Chapter 12 - Waves 2

Chapter 12 - Waves 2

1. Superposition of Waves

   • Overlap of two waves at a point in space
     • Resultant displacement is the sum of displacements of the original two waves
   • Constructive interference - Two waves in phase create a resultant wave with increased amplitude
   • Destructive interference - Two waves in antiphase create a resultant wave with decreased or zero amplitude

2. Interference

   • Coherence - Waves emmitted from two sources with constant phase difference and frequency
   • Maxima - Loud spots created from waves arriving in phase, constructively interfering. Path difference of whole number of wavelengths (0, λ, 2λ)
   • Minima - Quiet spots from waves arriving in antiphase, destructively interfering. Path difference of half number of odd wavelengths (0.5λ, 3/2λ, 5/2λ)
   • A signal generator with two speakers can be used to create an interference pattern, dected by a microphone
   • A microwave source can be used along with double slits, to create an interference pattern detected with a receiver

3. Young double slit experiment

   • It demonstrates the wave nature of monochromatic light. The slits have to be narrow enough to diffract the light, which creates an interference pattern in the form of light and dark fringes
   • Seperation between slits = d
   • Difference between screen and slits is L, where L>>d
   • Two bright fringes observed at P and C, with separation between them of y
   • Path difference s = λ, since L>>d
   • The two dashed rays of light are almost parallel
   • Sinθ₁ = sinθ₂ = tanθ₂
   • Where Sinθ₁ = s/a and tanθ₂=x/L (s=λ)
   • Therefore λ/a = x/L
   • (In the formula book, distance to screen L is represented by D)

4. Stationary Waves

   • Formed when two progressive waves travelling in opposite directions with the same frequency, superpose
   • When the two waves are in antiphase, they create a node, where displacement is always zero.
   • When the two waves are in phase, an antinode is formed, where amplitude and intensity is greatest
   • In a stationary wave, the seperation between two nodes/antinodes is 1/2λ. The stationary wave has the same frequency, but doesn't transfer energy
   • By reflecting microwaves off a metal sheet, a stationary wave can be formed. The wavelength can be detected by moving a receiver, until it detects nodes and antinode. The distance between successive nodes is λ/2, λ being wavelength of the microwaves

5. Harmonics

   • Fundamental frequency - minimum frequency of a stationary wave needed to create half a wavelength (depending on string's mass, tension and length)
   • Harmonics are produced, increasing in integer multiples of F₀
   • V=fλ shows that as frequency increases, wavelength decreases. This is because at a fixed tension, v remains constant
   • This wave can be created with a vibration generator, using a string attached to a fixed point

6. Stationary waves in air columns

   • In a tube closed at one end, there is an antinode at the open end, the closed end is a node, . Harmonics increase by 1/2λ e.g 1/4λ , 3/4λ, 5/4λ. The harmonics are always in odd multiples of the fundamental frequency, 3F₀, 5F₀ etc.
   • In an open tube, there are antinodes at both ends, and harmonics increase by 1/2λ, e.g 1/2λ, λ, 3/2λ. Harmonics are all integer multiples of F₀, 2F₀, 3F₀ etc.
   • A tuning fork can be used to create a stationary wave in a resonance tube. If f (tuning fork) matches F₀, the sound gets louder and resonates.
   • The length of tube above the water must be L=1/4λ. The speed of sound can be calculated with v=fλ =f*4L

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