GRE® Physics > Optics and Wave Phenomena > Flashcards

Optics and Wave Phenomena Flashcards

Interpret and predict wave behaviors such as interference, diffraction, and polarization in optical and mechanical systems. (64 cards)

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1
Q

For a harmonic mechanical wave on a string, what wave property primarily determines the amount of energy transported?

Assume constant wave speed and mass density.

A

The amplitude squared (E ∝ A²).

While energy in a wave can depend on multiple factors (such as amplitude, frequency, and wave speed) for a harmonic transverse wave on a string, energy is proportional to the amplitude squared when other parameters are held constant.

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2
Q

What is the result when two waves of equal amplitude and frequency traveling in opposite directions interfere?

A

A standing wave.

Nodes and antinodes form due to a fixed spatial interference pattern.

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3
Q

Explain why standing waves occur only at specific frequencies on a string fixed at both ends.

A

Only wavelengths that fit an integer number of half-wavelengths between the fixed ends satisfy the boundary conditions, leading to discrete resonant frequencies.

The condition L=nλ/2 (with n=1,2,3,…) ensures nodes at both ends and defines the allowed standing wave modes.

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4
Q

What feature distinguishes a transverse wave from a longitudinal wave?

A

The direction of particle oscillation relative to propagation.

  • Transverse: perpendicular
  • Longitudinal: parallel to wave direction

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5
Q

Fill in the blanks:

The quantity ω/k gives the wave’s ______ ______.

A

phase velocity

vₚ = ω/k describes how a point of constant phase moves.

Note that the phase velocity need not be the same as the group velocity.

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6
Q

In a dispersive medium, how does group velocity differ from phase velocity?

A

In a dispersive medium, these two velocities are generally not equal.

Group velocity describes the speed at which a wave packet (and energy/information) travels, while phase velocity is the speed of the individual wave crests.

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7
Q

What happens to a wave pulse when it reflects from a fixed end?

A

The reflected pulse travels in the opposite direction and undergoes a 180-degree phase shift.

The phase change is due to the boundary condition. At the fixed end, the medium cannot move.

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8
Q

List 3 different wave phenomena that inherently rely on the principle of superposition.

A
  • Interference
  • Diffraction
  • Beats

Each of these phenomena arises from the overlapping and subsequent superposition of waves, illustrating the principle’s foundational role in wave mechanics.

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9
Q

A string of fixed length L is clamped at both ends. Derive the expression for the allowed frequencies of standing waves on the string.

A

Only standing waves that fit integer half-wavelengths between the fixed ends are allowed.

  • We can then write L = n(λ / 2)
  • So, the allowed wavelengths (indexed by n) are λₙ = 2L / n.
  • Then the corresponding frequencies are easily found from fₙ = v / λₙ.

Note that the speed of the wave depends on the tension in the string and the mass per unit length.

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10
Q

True or False:

In a wave on a stretched string, the phase velocity increases if the string’s tension is increased.

A

True

v = √(T / μ), where T is tension and μ is linear mass density.

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11
Q

A string supports two frequencies that differ by a perfect fifth. What is the length ratio required between the two segments of the string?

A

A perfect fifth corresponds to a frequency ratio of 3:2.

Since 𝑓∝1/𝐿 for the same tension and mass per unit length, length scales inversely. The lower-pitched segment is longer.

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12
Q

Derive the dispersion relation for waves on a string of mass per unit length μ and tension T.

A

A dispersion relationship is the relationship between the angular frequency and the wavenumber.

  • Assuming that the displacement of the string is small compared to its length, the waves on the string follow the wave equation ∂²y/∂t² = √(T / μ) ∂²y/∂x².
  • Plug in y(x,t) = A sin(kx - ωt)
  • Take the derivative and simplify
  • Arrive at ω = (√(T / μ))k.

Note that the velocity is √(T / μ), so this result is not a surprise.

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13
Q

An open-closed tube supports a fundamental frequency 𝑓. What is the next higher allowed frequency for resonance?

A

𝑓₃=3𝑓

Open-closed tubes support only odd harmonics: 𝑓𝑛=𝑛𝑣/(4𝐿), where 𝑛=1,3,5,…

This follows from the boundary conditions. Even harmonics require nodes (or antinodes) at both ends.

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14
Q

What happens to the wavelength of a wave if the frequency increases but the speed remains constant?

A

Wavelength decreases.

λ = v/f

Wave speed is constant in a medium, so frequency and wavelength are inversely related.

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15
Q

What is the wave impedance of a stretched string and what does it represent?

A

The wave impedance 𝑍 represents the resistance the string offers to wave motion at a particular velocity.

Impedance matching determines reflection/transmission at boundaries.

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16
Q

Two waves y₁=Acos(kx−ωt) and y₂=Acos(kx−ωt+ϕ) superpose.

Find the resulting amplitude.

A

From trigonometric identity: cosa+cosb=2cos[(a−b)/2]cos[(a+b)/2].

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17
Q

A string is driven at one end with a fixed frequency.

Under what condition will resonance occur?

A

When the frequency matches a normal mode of the string.

Boundary conditions define allowed standing wave frequencies.

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18
Q

Fill in the blanks:

Constructive interference occurs when the phase difference is an ______ ______ ______.

A

integer multiple of 2π

Phase difference of Δϕ=2nπ (n is an integer) means that the waves arrive ‘in step’.

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19
Q

What is the beat frequency between two waves of close frequencies f1 and f2?

A

Beats are the modulation envelope of interfering waves of slightly different frequencies.

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20
Q

Fill in the blank:

The phenomenon of wave ______ occurs when waves overlap and combine to form a resultant wave.

A

interference

Interference can be constructive or destructive, depending on the phase difference between overlapping waves.

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21
Q

Fill in the blank:

For destructive interference, the phase difference must be an ______ multiple of π.

A

odd

Δϕ=(2n+1)π gives cancellation since this means that the waves arrive ‘out of step’.

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22
Q

How does introducing a thin dielectric in one slit affect a double-slit interference pattern?

A

It shifts the entire pattern due to a phase shift.

Optical path length increases, thereby introducing an additional phase shift.

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23
Q

Why are interference patterns difficult to observe with white light?

A
  • Broad wavelength spectrum causes fringe overlap.
  • White light has a short coherence length.

Different colors interfere at different positions, washing out pattern. A long coherence length is generally required to observe interference (otherwise a fixed phase relationship is not maintained).

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24
Q

What interference pattern results from two point sources emitting spherical waves in phase?

A

Hyperbolic fringes of alternating constructive and destructive interference.

Path difference defines loci of constant phase difference.

In the effective 2D space, the set of points where the difference in distances to two fixed points is constant forms a hyperbola.

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25
What is the effect of **increasing wavelength on fringe spacing** in a double-slit setup?
It increases the **spacing**. ## Footnote Δy=λL/d , so therefore spacing grows with λ.
26
# True or False: **Fringe shift** can be used to measure changes in refractive index.
True ## Footnote Refractive index affects optical path length, hence the position of interference fringes.
27
Derive an expression for the **angular width** of the central maximum in a single-slit diffraction pattern, for slit width 𝑎 and incident light of wavelength λ.
## Footnote * The first minima occur at sin𝜃=±𝜆/𝑎, so the angular width is twice that angle. * If λ≪a, the angular width is simply 2(λ/𝑎).
28
In **Michelson interferometer**, what is observed when one mirror is moved by a quarter wavelength?
The **bright fringes become dark** and the dark fringes become bright. ## Footnote The length of one of the paths is changed by half wavelength (due to the round trip in a Michelson interferometer). This means that we have a phase difference of 180 degrees now.
29
How does **increasing slit width** affect the diffraction pattern in a single-slit setup?
It creates a **narrower central maximum**. ## Footnote Inverse relation: larger slit → smaller angle. Δθ≈2arcsin (λ/𝑎).
30
Why does the **color of a soap film** change as we move our head while observing it?
This is due to **interference**. ## Footnote Light reflects off the outer layer of the thin film. Some light also passes through the film and reflects off the inner layer. There, two rays interfere. As we move our head, the path difference between them changes, so different wavelengths constructively and destructively interfere as we move our head. This changes the colors we observe.
31
What pattern is produced by **monochromatic light** through a narrow slit?
A **central bright fringe** with maxima and minima on either side. ## Footnote The maxima are smaller peaks, whereas the minima represents points where the intensity is zero. This arises due to interference between light waves emerging from different parts of the slit.
32
Explain the condition for **destructive interference** in a single-slit diffraction pattern.
asinθ=nλ | where n is an integer. ## Footnote To see this physically, observe that the slit can be divided into zones that **cancel pairwise**. For example, for n = 1, light from the top half of the slit interferes destructively with light from the bottom half. Similarly, for n = 2, we can divide the slit into four parts and then form two pairs of zones. For each pair, we have destructive interference. And so on.
33
How does the diffraction pattern change when using a **circular aperture instead of a slit**?
A central **airy disk (bright circular region)** with concentric dark and bright rings appears. ## Footnote The pattern has circular symmetry. The dark and bright rings are due to destructive and constructive interference of light from different parts of the circular aperture. The first minimum occurs at angle 1.22λ/D (assuming that we are far from the aperture and the diameter of the aperture D is much bigger than the wavelength).
34
What **limits the resolution** in optical systems like telescopes?
**Diffraction** through the aperture. ## Footnote Rayleigh criterion for a circular aperture: the angular resolution is given approximately by 1.22λ/D where D is the diameter of the lens' aperture.
35
How can **diffraction gratings** be used to separate light into components?
Each wavelength diffracts at a **different angle**. ## Footnote d sinθ=nλ; each wavelength satisfies this equation at a different angle.
36
What happens to the **diffraction pattern** if the wavelength of light incident on a diffraction grating increases?
The fringe spacing **increases**. ## Footnote From grating formula **dsinθ=nλ**, sin θ increases. This means that the distance between the fringes increases.
37
# True or False: The **focal length** of a lens is dependent on the wavelength of light used.
True ## Footnote This is due to **chromatic aberration**, where different wavelengths of light refract at slightly different angles (dispersion), causing a variation in focal length.
38
How do you find the image position using the **paraxial approximation** for a light ray entering a spherical glass surface from air?
Use the **spherical refraction formula**. ## Footnote * s and s' are the object and image distances, respectively. * R is positive if the center of curvature is on the outgoing side. * n₁ is the refractive index for air, n₂ for the glass.
39
# True or False: A concave mirror always forms a **real image**.
False ## Footnote If the object is between the mirror and the focal point, the image is virtual, upright and magnified.
40
# Fill in the blank: A ray incident at the **critical angle** from a denser to a rarer medium undergoes \_\_\_\_\_\_ \_\_\_\_\_\_ \_\_\_\_\_\_.
total internal reflection ## Footnote Occurs when θᵢ = θc = arcsin(n₂/n₁), with n₁ > n₂.
41
How do you determine the **focal length** of a thick convex lens analytically?
Use the **lensmaker’s equation** with thickness correction: ## Footnote This applies for **thick lenses** of finite thickness d and takes the curved surfaces into account via R₁ and R₂.
42
Describe the image formation for an object placed between **two mirrors inclined at an angle θ**.
**Multiple images form** in a circular arc around the object. ## Footnote This happens due to repeated reflections between the mirrors. Number of images = 360°/ θ - 1 if θ is an exact divisor.
43
How do you find the **effective focal length** when a diverging lens is placed in front of a converging lens?
Use the **lens combination** formula. ## Footnote d is the separation between the lenses; sign convention applies. For a converging lens, the focal length is positive, while it is negative for a diverging lens.
44
What is the nature of the image when an object is placed at the **center of curvature of a concave mirror**?
* Real * Inverted * Same size * At the center of curvature ## Footnote For spherical mirrors: 1/s + 1/s' = 2/R.
45
# Fill in the blank: A **convex lens** forms a real, inverted image only when the object is placed \_\_\_\_\_\_ the focal length.
outside ## Footnote Inside the focal length, the image is virtual, upright, and magnified.
46
What happens to the **apparent depth** of an object submerged in water when viewed from air?
It **appears shallower** than its actual depth. ## Footnote The difference is due to refraction. Apparent depth = real depth × ( n₂/n₁), with the refractive indices n₁ for water and n₂ for air.
47
# Fill in the blank: A \_\_\_\_\_\_ polarizer transmits **only one component of the electric field** vector, aligning it along a specific direction.
linear ## Footnote This is crucial for applications requiring specific linear polarization states.
48
A **linearly polarized light** passes through a polarizer at 45° to its axis. What fraction of the intensity is transmitted?
## Footnote Follows Malus’ Law: I = I₀ cos² θ. Intensity is proportional to the electric field squared, and the polarizer picks out a component of the electric field.
49
# Fill in the blanks: **Unpolarized light** incident on an ideal polarizer results in transmitted light that is \_\_\_\_\_\_ \_\_\_\_\_\_.
linearly polarized ## Footnote The **intensity is halved** since the average of cos² θ is 1/2.
50
Explain how birefringent materials create **double refraction**.
Unpolarized light falling on a birefringent material splits into **ordinary and extraordinary rays**. These rays travel in different directions. This splitting takes place because the refractive index depends on the polarization and the propagation direction of light. ## Footnote The ordinary and extraordinary rays are polarized at right angles to each other. The ordinary ray sees a constant refractive index, while the refractive index seen by the extraordinary ray depends on direction.
51
# True or False: **Circularly polarized** light can be produced by a polarizer alone.
False ## Footnote A quarter-wave plate is needed after a polarizer to introduce the required phase shift between the orthogonal components of the electric field.
52
What is the **difference** between circular and elliptic polarization?
* **Circular polarization**: The electric field vector rotates (in a plane perpendicular to the propagation direction) in a circle. * **Elliptic polarization**: It rotates in an ellipse. ## Footnote This means that the magnitude of the electric field vector remains constant for circular polarization, but it changes for elliptic polarization.
53
How can a **quarter-wave plate** transform linear polarization into circular polarization?
It introduces a **90° phase** shift between orthogonal components of the electric field. Light along the slow axis travels slightly slower, and this leads to the phase shift. ## Footnote Incident light must be polarized at 45° to the fast and slow axes, otherwise the components along these axes will not be equal.
54
The Jones vector of light polarized horizontally is (1 0). **What is the Jones matrix for a linear polarizer with axis of transmission vertical?**
## Footnote Acting this matrix on the vector (1 0) gives us zero, while acting it on (0 1) gives us (0 1) back. (0 1) corresponds to vertically polarized light.
55
Describe the polarization of light reflected at **Brewster’s angle.**
Reflected light is perfectly linearly polarized **perpendicular to the plane** of incidence. ## Footnote Brewster's angle is given by θ = arctan(n₂ / n₁).
56
A beam passes through **three polarizers** with the second at 45° to the first and the third orthogonal to the first. What happens to the **transmitted light**?
Some light is transmitted due to the **intermediate polarizer**. The transmitted intensity is one-fourth that of the light exiting the first polarizer. ## Footnote **Malus’ Law** can be applied twice. After the second polarizer, the intensity is halved. After the third, it is halved again.
57
What are the **Stokes parameters**?
These are four numbers that fully characterize the **polarization state of light**. ## Footnote * The first number is the total intensity of light. * The second compares horizontal and vertical polarization. * The third compares polarization at 45° and -45°. * Finally, the last number compares the left-hand circular polarization and right-hand circular polarization.
58
Describe the **observed frequency** if a source approaches a stationary observer.
It increases. ## Footnote This is the Doppler effect. f' = f(v/(v - vs)). For light, one has to use a relativistic treatment to obtain f′ = f √((1 + v/c) / (1 − v/c)).
59
A source of waves moves away from an observer. **Does the observed frequency increase or decrease?**
It decreases. ## Footnote A longer wavelength is received as source recedes.
60
# Fill in the blanks: In the relativistic **Doppler effect**, time dilation affects the observed \_\_\_\_\_\_ and \_\_\_\_\_\_.
frequency; wavelength
61
How would you treat the Doppler effect for **sound** and **light** differently?
* Since the speed of sound is much smaller than the speed of light, the simpler, non-relativistic expressions can be used in the analysis. * For light, a relativistic treatment has to be used.
62
A moving observer approaches a **stationary source**. What happens to the observed frequency?
It increases. ## Footnote f' = f(v + vo)/v For light, this expression is modified.
63
# True or False: There is no **Doppler effect** when **source and observer move together** at constant velocity.
True ## Footnote Doppler shift depends on relative motion, not absolute. This goes back to a central idea in physics: rest and uniform motion are equivalent.
64
A source emitting light of frequency 𝑓 moves toward a mirror with speed 𝑣. What **frequency is observed by the source** after reflection?
## Footnote The mirror receives a higher frequency. It then acts as a source. Since the original source is moving towards the mirror, we have another shift to higher frequency. Combine these two shifts to get the required expression.
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