AHL Waves

  2. Oscillations and waves
  3. AHL Waves
    4. 20'

In Standard Level Waves, you learnt the basics of everything we will observe here, but with a few notable extensions.

What is the Doppler effect? What are the effects of a moving source, observer and medium? What are the equations for simple harmonic motion? How is the diffraction pattern formed for light passing through a single slit? What affects the interference patterns for two or more slits? What is thin film interference?

Key Concepts

Doppler effect

The Doppler effect is the change in frequency in observed sound due to relative motion between source and observer. 

For a moving source and stationary observer, the observed frequency can be calculated using the following equation:

\(f'=f({v\over v\pm u_s})\)

  • \(f'\) is the observed frequency
  • \(f\) is the frequency emitted by the source
  • \(v\) is the speed of sound (ms-1)
  • \(u_s\) is the speed of the source (ms-1), where \(u_s>0\) for the source moving towards the observer

If the source is stationary and the observer moves, we adapt the equation:

\(f'=f({v\pm u_o \over v})\)

  • \(f'\) is the observed frequency
  • \(f_0\) is the frequency emitted by the source
  • \(v\) is the speed of sound (ms-1)
  • \(u_o\) is the speed of the observer (ms-1), where \(u_o>0\) for the source moving towards the observer

A sonic boom is produced when the relative speed of the source exceeds the speed of the wave in the medium.

The Doppler effect affects light waves and can be used to determine the relative speeds of objects in the universe. The light of receding objects is red shifted. The light of approaching objects is blue shifted.

\({\Delta f \over f}={\Delta \lambda \over \lambda} \approx{v\over c}\)

  • \(\Delta f\) is the change in observed frequency (Hz)
  • \(f\) is the frequency emitted by the source (Hz)
  • \(\Delta \lambda\) is the change in the oberved wavelength (m)
  • \(\lambda\) is the wavelength emitted by the source (m)
  • \(v\) is the speed of the source (ms-1)
  • \(c\) is the speed of light (ms-1)

 

Simple harmonic motion

Simple harmonic motion is defined as oscillations where the acceleration is directly proportional to the displacement from a fixed point and always acts towards that point.

\(a  = -ω^2x\)

  • \(a\) is acceleration (ms-2)
  • \(\omega\) is angular frequency: \(\omega=2\pi f={2\pi\over T}\) (Hz)
  • \(x\) is displacement (m)

We can also calculate the displacement and velocity.

 \(x=x_0\cos{\omega t} \Rightarrow v=-\omega x_0\sin \omega t \)

\(v=\pm \omega\sqrt{{x_0}^2-x^2}\)

  • \(x_0\) is amplitude (m)
  • \(t\) is time (s)
  • \(v\) is velocity (ms-1)

When an oscillating system is modelled as a mass on a spring, we can calculate the time period (and therefore frequency) of the oscillations:

\(T=2\pi \sqrt{m\over k}\)

  • \(T\) is time period (s)
  • \(m\) is mass (kg)
  • \(k\) is spring constant (N m-1)

Simple pendula have a similar equation, but with different variable quantities:

\(T=2\pi\sqrt{l\over g}\)

  • \(T\) is time period (s)
  • \(l\) is the length of the string
  • \(g\) is gravitational field strength (N kg-1)

The total energy of an oscillating system is the sum of the kinetic and potential energies. Potential energy falls to zero when kinetic energy is maxium, and vice versa.

\(E_k={1\over 2}m\omega^2({x_0}^2-x^2)\)

\(E_T={1\over 2}m\omega^2{x_0}^2\)

 

Single-slit diffraction

Diffraction is the spreading out of a wave when passing through a gap in a boundary.

We can determine the position of the first minimum using the approximation equation:

\(\theta = {\lambda \over a}={y\over D}\)

  • \(\theta\) is the angle subtended at the slit by the first minimum and the centre (rad)
  • \(\lambda\) is the wavelength of the light (m)
  • \(a\) is the width of the slit (m)
  • \(y\) is the distance between the centre and the first minimum (on either side since symmetrical) (m)
  • \(D\) is the distance from the slit to the screen (m)

The larger the wavelength, the more spread the pattern becomes. Diffraction separates the colours of white light in the outer maxima, with the red light subtending larger angles than blue.

 

 

 

Essentials

Resolution

Two sources of light become resolved (distinguishable as separate) when the Rayleigh criterion applies: the principal maximum of one diffraction pattern coincides with the first minimum of the other.

In the instance that the Rayleigh criterion applies, we can calculate the separation of the sources:

\(\theta=1.22{\lambda\over b} ={\text{separation}\over D}\)

  • \(\theta\) is the angle subtended by the centre and the first minimum (rad)
  • \(\lambda\) is the wavelength of light (m)
  • \(b\) is the diameter of the aperture (m)
  • \(D\) is the perpendicular distance from the centre of the two objects to the slit (m)

Diffraction gratings can be used to resolve two wavelengths of light.

\(R={\lambda\over \Delta \lambda}=mN\)

  • \(R\) is the resolvance of the diffraction grating
  • \(\lambda\) is the mean wavelength (m)
  • \(\Delta \lambda\) is the smallest resolvable wavelength difference (m)
  • \(m\) is the order of the diffraction
  • \(N\) is the total number of slits illuminated

 

Multiple-slit interference

Light interferes when it passes through two or more slits in a boundary. The effect is observable for coherent light beams (constant phase difference) with the same amplitude.

Young (pictured above) observed the interference pattern for a double slit.

The fringe spacing can be calculated as follows:

\(s={D\lambda\over d}\)

  • \(s\) is the distance between consecutive maxima (m)
  • \(D\) is the distance from the slits to the screen (m)
  • \(\lambda\) is the wavelength of light (m)
  • \(d\) is the distance separating the centres of the two slits (m)

The two-slit interference pattern is modulated by the one-slit diffraction effect.

When light shines through multiple slits, an interference pattern forms similar to that through a double slit.

\(n\lambda = d \sin \theta\)

  • \(n\) is the number (order) of the maximum, where the central maximum is order 0
  • \(\lambda\) is the wavelength of the light (m)
  • \(d\) is the distance separating the slits (m)
  • \(\theta\) is the angle subtended by the maximum and the centre

The more slits added, the more narrow and intense the peaks become. A diffraction grating is an array of identical, equally-spaced slits.

 

Thin film interference

Thin film interference is the effect observed on soap bubbles, oily puddles and peacock feathers.

When a wave reflects off a loose boundary, it experiences no phase change. However, when a wave reflects off a fixed boundary, it experiences a phase change of \(\pi\) radians.

Thin film interference is a result of the combination of the different wavelengths of light and the different distances travelled through the film.

The condition for constructive interference is \(2dn=(m+{1\over 2})\lambda\):

  • \(d\) is the thickness of the film (m)
  • \(n\) is the refractive index of the film
  • \(m\) is an integer value
  • \(\lambda\) is the wavelength of light (m)

The condition for destructive interference is \(2dn=m\lambda\).

 

 

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