumum? Where is the potential energy maximum?

It can be shown that…. Ek = ½mω2(xo2 – x2) ET = ½mω2xo2 Ep = ½mω2x2 where ω = 2πf = 2π/T

Resources https://www.youtube.com/watch?v=SRh75B5iotI - Walter Lewin’s famous Pendulum derivation and demo Great Intro: http://helenkellerramirez.weebly.com/pendulum.html

Focault Pendulum Sixty Symbols description: https://www.youtube.com/watch?v=sWDi-Xk3rgw Wolfram Alpha model: http://demonstrations.wolfram.com/FoucaultsPendulum/ Sydney Uni: http://www.animations.physics.unsw.edu.au//jw/foucault_pendulum.html

Damping In most real oscillating systems energy is lost through friction. The amplitude of oscillations gradually decreases until they reach zero. This is called damping

Underdamped The system makes several oscillations before coming to rest

Overdamped The system takes a long time to reach equilibrium

Critical damping Equilibrium is reached in the quickest time

Natural frequency All objects have a natural frequency that they prefer to vibrate at.

Forced vibrations If a force is applied at a different frequency to the natural frequency we get forced vibrations

Resonance If the frequency of the external force is equal to the natural frequency we get resonance YouTube - Ground Resonance - Side View YouTube - breaking a wine glass using resonance http://www.youtube.com/watch?v=6ai2QFxStxo&feature=relmfu