Module 9- Wave Phenonema

Module 9- Wave Phenonema

1. 9.1- Simple Harmonic Motion

   • Angular Speed
     • Constant speed in a circle of radius x
     • Velocity is always changing direction
       • Acceleration towards the centre of the circle
     • Angular speed is measured in radians per second
       • An alternate way of measuring it is using radians
         • Pi rad = 180 degrees
         • 2 pi rad = 360 degrees
     • Angular Speed = angle/time
     • Angular Speed = 2pi/T (period)
   • Oscillating Mass-Spring System
     • Mass which begins at x=+x0 and is released from rest at t=0 seconds
       • Displacement = x=x0cos(theta)
         • x=x0cos(wt)
       • Velocity= v=-v0sin(theta)
         • v=-x0wsin(wt)
       • a=-a0 cos(theta)
         • a=-x0w^2cos(wt)
     • Mass which begins at x=0 and is given a positive velocity v0 at t=0
       • x=x0sin(wt)
       • v=x0wcos(wt)
       • a=-x0w^2sin(wt)
     • Relationship between x and v
       • v=+-sqrt (x0^2-x^2)
     • Period of a mass/spring system
       • T=2pi sqrt(m/k) where k is the spring constant
   • Defining Equation=a=-w^2x (where w is angular momentum)
   • Simple Pendulum
     • T=2pi sqrt(L/g) where L is length

2. 9.2 Single Slit Diffraction

   • Diffraction
     • If a wave meets a whole in a wall that is of comparable size to its wavelength then the wave will be diffracted
       • The smaller the aperture b the more pronounced the diffraction effect
     • Christian Huygens- Every point on a wavefront emits a spherical wavelet of the same velocity and wavelength as the original wave
       • The waves can also interfere with each other
     • Waves turn by the incoming wave transmitting a disturbance causing the medium to vibrate
       • Whenever the medium vibrates it becomes the centre of a new wavefront
   • Interference
     • Coherent is in phase and with the same frequency
     • Constructive Interference- Path difference is n lambda
     • Destructive Interference- Path difference is (n+1/2) lambda
   • Single Slit Interference
     • Location of first minium in single slit diffraction- theta = lambda/ b where b is the slit width
     • Relative intensities of the maximums and positions
       • Maximum 1- I0
       • Maximum 2- I0/20, lambda/b
       • Maximum 3- I0/50, 2 lambda/b
       • Maximum 5- I0/100, 3 lambda/b
     • The bigger b is the closer together the maximums and minimums will be and the smaller the central maximum will be
     • The smaller lambda is the closer together the maximums and minimums will be

3. 9.3 Interference

   • Double Slit
     • s=lambda D/d. Where s is separation of the maxima, D is distance to screen and d is slit separation
   • n slits (Diffraction Grating)
     • Intensity of central maximum for N slits- IN=N^2 I1 where I1 is the intensity of the single-slit central maximum
     • Intensity increases with increase in slits
     • Double slit separation remains the same for multiple slits
     • Used to make optical spectra
     • A typical diffraction grating consists of a large number of parallel, equally spaced lines
     • Diffrerent wavelengths are diffracted at different angles producing interference maxima
       • Diffraction grating maximum locations- n lambda = d sin(theta). d is slit separation
   • Thin Film Interference
     • Examples are soap bubbles and gasoline spills on water
     • At a boundary of a heavy rope and a light rope
       • Reflection and transmission are in phase if the pulses are all on one side of the rope
       • Reflection is out of phase and transmission is in phase if the pulse before the boundary is on the opposite side of the rope
     • Wave entering boundaries
       • Wave enters a boundary from a medium where wave speed is lower it will reflect in phase
       • Wave enters a boundary from a medium where wave speed is higher it will reflect out of phase
     • Destructive- 2dn = m lambda
     • Constructive- 2dn=(m+1/2) lambda
     • d is film thickness

4. 9.4 Resolution

   • Two distant objects that are close together will look like one object if they are far enough away
   • Diffraction gratings
     • More slits= higher the intensity of primary maxima
     • More slits = narrower primary maxima
     • Grating resolution is proportional to n
       • Resolvance is lambda/ difference in lambda
       • Resolvance is Nm
     • Light natural diffracts to different colours
   • Raleigh Criterion
     • Raleigh Criterion- Two sources are just resolved if the first minimum of the diffraction pattern of one of the sources falls on the central maximum of the diffraction pattern of the other source
       • Square Aperture- theta = lambda/b
       • Circular Aperture= theta = 1.22 lambda/b

5. 9.5 Doppler Effect

   • The Doppler Effect explains frequency change caused by moving sources or moving observers
   • If something approaches you then you will hear a higher frequency
   • Moving Source
     • f'=f[v/(v+-s)] +s when receding, -s when approaching. v is speed of sound
   • Moving Observer
     • f'=f[(v+-s)/v] -s when receding, +s when approaching. v is speed of sound
   • Light
     • Change in frequency/f = change in wavelength/lambda = v/c