Waves and Oscillations Flashcards
Oscillations
Involve any motion that regularly repeats itself about a fixed (equilibrium/mean) point: this mean point is usually where it comes to rest
Ex. mass moving on vertical spring; simple pendulum (whatever is attached to string is called a bob [won’t likely see b/c too simple]); light object bobbing up and down in water; diving board; swing; ruler; etc.
Displacement (x)
The instantaneous (in the course of the journey, displacemt. can change [so, at an instant]) distance of the oscillating object from the mean point in a given direction (unit: m)
Amplitude (A)
Max displacement of object from mean position. There can only be one amplitude (unit: m)
Frequency (F/f)
of oscillations (cycles [full circle/full path and ) completed by the object in one second (unit: Hz)
Period (T)
Time taken to complete one oscillation (unit: s).
F = 1/T
Inversely proportional (do inverse to find other).
Phase difference (Φ)
Angle difference between two oscillations that are out of phase. In ex. (a?), phase difference 180º/π (half a cycle, which is 360º).
In-phase: same side, same point
Out-of-phase: different sides, same point
Simple harmonic motion (SHM)
Oscillation where acceleration caused by a restoring force is always proportional to its displacement, and it is always directed towards the mean position (opposite the displacement)
Defining equation:
a ∝ -x
a = -w^(2)x
The acceleration is caused by a restoring force (F): since a ∝ -x, F ∝ -x
Negative sign signifies that a always points back towards mean position (opposite displacement)
SHM graph
a on y, x on x: when one positive, other negative (negative, linear [passes through origin])
*See note for a, v, x and, potentially, other formula (when x max, a max and opposite [v zero])…
W/ x and v, just remember free fall (as it falls, x decrease and v increases: as it rises, x increases and v decreases)
Energy during SHM
*Refer to note if needed…
Energy never negative, so above x-axis (t on x-axis): total E fixed, but KE and PE change throughout journey (see graph)—when one max, other zero (equal at two points)
PE decreases as going down; KE increases as going down
Wave pulse
Single disturbance that travels through a medium (one loop is a pulse and two is a cycle)
Continuous/Progressive wave
Repeated disturbance that propagates through a medium, transferring energy (going in same direction) from one place to another, w/o the net motion of the medium (medium doesn’t travel w/ the wave [remember pink Post-it on slinky, the medium]
Particles of medium oscillate abt mean position, returns to mean (no net movt of medium); oscillation of medium particles is SHM
Transverse wave
Wave whose oscillation of medium particles is perpendicular to the direction of energy transfer
Ex. water waves, all EM waves (radio, micro, gamma, x-, infrared, visible)
Almost all waves we know of are transverse (few are longitudinal [sound the most pervasive ex.])
Longitudinal wave
Other category (based on relationship between oscillation of medium particles and direction of energy transfer)
Wave whose oscillation of medium particles is parallel to the direction of energy transfer
Ex. sound waves, earthquake, ultrasound, etc.
W/ sound, air molecules compressed together (compression) or spread apart (rarefaction)
*See note, ofc (sine graphs can represent any wave [only labels change])
Wavefronts
One of two concepts used in wave analysis
Lines representing all particles of waves that are in phase w/ equal amt of wavelength (full cycles)
Ray
Other concept
Line extending outwards from the source of a wave, representing every part of the wave (shows direction of energy flow/travel direction)—used to describe characteristics of waves
*Perpendicular to wavefront
*when x-axis x, wavelength; when t, period
Wavelength (λ)
Shortest distance between two points that are in phase OR distance from crest to adjacent crest OR trough to adjacent trough (unit: m)
*Three
Wavespeed (c or v)
Speed at which energy is transferred by a wave through a medium (unit: ms^[-1])
c = fλ
Intensity (I)
Power per unit area of wavefront (unit: Wm^[-2])
I ∝ A^2
*A is amplitude
Electromagnetic (EM) waves
Oscillation electric and magnetic fields that are perpendicular to the direction of energy transfer/propagation of the waves (the two fields are at right angles to each other)
Move at the speed of light (3 x 10^8 ms^[-1])
They’re transverse waves and are classified according to their wavelengths or frequencies (classification called the EM spectrum)
E ∝ f
E = hf
f ∝ 1/λ
Through a vacuum
Visible light is of the range 400-800 nm (or 400-700 nm [range, so doesn’t really matteer]) and has the colours ROYBGIV (increasing f, decreasing λ)
*Need to know the actual values…
Sound waves
Disturbances that propagate through the aire, compressing and expanding the air molecules as they move (in other words, it changes the air pressure/density as it moves)
Direction of the oscillating of the molecules is parallel to the propagation of the waves, making it a longitudinal wave
Reflect off a wall/barrier, resulting in an echo (why one’s voice sounds differently in the room from the outside)
Travel the fastest in solids (molecules closely packed together)
Travel faster in gases of higher temp (greater particle collision) than those of lower temp
W/ ex., tuning fork produced sound, set air molecules in motion, which then carried the sound to your ears
Reflection
The bouncing off of a wave on hitting a barrier
Reflection at fixed end
Wave inverted as it bounces off
Reflection at loose end
No inversion
*Loose meaning not tightly attached
Diffraction
Another property of all types of waves
The spreading out/scattering (particles can scatter/diffract as well) of waves when they pass through apertures/holes/slits or meet obstacles (don’t say ‘bending,’ as it may not be precise enough/advanced enough [?])
Try to squeeze themselves through, spread out
*Regardless of shape, those lines are wavefronts
For circular apertures, wavefronts will become circular (once squeeze through, will spread out on other side)
For obstacles, only place waves can pass through is after obstacle (all will try to squeeze through upper part [in process, scatter])
Factors that affect diffraction
Aperture size - the smaller the aperture, the greater the diffraction/spreading (and vice versa)
*For big aperture, more or less horizontal in middle (diffraction less pronounced [very wide would mean hardly any diffraction])
Wavelength of wave - the greater the wavelength, the more the diffraction
Applications of diffraction
Dispersion of light into its component colors by compact disc (CD), polished surfaces, etc. (little holes in CD covered in plastic film)
Measurement of blood flow (complex device at big hospitals makes use of diffraction)
Hearing of sounds when the source isn’t seen
Refraction
Change in speed/wavelength (directly proportional) of a wave as it goes from one medium to another
Results in the bending of the wave
Frequency of wave constant during refraction (v = fλ [if two variables change, one must remain fixed])
Angle, λ, speed, # of particles/density (of medium), direction—all these are changing
*See bonding…
Less dense to more
Refracted ray bends towards normal
n2 > n1
*Refractive index is a function of medium density (and directly proportional])
More dense to less
Refracted ray bends away from normal
n1 > n2
NOTE (refr.)
When moving from less dense to more dense, speed decreases, wavelength decreases, frequency stays constant, and light bends towards normal
When moving from more dense to less dense, speed increases (less restriction by particles), wavelength increases, frequency stays constant, and light bends away from normal
Density of medium directly proportional to refractive index (ability of a medium ot bend a wave)
Refraction of wavefronts
W/ wavefronts, forget abt normal; just focus on boundary (other way around for ray)
Less dense to more dense:
Here, bends away
Angle between wavefront line and boundary (same for each line)
Wavelength decreasing
More dense to less dense:
Wavelength increasing (moving closer to boundary, away from normal)
Snell’s Law
States that when a ray is refracted between two different media, n = sin i/sin r, where n is a constant (refractive index)
Law applies to when light moves from air/vacuum to another medium (medium 1)
sin i/sin r = n2 (refractive index of medium 2)/n1 (refractive index of medium 1)
Refractive index of air/vacuum is approximately 1
NOTE
Refraction does not occur if light strikes a boundary at 90º (has to be less)
n = v1 (speed of incident ray)/v2 (speed of refracted ray) = λ1 (λ of incident ray)/λ2 (λ of refracted ray)
n2/n1 = sin i/sin r = v1/v2 = λ1/λ2
*λ part not in db
W/ interference of ex. water waves (generally, for liquids), where the wavefronts intersect, constructive, and anywhere they don’t, destructive (different types)
See single-slit graphs and others….
For polarization using a polaroid/polarizer, a polaroid is a substance used to polarize light
Applications of refraction
Objects in water appear to be in different positions
Splitting of whit light into ROYGBIV
Interference (Superposition)
The combination of two or more identical (same wavelength, frequency, same type) waves traveling in opposite directions to form a resultant wave
Displacements combine (don’t just say A’s b/c all x’s combine, whether A or not)
Superposition Principle
States that when two or more waves overlap (combine) , the overall displacement of the resultant wave is a vector sum of the displacements of the individual waves
Constructive interference
The individual waves on same side of the mean position combine to form a resultant wave w/ a bigger displacement (after, continues journey in same direction)
Destructive interference
The individual waves on opposite sides combine to produce a resultant wave whose displacement is smaller
Complete destructive interference
A special kind of destructive interference that involves two waves w/ same displacement
Law of Reflection
States that when an incident ray strikes at an angle on a boundary, the individual angle equals the reflected angle (i = r)
*On diagram w/ fixed boundary, wave reflected and inverted back, but some was transmitted/refracted and had less amplitude/energy)
So, you’ll have both when the angle is less than 90
Greater the incident angle, greater the refracted angle (proportional [n2 = sin i/sin r])
Critical angle
The incident angle at which the refracted ray (critical ray) has an angle of 90 (a little further [increase that passes]?)
*Partially refracted angle still present: when 90 passed, only one present (when refracted angle 90, disappears [no refraction])
Total internal reflection
Only occurs when more dense at top
Occurs when the incident angle > the critical angle (at this angle, refraction cannot occur anymore [incident ray will be totally reflected])
*See formula, remembering that medium 2 is not your air/vacuum, the refracted angle is 90, and i is now your critical angle
Polarization
Polarized light is a light for which the electric field is oscillating in only one plane
Only electric fields blocked (magnetic still goes through [doesn’t affect polarization]): only that which aligns w/ polarizer passes (can be horizontally or vertically polarized)
Originally, changes plane continuously (? [electric and magnetic, and at right angles to each other—2)
No light at all when everything filtered out (even if magnetic going through, if no electric present, no longer light [light is a combo])
All the different planes…
Non-polarized light is light whose electric fields vibrate in different directions
Methods of polarization
Using a polaroid/polarizer, by reflection, by refraction, by scattering
Polarization by reflection
Natural (don’t do anything to make it happen): when light is incident on a transparent object (ex. glass, water, plastic, non-metallic surface), it is horizontally polarized
Application: if a horizontally polarized light hit polarized sunglasses with a vertical transmission, no light would pass through
If diagonal, component (some still go through b/c other things happening to light [as long as not 90])
Analyzer
Type of polarization used to detect direction of unpolarized light (if you didn’t know at first, can find out)
If analyzer is parallel or at an angle to the polarized light, light will get through
But if the analyzer is placed perpendicular to polarized light, there’ll be a zero transmission
ex. unpolarized through vertical polarizer (only vertical passes through), then parallel analyzer (goes through), then horizontal analyzer (no light)
Malus’ Law
States that an analyzer only allows a component of polarized light in same direction w/ it to be transmitted through
Applies when the polarized light strikes the analyzer at an angle
See formulae…
Double-slit/Two-slit interference
Conditions necessary for a clear interference pattern for double-slit:
- Sources must be coherent (must maintain a constant phase difference)
- The two sources must have the same amplitude/intensity and frequency/wavelength
See equations and graphs…
s = distance between two adjacent maxima
D = distance to screen
d = separation of slits
b = slit size
x = width of first max
Inverse Square Law of Radiation
States that for a given area of a medium (?), the intensity of light received is inversely proportional to the square of distance from the point source
I = P/Area = P/4πr^2
Diffraction/Interference of light (or sound)
Through a single slit:
Path length - distance traveled by wave (ray) from source to location
Path difference (∆L) - difference in path length between two waves
Conditions for constructive interference
- Phase difference = 0
- Path difference = nλ, where n = 0, 1, 2, 3,…
Conditions for destructive interference
- Phase difference = π (or 180º)
- Path difference = (n +1/2)λ, n is a positive integer
Single-slit diffraction/interference
See diagram…?
Series of bright and dark bands on screen (fringes)
- When light is incident on a single slit, both diffraction and interference occur as a result
- Constructive interference will occur if path difference is nλ (produces bright bands on screen)
- Destructive interference will occur if path difference is (nλ + 1/2)—produces dark bands on screen
- λ of light must ≈ the slit size
See diagram…
Central brightest
Goes by half λ in either direction
Standing/Stationary waves
Special kinds of waves that occur as a result of reflection and interference (have to happen at the same time) of the following:
- two identical waves
- of same speed, λ, and A
- traveling in opposite directions
Resultant wave is a now fixed wave (in one place, but still moving, ofc), whose amplitude varies (different spots it can be at as it vibrates)
Two fixed ends reflect and then interfere (new wave)
Change f to change # of loops
Max fatness (width) multiples of first harmonic (?)
Three nodes and two antinodes make up a λ
Antinode (A) and node (N)
A - point of max displacement (constructive interference occurs)
N - points of zero displacement (complete destructive interference occurs)
Comparing standing and traveling waves
Amplitude variable for standing, but all points have same for traveling
Standing has energy, but it’s non-transferrable (trapped); w/ traveling, energy can be transferred
W/ standing, waves don’t travel (stationary); w/ traveling, waves travel from one point to another
W/ standing, λ is twice the distance between two consecutive nodes/antinodes; w/ traveling, it’s the distance between two points in phase
W/ standing, velocity direction may vary (string may be going upwards or downwards); w/ traveling, velocity direction of wave is the same (wave travels in one direction)
Modes of vibration (harmonics)
Sound means vibration of air molecules: b/c longitudinal, vibrate left/right (front/back).
F= nF (subscript 0 [or 1?]: fundamental frequency [frequency of first harmonic])
λ = 2L/n
and then use
f = v/λ
So, frequency of new will just be a multiple
n = # of harmonics
L = length of string
Have to consider however many loops make up one length: how many lengths to get two loops?
ex. 3rd harmonic: 3x = 2
Don’t forget subscripts to represent harmonics
n for harmonics on strings w/ two fixed ends can represent either # of harmonics or loops
For a pipe open at both ends
B/c two ends open, can’t have a node at each end
Same as string fixed at two ends
Just have to consider that loops at end are half (two make one)
Length is just length of pipe
A still at highest points, ofc
NOTE:
- For diagram, dots represent molecules in air of pipe, double-headed arrows represent how far the air molecules oscillate back and forth (A of the oscillations); molecules at antinodes oscillate most (molecules at nodes don’t oscillate at all [only dots; no arrows])
- Harmonics represents changes in sounds (different sounds/tunes correspond to different harmonics [why they have different frequencies—like plucking a guitar string])
For a pipe closed at one end
Same as string fixed at one end
Loop # half of harmonic # (yes, decimal)—half at open end
NOTE:
- Only odd harmonics present (n = 1, 3, 5,…)
