Waves 2

• Principle of Superposition

  When two waves meet at a point the resultant displacement is equal to the sum of the displacements of the individual waves
• Constructive interference occurs when two waves are in phase and the maximum positive displacements align creating a resultant wave with an increased amplitude
• Destructive interference occurs when two waves are in antiphase so the resultant displacement is small than for each individual wave
• Intensity is proportional to the square of amplitude
• Coherence is when waves from two sources have a contant phase difference and the same frequency
• Filament lamps do not provide a coherent source as they have a range of frequencies and an ever changing phase difference
• path difference is the difference in distance travelled by waves from their source
• Constructive interference

  if the path difference of a path is 0 or a whole number of wavelengths then the two waves will always arrive at that point in phase
• destructive interference

  if the path difference is an odd number of half wavelengths then the waves will always arrive in antiphase, and the resultant wave will always have minimal amplitude
• at the first order minimum,the path difference will be a half wavelength and there will be a phase difference of 90 degree or pi radians
• For an interference pattern, 2 coherent sources of waves are needed, which can be created from a filament lamp by using a colour filter that only allows a specific frequency through, and narrow slits to diffract the light
• Young's double slit expriment proved that light was a wave because if light were made of particles, only 2 bright bands would appear on the screen. The existence of muliple fringes proved that light waves from the slits interacted with each other
• Young double slit equation

  $\lambda =$$ax/D$

  a is the distance between the slits

  x is the fringe separation

  D is the distance between slits and the screen
• For the Young double slit equation to be valid, the distance between the slits should be much smaller than the distance from the slits to the screen
• $n\lambda=$$dsin\theta$

  n - number of orders

  $\lambda$ - wavelength

  d - 1/diffraction grating

  $\theta$ - angle made between central maxima and the beam
• Determining wavelength of light with a double slit

  Set up Young's double slit experiment (a laser through a single slit then through double slits to ensure that the waves are coherent)

  Use Young's double slit equation to calculate wavelength ($\lambda =$$ax/D$)
• Stationary wave (also known as a standing wave)

  A wave that remains in a constant position with no net transfer of energy and is characterised by its nodes and antinodes
• A stationary wave is formed when two progressive waves with the same frequency (and ideally amplitude) travelling in opposite directions are superposed. As they have the same frequency, at certain points they are in antiphase so their displacements cancel out, creating a node
• The separation of 2 node or antinodes is half the wavelength of the original progressive wave
• All particles oscillating between adjacent nodes are in phase as they reach their maximum positive displacement at the same time, however their amplitudes differ to the maximum amplitude at the antinode
• All particles on different sides of a node are in antiphase as the particles on one side of a node reach their maximum positive displacement at the same time as those on the other side reach ther maximum negative displacement
• A stationary wave on a string is formed by a wave being created by an oscillator and reflected back onto itself as one end of the string is held in place, forming the node
• The fundamental frequency is the lowest frequency at which an object can vibrate to form a standing wave
• A harmonic is a whole number multiple of the fundamental frequency
• for harmonics on a string (a node on either end) or a tube open on both ends (an antionode on either end):

  $\lambda_n =$$2L/n$

  n - harmonic

  L - length of the string
• for harmonics on a string (a node on either end) or a tube open on both ends (an antionode on either end):

  $f_n =$$nf_0$

  n -harmonic

  $f_0$ - fundamental frequency
• For a tube open on one end (one node and one antinode):

  $\lambda_n =$$4L/(2n-1)$

  L - length of string

  n - harmonic
• For a tube open on one end (one node and one antinode):

  $f_n=$$(2n-1)f_0$

  n - harmonic

  $f_0$ - fundamental frequency
• Using microwaves to demonstrate stationary waves

  Reflecting microwaves off a metal sheet so the microwaves of the same frequency travel in opposite directions. A microwave receiver between the source and the metal sheet can be moved to to view the change and intensity.

  The distance between the transmitter and metal sheet has to be adjusted until the receiver detects a series of nodes and antinodes. The distance between a node and antinode is $\lambda/4$
• It is not possible for there to be a harmonic of $2f_0$ in a tube with an open end as the open end is always an antinode so the frequencies of harmonics are always an odd multiple of the fundamental frequency (2n-1)
• Calculating the speed of sound using a resonance tube

  Hold a tuning fork above a resonance tube partially submerged in water and slowly lift the tube and fork until the sound becomes louder and the air in the tube resonates. Measure the length of the tube above the water = $\lambda/4$

  Repeat for tuning forks of different frequencies and plot a graph of wavelength against 1/frequency and the gradient is the speed of sound