SUPERPOSITION

when >1 wave w/ same freq overlap => interference (illustrated by ripple tank)

2 sources A & B w/ equal freq giving circular waves
principle of superposition can explain interference pattern
along RS: Acrest and Bcrest meet at same time > constructive interference > large wave along RS
along XY: Acrest and Btrough meet at same time > destructive interference > no wave along XY(> every pt on XY: l/2 nearer to A)
reinforcement: constructive interference- at pt C when path diff AC - BC = kl (k Î Z+)
cancellation: destructive interference- at pt C when path diff AC - BC = (k+½)l (k Î Z+)

Principle of superposition:
when 2 waves travel through, the resultant displacement at any pt = vector sum of separate displacement due to the 2 waves


Interference of microwaves

max & min detected along AB due to inter. of wave from PQ
Interference of sound waves

2 loudspeakers in // w/ audio freq oscillator microphone/ ear moved along MN: alternate loud & soft

Conditions for 2-source interference
-coherent sources: same freq (=> same l), in phase/ w/ const phase diff
-~ same amplitude (or else no min)
-for transverse waves: -unpolarised / -polarised in same plane
-for light: sources v.clost as l of light is v.small (not close: light & dark bands v.close => can't be seen)

Young's double slit experiment

A, B, C: narrow slits, A & B: v. close together
bright & dark bands seen on both sides of O (perpendicular bisector of AB)
[width of slits = 0.2mm, separation of double slit = 0.5mm]
white light used: diff colours > diff l

P = mth bright line, BP - AP = ml
PA = PN = > BN = BP - AP = ml
AB is v.small > PM >> AB => AN & PN meet at ~90° => ÐPMO » ÐBAN = q
DBAN: sinq = BN/AB = ml/a
DPMO: tanq = PO/MO = xm/D
q is v.small, tanq » sinq => xm/D = ml/a => xm = mDl/a
Q: neighborring bright fringe- OQ = xm-1 = (m-1)Dl/a
\fringe separation, x = Dl/a [D: separation of double slit & screen, a: separation of double slit, l: wavelength of wave]

llight: 10-6m
source slit closer to double slit: x unaffected (only brighter)
source slit widened & widened > fringes gradually disappear (large slit acts as several narrow slits > many interference patterns at diff places > dark & bright fringes overlap > uniform illumination)
white light used: central fringe = white, other fringes = colour
blue fringe (smallest l) closest to central, red fringe (largest l) furthest from central


Diffraction = spreading of wave when wave pass through aperture / around obstacles



generally: smaller aperture (compared to l) > more diffraction
llight= 6 × 10-7m => small objects/ aperture: sig diffraction of light

sound waves can diffract around wide openings as opening comparable to sound
diff µ l (low notes heard easier than high notes)

Single slit diffraction



A & B: on plane wavefront & can be considered as 2 secondary sources of light, since from same wavefront => coherent w/ identical amplitude =>interference pattern as long as slit is narrow (rel to l)
short dist beyond M & N (projections of A & B): alternate bright & dark fringes
central fridge: twice as wide as & intensity v.much greater than other bright fringes
sin q = l/d (d: separation of double slit)



Increasing # of slits





each slit: similar diffraction effect in same direction > pattern similar to 1 slit w/ intensity variation (but crossed by intermediate fringes due to interference bet slits)
more // equidistant slits; intensity & sharpness of: -principal max increases, -subsidary max decreases
100's slits/mm: only a few sharp principal maxima

diffraction grating: glass/ metal w/ large # of close // equidistant slits ruled on it

grating w/ n lines/mm: d = 1/(n × 104)m
BC = dsinq = path diff bet 2 adjacent slits
for bright fringe: BC = k l (k = whole #/0)
=> sinq = BC/AB = k /d => dsinq = kl
1st order bright fringe = 1st adjacent fringe to central fringe


more slits > less # of bright fringes which are brighter

Measurement of wavelength

find 1st order of diffraction on each side => angular diff = 2q => l = dsinq (k = 1)

Stationary /standing waves
-formed when 2 identical waves (same freq + amp) travel in opp directions in a medium
-wave-like profile doesn't move along medium

N-nodes = pts of permanent zero displacement
A-anti-nodes = pts ½ way bet 2 nodes w/ max amplitude of vibration (vib: 0 £ A £ 2 amp.original)
dist bet 2 nodes, NN = l/2, AA = l/2, AN =l/4

Modes of vibrations of stretched string

fundamental freq, f0 (1st harmonic): l =l/2 => f0 = v/l = v/2l

1st overtone, f1 (2nd harmonic): l =l1 => f0 = v/l1 = v/l = 2f0

2nd overtone, f2 (3rd harmonic): l =3l1/2 => f0 = v/l2 = 3v/2l = 3f0

nth overtone: (n+1)f0

Stationary waves in air columns
closed pipe- open end always anti-node, closed end = node
open pipe- both ends always anti-node
  closed open
fundamental freq, f0 (1st harmonic) l =l/4 => f0 = v/l = v/4l l =l/2 => f0 = v/l = v/2l
1st overtone, f1 (2nd harmonic) l =3l1/4 => f1 = v/l1 = 3v/4l = 3f0 l =l1 => f1 = v/l1 = v/l = 2f0
2nd overtone, f2 (3rd harmonic) l =5l2/4 => f2 = v/l2 = 5v/4l = 5f0 l =3l2/2 => f2 = v/l2 = 3v/2l = 3f0
nth overtone (2n+1)f0 (n+1)f0
Measuring l of sound using stationary waves

sound waves of const freq emitted from loudspeaker aimed at reflecting surface > sound reflected > interfere w/ incident waves > stationary waves set up
microphone, connected to y-plates of oscilloscope, moved through region of standing waves
mic placed at 2 successive max pts (anti-nodes- loudest sound) => dist, x, bet pts = l/2 => l = 2x
anti-node >(move)> node: loud > soft sound, node >(move)> anti -node: soft > loud sound

Calculate velocity of sound in air by Kurdt's tube


freq of sound varied, resonance occurs when f equals natural freq of vib of air column in tube > stationary waves formed > lycopdium powder collects in heaps at nodes (no disturbance)
av. dist bet 2 heaps/nodes = l/2

Resonance tube experiments

A: anti-node actually slightly above rim, dist 'c' above rim
at 1st resonance: l1 + c = l/4
at 2nd resonance: l2 + c = 3l/4
l2 - l1 = l/2


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