Waves 2

• The speed of a wave on a string is affected by the tension in the string (T), measured in Newtons and the mass per unit length (𝝁) measured in kgm-1 . This is linked to the equation ( v = squrt T / u )
• The speed of the surface waves depends on the elastic properties and mass of the particles, as well as the depth of liquid.
• Interference can happen when 2 waves are moving through the same medium. The effect is most obvious with any combination of coherent waves (same f and constant phase relationship)
• The two-slit interference is described by Thomas Young as evidence for the wave nature of light. The double slit produces coherent wave patterns which then create an interference pattern
• This is the interference between two waves

  
• Use a ruler to measure the wavelength. Measure the distance of the following points from S1 and S2 and subtract one distance from the other. This is the PATH DIFFERENCE P0, P1, P2, P3, P4
• Constructive interference is when crest meets crest or trough meets trough. Path difference = nλ  (n=0,1,2,3…)
• Destructive interference is when trough meets crest or crest meets trough.

  
• Two-slit interference can be shown through the equation Where S is the slit width, x is the fringe separation and D is the distance from the slits to the screen

  
• Remember that 1 complete wavelength = 2π radians of phase.

  As the waves travel to a certain point, they are cycling through wavelengths

  Therefore path difference can be converted to phase difference: Points where the path difference is equal to nλ exactly, will have a phase difference of 2nπ exactly

  
• When continuous (progressive) waves are travelling in opposite directions, they will superpose one another. If the waves have the same speed and frequency, with similar amplitudes and a constant phase relationship, this will create STANDING WAVES (Also known as STATIONARY WAVES) Waves that share the same frequency, with a constant phase relationship, are known as coherent waves.
• Depending on the frequency of the wave that is producing the standing wave, the standing wave that is created will have a wavelength that will satisfy v = fλ. The lowest frequency that produces a harmonic (resonance) is called the fundamental frequency (f0). The wavelength where this occurs, as compared to the length of the medium (ie string, air column, etc) depends on the scenario. Each integer multiple of the fundamental frequency will produce further harmonics - these additional harmonics are called overtones
• • In a scenario where both ends are fixed (string clamped at both ends/ air column sealed at both ends), the end points must always be nodes. This is because vibration cannot occur at fixed points. 
  • The first harmonic will occur when the waveform matches this rule.
  • The fundamental frequency in this situation is when the length L of the vibrating medium is ½ λ

  
• • In a scenario where both ends are open (eg. Woodwind instrument), the end points are always antinodes. 
  • The fundamental frequency in this situation is when the length L of the vibrating medium is ½ λ

  
• Waves on a String

  
• Diffraction - It is the SPREADING of waves when they pass through a gap (aperture/slit). It happens to both TRANSVERSE and LONGITUDINAL waves
• Huygens' principle states that every point on an advancing wavefront is a new centre of disturbance from which emanates (issues from a source) independent waves in all directions
• Centre of disturbance is when waves will radiate outwards from a centre point in all directions
• You can Huygens' principle to predict the different diffraction patterns of different shapes of barriers
• The extent of the spreading (diffraction) depends on how the width of the gap compares to the wavelength of the waves. The wavelength is unchanged after diffraction. A gap width similar to the wavelength of the waves passing through causes a lot of spreading, eg sound waves passing through a doorway.
• Single slit diffraction - Passing a laser beam through a single slit produces a specific diffraction pattern, consisting of a central maximum and then repeating maxima and minima.
• Diffraction grating

  
• The diffraction pattern shows a central maximum edged by a series of lower-intensity maxima and minima as opposed to the regular pattern of interference from a double slit. The central maximum will broaden when the slit width is reduced: sinθ = λ/a where a is the slit width, the angle theta between the central maximum and the first minimum.
• Diffraction gratings are when light is reflected from a surface with a thousand of equally spaced, parallel grooves or transmitted through thousands of equally spaced, microscopic gaps, a diffraction pattern is produced
• nλ = d sinθ where d is the slit separation and n is the order of the maximum
• In 1912, Max von Laue suggested that if X-rays had a wavelength similar to atomic separations they should produce diffraction patterns when fired through single crystal materials
• In von Laue's time, electrons had been shown to be particles of mass 9.1 x 10-31 carrying a charge of 1.6 x 10-19
• λ = h/mv where h is Planck's constant (6.6 x 10-34 Js). This is often referred to as the 'de Broglie wavelength' of the electron. This equation links the wavelength to the momentum (mv) of the 'particle'. The relationship was subsequently verified using electron diffraction
• Diffraction is the wave phenomenon and for electrons to be diffracted, they must be behaving as a wave. This is true for both diffraction through a grating and reflecting off a grating
• In 1924, Prince Louis de Broglie first suggested a method for calculating the wavelength of the perceived electron wave. He suggested that the wavelength is inversely proportional to the momentum of the electron when it is considered a particle
• The equation that supported de Broglie's hypothesis was: electron wavelength(m) = Planck's Constant (Js) / momentum (Kgm/s)
• The kinetic energy of a particle = charge of the particle x accelerating pd. KE = qV
• Electron interference - It was first suggested in 1965 that electrons should also be expected to produce the same 2 slit interference patterns that we observe with light
• Electron microscopy is used because the wavelength can be controlled (through varying the accelerating voltage), the wavelength can be made to be many times smaller than that of visible light and therefore, much smaller subjects can be imaged
• Huygens' principle (light as a wave) - This principle is to predict the future motion of a wave with a known wavefront. It considers every point on the wavefront to be a source of a new circular wave. When the new circular waves are plotted and their superposition considered, the resultant wave will be the new wavefront.

  
• The evidence that suggests that light is a wave is diffraction, superposition and polarisation
• The evidence that suggests that light is a particle: Max Planck suggested in 1901 that light could exist as quantised (minimum possible) packets of energy called photons. Einstein built on this idea to describe the photoelectric effect - which earned him the Nobel Prize in 1921. This observed phenomena cannot be explained using wave theory. It can be described if light consists of photons whose energy is proportional to the frequency of the wave.
• For EM radiation, the energy of a photon can be calculated by multiplying the frequency by Planck's constant (h). THe value of h represents the minimum possible step in energy and is 6.63 x 10^-34 Js
• Photon energy (J) = Planck's constant (Js) x frequency (Hz), which is also represented by the equation E = hf
• Electrons are particles as we can quantify the mass and charge of an electron by measuring the related changes when an atom is changed into an ion