Waves and Oscillations

Question 1

Oscillations

Answer 1

More technical/less general physical term for vibrations. Involve any motion that regularly repeats (back and forth) itself about a fixed (equilibrium/mean) point. This ‘mean point’ is usually where it comes to rest.

Ex. mass moving on a vertical spring, simple pendulum (weight called a ‘bob’), object bobbing up and down in water, diving board, etc.

Question 2

Displacement (commonly associated)

Answer 2

Symbol: x. The instantaneous distance of the oscillating object from the mean point in a given direction (away from the mean point at any time [lots of possibilities]: to the left = negative [but we can forget the sign]. Linear, so the unit is meters [in a peculiar situation—a simple pendulum, for example—, we would go with degrees]).

Question 3

Amplitude (commonly associated)

Answer 3

Symbol: A. The maximum displacement of the object from the mean position (unit is meters): distance from the crest/trough to the equilibrium position OR maximum displacement of the medium from the mean point.

Question 4

Frequency (commonly associated)

Answer 4

Symbol: F (sometimes, a lowercase f is used, but it doesn’t really matter). The number of oscillations (cycles [amplitude to opposite and back or mean to amplitude to opposite and back: to make a cycle, make a circle. So, 360 degrees.]) completed by the object in a second (can be measured by counting cycles for a specific amount of seconds and then dividing by that amount of seconds). Unit is hertz (Hz).

Question 5

Period (commonly associated)

Answer 5

Symbol: T. The time taken to complete one oscillation/cycle. Unit is seconds, and the formula is F = 1/T (note: we can convert).

Question 6

Phase difference (commonly associated)

Answer 6

Symbol: ɸ (phi). The angle difference between two oscillations that are out of phase (‘in phase’ means on same side of equilibrium).

Question 7

What are sine graphs used for?

Answer 7

Sine graphs are used to show cyclatory movements. All waves can be represented using a sine wave.

Question 8

Simple Harmonic Motion

Answer 8

This is an oscillation (bob) whose acceleration (caused by a restoring force [remember: always in the same direction as the net force]) is ALWAYS PROPORTIONAL to its displacement, and it is ALWAYS DIRECTED TOWARDS THE MEAN POSITION (opposite the displacement). When you let it go, where does it go?

In the middle, tension force is the main; you have to consider what forces are at play in a given position (tension, gravity).

Formula: a∝-x
a = -w^2x
This is the defining equation for SHM: the negative shows that we’re going towards the mean position (you pull it to the right and it goes to the left). The a is the acceleration. The x is the displacement. In Physics, whenever you replace a proportional symbol with an equal symbol, you always get a constant. The ‘w’ is a constant called angular frequency. The negative sign signifies that acceleration always points towards the mean position (opposite the displacement).

Important:
The acceleration is caused by a restoring force (F)
a∝-x
∴ F∝-x
When one is positive, the other is negative (directly proportional [straight line and has to start from origin], but there’s a negative).

Question 9

Other SHM equations

Answer 9

Simple pendulum: T = 2πxroot(l/g)
T is period, l is the length of the string, and g is the acceleration due to gravity
Mass-spring system: T = 2πxroot(m/k)
T is period, m is the mass, and k is the spring constant

Question 10

Hooke’s Law

Answer 10

The extension of a spring is directly proportional to the the force that produced it.

Formula:
F∝x
F = kx
*Where k is the spring constant (unit: N/m)

Question 11

Waves

Answer 11

Periodic/Repeated disturbances that transfer energy as they move from one place to another: caused by disturbances. Waves carry energy along with them when they propagate.

Whatever the wave passes through is the medium (carrier of the waves [ex. air is the medium through which sound waves pass through]). Most waves travel through a medium (electromagnetic waves don’t [can travel through a vacuum: also, they don’t lose energy because there are no collisions. On earth, there’s absorption]). Although waves can travel through a medium, the medium does not travel with the waves, but only vibrates back and forth (oscillates as waves travel through it [left to right or forwards and backwards depending on the type of wave]) and returns to its original position. Particles travel right to left (medium [again, substance/material that the wave passes through] moves up and down).

Examples of waves include sound waves, EM waves, water waves, seismic waves, earthquakes (waves in the earth’s crust), etc. (also slinky waves [but they’re not real waves: slinkies don’t produce real waves. They produce something very close to them, and this helps us understand them]).

Waves can be categorized into two based on the direction of medium oscillations: transverse waves and longitudinal waves.

Question 12

Pulse

Answer 12

Waves are a bunch of disturbances; a pulse is a single disturbance (a wave is made up of several pulses).

• Radio, etc. under EM
• Propagating: technical term for when waves spread out
• When waves make contact, they can be reflected, scattered, or pass through/transmitted?

Question 13

Transverse waves

Answer 13

Waves whose medium oscillations are perpendicular to the direction of energy transfer.

A transverse wave can be represented by a sine graph:

Crest: the highest point above the equilibrium.
Trough: the lowest point below the equilibrium.

Ex. everything except sound waves (and those related to sound waves). So, water waves, EM waves, and slinky waves (again, not real waves).

*Note the double-headed arrow in diagrams!

Question 14

Longitudinal waves

Answer 14

These are waves whose direction of medium oscillation is parallel to the direction of energy transfer/wave travel.

*When sound waves move, they push air molecules together.

Can also be represented using a sine graph.

Crest of transverse = compression of longitudinal
Trough of transverse = rarefaction of longitudinal

Ex. sound waves, earthquakes, etc.

Question 15

Wavelength (characteristic)

Answer 15

Symbol: λ. The distance between two adjacent corresponding points of a wave. It can be measured in the following ways: crest to crest, trough to trough, distance for one cycle. Unit is meters.

*Lines must be drawn from CENTER TO CENTER!

Question 16

Wave speed (characteristic)

Answer 16

Symbol: v. Distance traveled by wave in a given amount of time (same def) OR speed at which energy is transferred by a wave.

Formula that relates wave speed, frequency, and wavelength:
v = fλ
v is speed, f is frequency, and lambda is wavelength.

In Physics, the only way to tell the relationship between two variables is to get them on separate sides of the equation:

f = v/λ
Therefore,
f = 1/λ
and/or
f∝1/λ
The two are inversely proportional.

Question 17

Other way to categorize waves

Answer 17

Waves can be categorized based on the requirement of a medium for energy transfer. They include…

Mechanical waves:

• Require a medium for every transfer
• Ex. water waves, sound waves, earthquakes, etc.

Electromagnetic waves:
- Do not require a medium for every transfer
- Ex. [visual] light, UV rays, infrared, microwaves, x-rays, ɣ-
rays (gamma rays), and radio waves

• EM waves are mostly rays (in the form of heat)
• Seismic waves are produced after earthquakes
• Light is traveling through nothingness to reach us
• We wouldn’t hear explosions on the moon (no gas for the sound waves to travel through)

Question 18

Behavior/Properties of waves

Answer 18

Common to all waves (waves must be able to do all these things). Certain properties are used to describe the behavior of waves. They include: reflection, refraction, diffraction, and interference/superposition.

Question 19

Reflection

Answer 19

The bouncing back of waves whenever they reach a boundary (ex. light reflecting off a flat mirror, etc.)

Incident wave: the incoming wave (original wave; the one that was sent first).

Reflected wave: the wave that bounces off/comes back.

Two types of reflections (based on what happens to the reflected wave at the boundary): reflection at a fixed boundary/end (the incident wave is inverted as soon as it bounces off) and reflection at a loose boundary/end (there’s no inversion as the incident wave bounces off the boundary).

There’s also that for the reflection of a light ray to a flat surface (see diagrams):
i = incident ray
r = reflected ray

Law of Reflection: States that whenever light strikes on a flat surface, the incident angle is always equal to the reflected angle (i = r).

Question 20

Refraction

Answer 20

The change in speed and direction of the wave as it moves from one medium to another. This change in speed results in the bending of the wave.

From a less dense to more dense (*see notebook [ns; incident and reflected ray; boundary; less dense/more dense; what i/r is [angles]; if i is greater than r; n1 on the top; normal [imaginary, perpendicular…helps us measure the angles]) medium, the refracted ray bends towards the normal and the speed decreases.

*Generally, for every refraction, frequency is constant, so whatever happens to the speed happens to the wavelength (v = f𝝀)

From a more to less dense medium, the refracted ray bends away from the normal and the speed increases (speed increases, 𝝀 increases, frequency constant [also, same diagram concepts]).

*Transparent or not, light can pass through it

Applications of refraction: objects in water appear to be in different positions, splitting of white light into ROYGBIV

n2/n1 = v1/v2 or 𝝀1/𝝀2
Therefore,
n2/n1 = sin i/sin r = v1/v2 = 𝝀1/𝝀2
*1 is for the incident rays.
Rarer means less dense.
Small refraction when light enters object (except i = 90)
Change of speed = change in direction in this case.
Refraction example: rays of sunlight having to pass through layers of air of increasing density.

Question 21

Refractive index

Answer 21

Density of medium is directly proportional to the refractive index (n [ability of a medium to bend a wave]): the denser the medium, the greater the refractive index.

Question 22

Snell’s Law

Answer 22

Snell’s Law states that the ratio of the sine of the incident angle i to the sine of the refracted angle r is a constant n (so, equal to refractive index of second medium [sin i/sin r = n]). The law applies to when light moves from air/vacuum to another medium.

Refractive index of air/vacuum is approximately one, so sin i/sin r = n2/n1 (refractive index of medium 2 over medium 1 [equal to other formula]).

Question 23

Total internal reflection

Answer 23

When light strikes a boundary at 90, refraction doesn’t occur (total internal reflection [depends on refractive index]).

Question 24

Diffraction

Answer 24

The bending or spreading out of waves when they pass through apertures (openings/slits) or meet obstacles. Waves squeeze themselves through the opening, are bent/spread out on the other side.

Factors that affect diffraction: width of aperture (narrower = more diffraction [wider = less curving/diffracting]), wavelength (longer = more diffraction [shorter = less spreading/diffraction]).

*See diagrams—also for a circular aperture (aperture/obstacle, direction of wave travel labeled [wavelength remains the same]).

Applications of diffraction: dispersion of light into its component colors by compact disc, polished surface, etc. (surfaces have apertures); measurement of blood flow (a special apparatus uses diffraction to measure speed of blood flow); hearing of sounds even when the source is not seen (the sound waves curve when they enter an opening and come to you).

Question 25

Interference (Superposition)

Answer 25

The combination of two or more identical waves (have to be the same kind of wave [same wavelength and frequency]), traveling in opposite directions, to form a resultant wave.

Superposition Principle: When two or more waves interfere, the overall displacement of the resultant wave is a vector sum of the displacements of the identical waves.

Forward means + (above equilibrium), below equilibrium means -.

Question 26

Constructive interference

Answer 26

The individual waves on same side of the mean position combine to form a resultant wave with a bigger displacement (*see diagrams [return to normal after—still traveling in their respective directions]).

Question 27

Destructive interference

Answer 27

The individual waves on opposite sides combine to produce a resultant wave whose displacement is smaller (*see diagrams).

*Tip: Partition first.

Question 28

Complete destructive interference

Answer 28

A special kind of destructive interference that involves two waves with the same displacement (*see diagrams).

Question 29

Electromagnetic waves

Answer 29

“Electromagnetic” implies two types.

Oscillations of electric and magnetic fields that vibrate perpendicularly to the direction of their travel (electric field waves moving perpendicular to magnetic field waves): they are, in other words, transverse waves.

There’s visible light, radio waves, microwaves, infrared, UV radiation, 𝛄-rays, and x-rays (seven). All EM waves travel at the same speed (speed of light: 3 x 10^8 m/s). They are classified according to their frequencies or 𝝀s (this classification is known as the electromagnetic spectrum).

Mnemonic: Raging Martians Invaded Venus Using X-ray Guns (increasing frequency going right, increasing 𝝀 going left).

Increased frequency = increased energy.
Their frequencies are inversely proportional to their 𝝀s
Visible light (also follows the above) is of the range 400-700nm and has the colors ROYGBIV.

Question 30

Sound waves

Answer 30

They’re longitudinal waves: this means that they’re medium vibrates parallel to their transfer of energy. They are produced by vibrations that set air molecules (carry sound along) in motion, forming regions of high (compression) and low (rarefaction) density/pressure (*see diagrams). Divided into three based on their frequency: audible sound, ultrasonic sound, and infrasonic sound.

Question 31

Audible sound waves

Answer 31

Sound waves that can be heard by the average human ear. Their frequencies range from 20 to 20000 Hz (quality of eardrums and loudness of sounds in environment you grew up in may affect [why we have a range]).

Question 32

Ultrasonic sound waves

Answer 32

Can be heard by dogs, bats, etc. (they have frequencies above 20000 Hz).

Question 33

Infrasonic sound waves

Answer 33

Sound waves with frequencies less than 20 Hz (can be heard by elephants).

Question 34

Pitch

Answer 34

How high or low a sound is. Pitch of a sound is directly proportional to its frequency.

Question 35

NOTE

Answer 35

• Away: ni > nr
• Towards: ni < nr
• Harmonic: pulses on a line
• If same displacement, it will be clear
• See diffraction at corner
• A pulse wave is characterized as one assault to the medium while periodic waves are characterized as a series of successive assaults on the medium

Question 36

Standing waves

Answer 36

A wave pattern that results when two waves of the same frequency, 𝝀, and amplitude traveling in opposite directions interfere. The resultant wave is known as a standing wave, and its amplitude varies.

Standing waves have nodes and antinodes.
Nodes (N): Points of zero displacement (where complete destructive interference occurs).
Antinodes (A): Points of maximum displacement (where constructive interference occurs).

• See diagram: loops are not empty!
• You’d have to draw the varying amplitudes
• See possible standing waves (harmonics): first, second, third,…
• Number of nodes differs with number of waves (?)

In music, going from one key to another is going from one harmonic to another (changing frequencies).