Waves
DOUBLE SLIT INTERFERENCE
How interference pattern occurs
Light travel past narrow slits and diffracts, as each slit acts as a point source e As light diffracts, lights from each source overlap and interfere
There are points where lights arrive in phase, and the path difference is a whole number of wavelengths (nλ).
Constructive interference occurs at this point, causing a bright spot (maxima). (the path difference is 1λ for 1st order maxima, etc)
In between the maxima, there are points where lights arrive out of phase and cancel out. The path difference is an odd number of half wavelengths (1+nλ), and destructive interference occurs - dark fringes are produced.
Relationship between dsin(theta) = nλ
Changing A (eg by using redlight instead of violet light)
λ(red)>λ(violet)
λ proportional to angle, hence larger the λ, larger the angle to reach maxima
Thus, using red light, maxima will be further apart, and maximum number of maxima on the screen less than that of violet
Changing d (slit spacing)
d is inversely proportional to sin(angle)
Thus, when d increases, the angle decreases
Thus, maxima are closer together, and maximum number of maxima on screen greater
Diffraction grating vs double slit Using doubleslit
Wider maxima - fewer sources combining in superposition, thus less perfect cancellation. occurs in between maxima (the brightness of fringes change gradually as the path difference changes gradually)
Dimmer maxima - less point sources of light, thus less energy combining at each maxima constructively
Using diffraction grating:
Narrower maxima - constructive interference occurs much less frequently as the diffraction grating creates many point sources combining in superposition; thus, more waves cancel out perfectly (fewer positions where all waves arrive together)
Brighter maxima - are created by constructive interference of many extra sources
For both: maxima are still located at the same angle/position, the path difference is still a wavelength integer at the same location, as slit spacing d has not changed Explain the pattern of white light shining through a diffraction grating (Spectrum)
white light contains all wavelengths of the visible light spectrum. AS white light passes through the diffraction grating, all light waves diffract and spread through 180 degrees. each slit acts as a point source for all wavelengths of the visible spectrum.
waves interfere with other waves of the same wavelength from other slits. interference is predominantly destructive due to the many closely spaced sources.
in the dark regions, destructive interference occurs for all wavelengths of visible light, thus no light is seen
individual colours are seen where constructive interference is occurring for that colour, but destructive interference occurs for all other waves of different colours (wavelengths)> constructive interference occurs at different positions for different wavelengths according to nλ = dsin(angle)- larger the λ, larger the angle the wave diffract at. (give a calculation comparing angle to 1st order maxima for red and violet light if provided wavelengths). thus different colours are seen at different positions.
the central line is white. as this is the central antinode for all colours, here constructive interference occurs for all waves of the same wavelengths thus the path difference is 0λ for all waves. therefore the white colour seen is a composite maximum for all colours of the visible spectrum.
this the overall pattern observed is a central white line with coloured bands on either side.
To calculate maximum number of orders
set angle = 90 degrees, or calculate by tan (x/2 / L)
Calculate for n, round down to the whole number of maxima (half n for minima)
total is 1+nx2 (1 for central into and n on either side of the centre)
standing waves
how standing waves are formed
progressive waves travel by plucking string/blowing pipe to an end
waves reflect when it reaches the end of the string/pipe
for closed-end: phase inversion of 180 degrees occurs, thus reflected wave and incident wave are out of phase; hence destructive interference occurs, fixed note at this point.
for the open end: no phase inversion, thus reflected wave and incident wave are in phase; hence constructive interference occurs, fixed antinode at this point giving maximum vibration
the incident waves and reflected waves travel past each other and interfere, as they’re travelling in opposite directions and have the same wavelength
where waves arrive in phase: constrictive interference and antinode for maximum vibration and amplitude
where waves arrive out of phase: destructive interference and node particles cannot move here, minimum amplitude
the positions of N and AN alternate and these positions are fixed. thus positions of destructive and constructive interference alternate and are fixed formed standing wave pattern.
explanation for the specific correct wavelengths
Explanation for correct wavelengths in closed-closed (C-C) end and open-open end (0-0)
for C-C, only waves with wavelengths that fit nodes at both ends can persist and resonate
for O-O, only waves with wavelengths that fir antinodes at both ends can persist and resonate
this wavelength is λ=2L(for the fundamental frequency). For increased harmonics, the pipe length would be a multiple of exact half number of wavelengths.
only frequencies that correspond to these wavelengths will cause significant vibration and be heard (form a standing wave)
why are there no even harmoics for open-closed end (O-C) pipes?
for O-C, the node needs to occur at the closed end and an antinode is required at the open end. this waves with wavelengths that can fit N on one end and AN on the other can produce standing waves in O-C pipe.
for fundamental f, pipe lengths L = λ/4
for even harmonics, the pipe length should be an even number of quarter wavelengths eg λ/2, λ, 3λ/2. This can only occur with a node or an antinode at both ends. thus even harmonics cannot be sustained in an O-C pipe.
hence only odd garmonics can be sustained in O-C pipe. odd harmonics require the pipe length to be an odd number of quarter wavelengths eg λ/4, only waves with these wavelengths can form standing waves in O-C pipes as they give an AN on one end and an N on the other end
thus frequencies that correspond to these correct wavelengths will form standing waves in O-C pipes, so sound Is heard.
effect od opening/closing the hole in the pipe
the length of pipe is regarded from the hole, as an AN is located at this points, causing maximum vibration
when the hole is closed the effective length of the pipe increases
for fundamental frequency, L = λ/2 (or λ/4 depending on O-O or O-C)
thus L is proportional to λ - when L increases, λ also increases
v=fλ thus f inversely proportional to λ
as λ increases, f decreases (v is constant)
this not sounds at a lower pitch when close hole.
difference between travelling waves and standing waves
travelling waves transfer energy, standing waves do not
standing waves require two sources, and travelling waves only require one source
standing waves require interference, travelling waves do not.
effect of changing the material of pipe/string
V of sound changes as it is now travelling in a different medium
at fixed f, λ changes
this new λ may no longer fit in the pipe/string, as it will be different to λ that equals 2L or 4L
therefore even if the frequency is unchanged, the frequency correspisnign to this new wavelength may not beable to produce standing waves in the pipe/string and hence doesn’t resonate to produce sound
Timbre: why is the note played at the same frequency but it sounds different for two instruments?
When a string is plucked, there is a sudden increase in the amplitude of the wave which dies away
When the pipe is blown, a standing wave builds in pipe with a more steady amplitude
The timbre of a note produced depends on the wave shape and the relative amplitude of the harmonics emitted by the instruments. This is caused by the difference in amplitude of overtones.
wave shapes and relative amplitudes are different in string and pipe, producing different timbres
The pitch depends on frequency. As the two instruments are playing the same frequency, the note heard is considered the same pitch, however, they sound different due to different timbres
Doppler effect
for a source moving in constant velocity
travelling towards the observer
As each wavefront is emitted, the source has moved in the same direction as the previous wave in front
so the wavefronts in front of the source are compressed together.
this causes the waves reaching the observer to have a shorter wavelength
The sound waves still travel at the same velocity through air, and therefore a higher apparent frequency as f=v/λ. This is observed as a higher-pitched sound.
travelling away from the observer
As each wavefront is emitted, the source has moved in the opposite direction as the previous wave behind
so the wavefronts behind the dog are spread out.
The sound waves still travel at the same velocity through the air, but the waves reaching the observer have a longer wavelength and hence a lower apparent frequency.
This is observed as a lower-pitched sound.
for sound source moving in constant acceleration
Same explanation as in constant velocity, however, frequency gradually increases/decreases
In constant velocity, the pitch of note (frequency) observed is steady, but it’s just higher or lower than the original frequency.
*V of sound does not change despite the acceleration of sound source as the medium is unchange
Things to remember for Doppler effect
person with the source/moving with the source does not experience change in f as the sound source I never moving away/towards them
No Doppler shift occurs unless the sound source is approaching/receding; thus no Doppler shift perpendicular to the motion of the sound source
a person observing must be close to the sound source to eliminate the effect of angle
to calculate vs on an angle: calculate vs normally using the Doppler shift equation the calculate actual Vs using trigonometry.
diagram for Doppler shift is always circular waves, bunched up towards the side it is moving
beats
beats is the regular variation in the loudness of sound heard
beats occur when two waves of slightly different frequencies interfere and superimpose
where they arrive in phase, constructive interference occurs, producing a louder sound
where they arrive out of phase, destructive interference occurs, producing a quieter sound
as the amplitude of waves reaching the observer changes regularly over time, and the interference b/w constructive and destructive alternates the sound heard by the observer varies regularly over time between loud and oft
the period of getting louder and ofter is the beat frequency.