Oscillations, Sound, and Light Practice Flashcards

Vocabulary-style flashcards based on lecture notes covering oscillations, sound interference, Doppler shift, standing waves, wave optics, diffraction, and ray optics including vision and lenses.

Constructive interference

Two or more waves that are in phase with one another at a specific location P, meaning they are at the exact same point in their cycle at every instant in time. This occurs when $\Delta(\text{cycles}) = m$ (where $m = 0, 1, 2, 3 \dots$).

Destructive interference

Two or more waves that are perfectly out-of-phase (half a cycle phase difference) with one another at a specific location P. This occurs when $\Delta(\text{cycles}) = \frac{1}{2}, \frac{3}{2}, \frac{5}{2} \dots$ and results in the waves cancelling each other out.

Beat frequency

The frequency ($f_b = |f_2 - f_1|$) at which an observer hears a sound alternate between loud and quiet due to the interference of two sound waves with similar but different frequencies.

Doppler shift

A change in observed frequency when either a source of sound or an observer moves in relation to each other, resulting in higher frequencies when moving toward one another and lower frequencies when moving away.

Standing wave

The sum of two traveling waves present in the same constrained medium at the same time, producing resonant frequencies where specific locations remain fixed (nodes) or reach maximum oscillation (antinodes).

Node

A location in a medium for a standing wave where the medium does not oscillate and remains at its equilibrium value for all time.

Antinode

A location in a medium for a standing wave in which the medium oscillates between its maximum and minimum values.

Electromagnetic (EM) wave

Coordinated electric and magnetic fields of oscillating strength that are perpendicular to one another and to the direction of wave travel (transverse waves).

Index of refraction (n)

A property of a material representing the factor by which an EM wave slows down in that material compared to its speed in vacuum, calculated as $v = \frac{c}{n}$.

Diffraction

The spreading out of a wave after it passes through a slit or moves around a barrier when the width is on the order of the wavelength.

Double-slit interference setup

The process where a single plane wave passes through two narrow slits to create a pattern of bright and dark spots (maxima and minima) based on the path difference $|\Delta r| = d \sin(\theta)$.

Surface normal

A line drawn perpendicular to an interface or boundary between two materials at the location where an incoming light ray intersects the interface.

X-ray diffraction

The use of x-rays on a solid to determine the positions of atoms or molecules, characterized by the interference condition $2d \sin(\theta) = |\Delta r| = m\lambda$.

Snell's Law

The principle that relates the indices of refraction of two media and the angles of incidence and transmission ($n_i \sin(\theta_i) = n_t \sin(\theta_t)$) when light crosses an interface.

Real object

An object from which light rays physically diverge as they move away, such as a light bulb, or objects that reflect light similarly like a pencil.

Real image

An image formed at the location where at least two rays of light originating from a single point on the object physically intersect.

Virtual image

An image formed where light rays appearing to originate from the object do not physically intersect at the image location, but appear to do so to an observer.

Focal point (f)

The point where light rays approaching parallel to the axis of a lens or mirror intersect (real) or appear to intersect (virtual).

Converging lens

A lens thicker in the center than at the edges that focuses incoming light rays toward a real focal point on the opposite side.

Diverging lens

A lens thinner in the center than at the edges that spreads incoming light rays farther apart, so they appear to intersect at a virtual focal point on the same side.

Diopter (D)

A unit of measure for the power of a lens calculated as the inverse of the focal length in meters ($D = \frac{1}{f}$).

Near point (N)

The closest distance to the eye where an object can be placed and still form a clear image on the retina; for a normal eye, this is $25\,cm$.

Far point (F)

The farthest distance from the eye where an object can be placed and still form a clear image on the retina; for a normal eye, this is infinitely far away.

Nearsightedness (Myopia)

A condition where the eye's far point is closer than infinity ($F_{ns} < F_{\text{normal}}$), often because the eye lens is too strong or the eye is too long.

Farsightedness (Hyperopia)

A condition where the eye's near point is farther than normal ($N_{fs} > N_{\text{normal}}$), often because the eye lens is too weak or the eye is too short.

Simple magnifier

A converging lens used to create an enlarged virtual image of an object placed at or inside the focal point ($M = \frac{N}{f}$).

Microscope

An optical instrument using an objective lens and an eyepiece to create highly enlarged images, with total magnification $M = - \frac{N \ell}{f_o f_e}$.

def’n period (T)

The amount of time required for an oscillating system to complete one cycle. The SI unit of period is the same as the SI unit of time: seconds (s).

def’n frequency (f)

The number of cycles an oscillating system completes in a given amount of time. The SI unit of frequency is Hz, which is 1/s. None (definition), but only applies to oscillating systems

def’n angular frequency ω ≡ 2πf

Angular frequency is like frequency – number of cycles per time, but with cycles converted into the angular unit of radians. 2π radians is one complete cycle, so if an object has a frequency of 15 cycles per second, its angular frequency is 30π radians per second.

def’n amplitude (A)

For something that oscillates, the amplitude is the maximum displacement from equilibrium. Therefore for simple harmonic motion, amplitude is a property of an oscillator – a constant that does not change as the object is oscillating. The displacement, however, changes constantly depending on the oscillator’s position.

def’n restoring force F_restoring

The restoring force exerts a force in the direction opposite its displacement – always pointing towards the equilibrium position. This directionality of the force causes oscillation. The elastic spring force given in Hooke’s Law is the most common example of a restoring force, but it is not the only possible restoring force.

F = −k(∆⃗x)

Hooke’s Law is a restoring force for an ideal mass-spring. The nature of the restoring force – always directed towards equilibrium– is shown by the negative sign between force and displacement. The strength of the force is linear – depends on displacement multiplied by a constant.

Ideal mass-spring pr simple harmonic oscillator

def’n simple harmonic oscillator

A simple harmonic oscillator is something that oscillates with simple harmonic motion. This is equivalent to saying that the net force on the oscillator is a restoring force with the form of Hooke’s Law. It is also equivalent to saying that the oscillator’s position vs. time graph is a sinusoid.

Simple harmonic oscillator angular/natural/resonant

frequency ω²spr = k/m

The angular frequency of a simple harmonic oscillator depends on the spring constant, k, and the mass m. The larger the spring constant, the higher the frequency. The larger the mass, the smaller the frequency. Because a given, ideal mass-spring can only vibrate at this one frequency, it is called the natural (angular) frequency.

Simple harmonic oscillator position equation: x = xeq + A sin(ωt + φ)

Because this equation includes a phase constant (φ), it can also be written as a cosine instead of a sine. Often we use − sin, sin, cos, or − cos forms without a phase constant, as determined by the specific situation given. A is the oscillator’s amplitude, xeq is the equilibrium position of the mass, ω is the angular frequency of the oscillator, and the variable t is time. We can substitute a value for time t and then solve to find out the position, x, of the oscillator at that time.

spring potential energy PE_spr = ½ k(∆x)

When a spring is part of a system, the system has energy due to the configuration of the spring. (∆x) is how far the spring is stretched or compressed from its natural length. k is the spring constant.

Ideal mass-spring

def’n damping force

A force applied to an oscillating system that results in the system losing energy (i.e. decreasing the amplitude of oscillation over time). Note that if a system is damped, it cannot be a simple harmonic oscillator. But it is still an oscillating system.

def’n driving force

A periodic force applied to an oscillating system. Usually this force causes an increase in the energy of the system. Note that if a system is driven, it cannot be a simple harmonic oscillator. But it is still an oscillating system.

def’n resonance

A state of an oscillator in which a driving force is applied with the same frequency as the natural frequency of the oscillator. This causes large increases in the system’s energy and can cause effects like a rich sound when well-made instruments vibrate and the breaking of a wine glass by a trained singer.

Oscillating system being acted on by a driving force

Pendulum angular frequency ω²pendulum = g/L

A simple pendulum, consisting of a massless rod attached to a mass that oscillates back and forth with small angles. As long as the angles are small (for our class, up to 30◦ =π/6 radians), the pendulum is a simple harmonic oscillator. The angular frequency of a simple pendulum depends on the acceleration due to gravity g and the length l of the pendulum arm. Note that it does NOT depend on the mass of the pendulum.

def’n traveling mechanical wave, source

A traveling mechanical wave is a wave that moves because of the movement of the medium. In other words, a mechanical wave would not exist were it not for the medium. Most traveling waves are mechanical waves, including sound waves, water waves, and waves on strings. This is different than light and other electromagnetic waves, which are NOT mechanical waves.

def’n transverse wave

For a mechanical wave, a wave in which the pieces of the medium oscillate in the direction perpendicular to the direction the wave travels. Note that medium particles do not move forward in space with the wave, but oscillate continually around some equilibrium position.

def’n longitudinal wave

For a mechanical wave, a wave in which the pieces of the medium oscillate in the direction parallel to the direction the wave travels. Note that medium particles do not move forward in space with the wave, but oscillate continually around some equilibrium position.

Speed of a wave on a string: v = squareroot(F_T/u)

The speed of a wave on a string depends on the tension in the string, FT , and the mass density μ of the string. The mass density, μ, of the string is just the string’s mass, m, divided by its length, L. Higher tension makes the wave travel more quickly, and strings with higher mass density cause waves travel more slowly. Note that wave speed is physically determined by tension and mass density – properties of the string (medium), and is therefore the same for waves of any frequency traveling on the string. The SI unit of wave speed is meters per second.

Mechanical wave speed: v = squareroot(elasticity/density)

In a general sense, a medium has some elasticity, or connectedness of adjacent pieces of the medium. In one dimension on a string, this is tension.

def’n wavelength (λ)

The distance the wave travels forward in space in one cycle. It can also be viewed as the distance in space between a location (in the medium) and the closest second location (in the medium) that are at the same point in their oscillation cycle. The SI unit of wavelength is meters.

Periodic wave (moving at constant speed)

Wave speed, wavelength, frequency relationship: v = λf

The distance d the wave travels in one cycle is its wavelength, λ. The speed (rate) the wave travels is v. The time it takes to complete one cycle is the period, T. However we move this to the same side of the equation as λ, and substitute the wave frequency f for 1/T (using the period-frequency relationship).

Periodic wave moving at constant speed

def’n wave reflection

If the reflection is off of a wall, or medium with a higher density than the original medium, the reflected wave has a phase shift of half a cycle forward. For a transverse wave on a string, this looks like a reflected wave pulse having the opposite orientation as the incident pulse. If the reflection is off a medium with lower density than the original medium, there is no phase shift for the reflected wave. For a transverse wave on a string, this looks like the reflected pulse having the same orientation as the incident pulse. Because reflection sends the wave back into the same medium it was already in, its wave speed is the same as the incident wave speed. This means that, because its frequency doesn’t change, v = λf tells us that its wavelength must also stay the same.

def’n wavenumber k ≡ 2π/λ

The wavenumber is defined as 2π (one complete cycle measured in radians) divided by the wavelength, λ. Its SI units are therefore inverse meters (1/m). It is sometimes called ”spatial frequency” because it describes the number of oscillations in a given distance in space. A system with higher wave number k means more oscillations per meter than a system with lower wavenumber.

def’n 3D pressure (sound) wave SA_sphere = 4πr2

A sound wave is a longitudinal wave that causes the pressure at a given location to oscillate around the equilibrium position (i.e. the equilibrium pressure of the air). If we were to take a snapshot in time, we would see places in the air with higher density than equilibrium (compressions), places in air with lower density than equilibrium (rarefactions), and places in the air with the equilibrium pressure.

Speed of sound in air v_sound,_air = (331 + 0.6T)

The speed of sound in air increases with temperature. Here temperature is measured in Celsius, and the sound wave speed is given in meters per second (m/s). Therefore, the speed of sound in air at 0 degrees Celsius is 331 m/s.

def’n intensity I ≡ P/A = E/ t · A

Intensity is defined as the power that crosses through a surface, per unit area. We can rewrite power as energy per time. Another way to say this is that the intensity of a sound at a specific location is the energy received at that location per time per area. We can think of this in terms of our eardrum (which has a fixed size). If intensity is larger, our eardrum will receive more energy per second than if intensity is small. Also, imagine two people have different sized eardrums. The person with the larger eardrum collects more sound energy per time at a given location than the person with a smaller eardrum because A (the area of the eardrum collecting the sound energy) is larger.

def’n intensity level ∆β = β2 − β1 ≡ (10dB) log(I₂/I₁ )

Intensity level is a comparison between two sound intensities that takes into account the huge range of sound intensities (from 10−12 W/m2 to hundreds of W/m2). We calculate the difference intensity level by comparing its a sound’s intensity (I₂) to a reference sound’s intensity (I₁). Often the reference intensity is chosen to be the threshold intensity of human hearing – the lowest volume sound perceptible to the average human (I0 = 10−12 W/m2 ). To determine difference in intensity level, β, we divide the two intensity values (I₂ divided by I₁) to find by what factor the intensity I₂ is bigger (or smaller) than the reference intensity I₁. Then we take the logarithm (log) of this ratio. Log is a math function that extracts the power of ten by which two numbers differ. So the difference in intensity level (∆β) is a shorthand to describe the power of ten by which the two intensity levels differ. For example, a ∆β of 90dB means that the sound (I₂)’s intensity in W/m2 is 9 orders of magnitude (9 powers of ten, or 10⁹ times) greater than the comparison intensity I₁. The SI unit of intensity level is the decibel (dB).

def’n sources in phase

If we imagine an oscillating source creating a wave, then directly in front of the source the wave (or medium) will oscillate at different points in its cycle (as time progresses). If sources are in phase, it means that at any specific time chosen, the sources are all emitting waves that are at the exact same point in their cycle. Note that sources can only be in phase if they have their waves have the same frequency (and therefore wavelength) – otherwise the waves take different times to complete and couldn’t be in phase over time.

def’n coherent sources

Two sources are coherent if they have a constant phase relationship to one another. For example, sources A and B are coherent if they both emit 550Hz waves and source A is always one quarter of a cycle ahead of source B. In other words, if we imagine any source emitting a wave, directly in front of the source, the wave (or medium) will oscillate at different points in its cycle as time progresses. For coherent sources, if we pick a time and measure how much farther ahead a wave is in its cycle at source 1 compared to source 2, the wave at source 1 is ALWAYS this far ahead in its cycle compared to the wave at source 2.

Fourier’s Theorem

In other words, any periodic wave of ANY shape can be created by adding together sine waves of various amplitudes and frequencies (Fourier series).

periodic wave/ sound

Wave interference contributions: ∆(cycles)location_X =

∆(cycles)sources + ∆(cycles)path+ ∆(cycles)reflection

Waves can interfere constructively, destructively, or something in between. Two waves can only interfere constructively or destructively if the wave sources are coherent. Waves from different sources can be at a different point in their cycles (relative to one another) at point P due to three causes which can occur in combination:

1. Difference in the wave’s phase at the source (i.e. sources not in phase)

2. Difference in the distance traveled between the each source and location P

3. Change in phase of one (or more) waves due to reflection after leaving the source and before arriving at P

2+ waves at same location over extended time from coherent sources

Standing waves (2)

(Mathematically, there are an infinite number of wavelengths that can fit in a medium that supports standing waves as the wavelength decreases.)

Fundamental standing wave: string or open-open tube

The longest sinusoidal standing wave that resonates on a string fixed at both ends fits half the wave on the string. This is because the endpoints of a fixed string must be nodes. The longest sinusoid that meets this condition fits one ”arc” of the sinusoid, which is half of the complete wavelength. To determine all other allowed standing waves, we sketch the image based on the requirements (ends are displacement nodes for strings / pressure nodes for open-open tube).

Fundamental standing wave: open-closed tube

The longest sinusoidal standing wave that resonates in a tube open at one end and closed on the other end. This is because the open end of the tube must match the pressure outside the tube (pressure node), and the closed end of the tube does not allow particles to oscillate back and forth (pressure antinode). The longest sinusoid with a node on one end of the tube and an antinode at the other end of the tube is one-quarter of the complete wavelength. To determine all other allowed standing waves, we sketch the image based on the requirements (open end is pressure node, closed end is pressure antinode).