Waves, Sound, and Physical Optics

• 14.1.A.1 Waves transfer energy between two locations without transferring matter between those locations.
  • 14.1.A.1.i A wave pulse is a single disturbance that transfers energy without transferring matter between two locations.
  • 14.1.A.1.ii A wave is modeled as a continuous, periodic disturbance with well-defined wavelength and frequency.
• 14.1.A.2 Mechanical waves or wave pulses require a medium in which to propagate. Electromagnetic waves or wave pulses do not require a medium in which to propagate.
• 14.1.A.3 The speed at which a wave or wave pulse propagates through a medium depends on the type of wave and the properties of the medium.
  • 14.1.A.3.i The speed of all electromagnetic waves in a vacuum is a universal physical constant, $c = 3.00 \times 10^{8}$ m/s.
  • 14.1.A.3.ii The speed at which a wave pulse or wave propagates along a string is dependent upon the tension in the string, $F_T$, and the mass per length of the string.
    • Equation: $v_{\text{string}} = \sqrt{\dfrac{F_T}{m/\ell}}$
  • 14.1.A.3.iii In a given medium, the speed of sound waves increases with the temperature of the medium.
• 14.1.A.4 In a transverse wave, the direction of the disturbance is perpendicular to the direction of propagation of the wave.
• 14.1.A.5 In a longitudinal wave, the direction of the disturbance is parallel to the direction of propagation of the wave.
  • 14.1.A.5.i Sound waves are modeled as mechanical longitudinal waves.
  • 14.1.A.5.ii The regions of high and low pressure in a sound wave are called compressions and rarefactions, respectively.
• 14.1.A.6 Amplitude is the maximum displacement of a wave from its equilibrium position.
  • 14.1.A.6.i The amplitude of a longitudinal pressure wave may be determined by the maximum increase or decrease in pressure from equilibrium pressure.
  • 14.1.A.6.ii The loudness of a sound increases with increasing amplitude.
  • 14.1.A.6.iii The energy carried by a wave increases with increasing amplitude.

• 14.2.A.1 Periodic waves have regular repetitions that can be described using period and frequency.
  • 14.2.A.1.i The period is the time for one complete oscillation of the wave.
  • 14.2.A.1.ii The frequency is the rate at which the wave repeats.
    • Equation: $T = \dfrac{1}{f}$
  • 14.2.A.1.iii The amplitude of a wave is independent of the period and the frequency of that wave.
  • 14.2.A.1.iv The energy of a wave increases with increasing frequency.
  • 14.2.A.1.v The frequency of a sound wave is related to its pitch.
  • 14.2.A.1.vi Wavelength is the distance between successive corresponding positions (such as peaks or troughs) on a wave.
• 14.2.A.2 A sinusoidal wave can be described by equations for the displacement from equilibrium at a specific location as a function of time. A wave can also be described by an equation for the displacement from equilibrium at a specific time as a function of position.
  • Equation: $x(t) = A\cos(\omega t) = A\cos(2\pi f t)$
  • Equation: $y(x) = A\cos\left(2\pi \dfrac{x}{\lambda}\right)$
• 14.2.A.3 For a periodic wave, the wavelength is proportional to the wave's speed and inversely proportional to the wave's frequency.
  • Equation: $\lambda = \dfrac{v}{f}$

• 14.3.A.1 A wave that travels from one medium to another can be transmitted or reflected, depending on the properties of the boundary separating the two media.
  • 14.3.A.1.i A wave traveling from one medium to another (for example, a wave traveling between low-mass and high-mass strings) will result in reflected and transmitted waves.
  • 14.3.A.1.ii A reflected wave is inverted if the transmitted wave travels into a medium in which the speed of the wave decreases.
  • 14.3.A.1.iii A reflected wave is not inverted if the transmitted wave travels into a medium in which the speed of the wave increases.
  • 14.3.A.1.iv The frequency of a wave does not change when it travels from one medium to another.
• 14.3.A.2 Transverse waves that are reflected from a surface, refracted through a medium, or pass through specific openings may be polarized.
  • 14.3.A.2.i Transverse waves can be polarized and oscillate in a single plane.
  • 14.3.A.2.ii Longitudinal waves cannot be polarized.
• 14.3.A.3 Polarization of a wave may result in a reduction of the wave's intensity.
  • 14.3.A.3.i Intensity is a measure of the amount of power transferred per unit area.
  • 14.3.A.3.ii The intensity of a wave is the average power per unit area over one period of the wave.

• 14.4.A.1 Electromagnetic waves consist of oscillating electric and magnetic fields that are mutually perpendicular.
  • 14.4.A.1.i Electromagnetic waves are transverse waves because the oscillations of the electric and magnetic fields are perpendicular to the direction of propagation.
  • 14.4.A.1.ii Electromagnetic waves are commonly assumed to be plane waves, which are characterized by planar wave fronts.
• 14.4.A.2 Electromagnetic waves do not need a medium through which to propagate.
• 14.4.A.3 Categories of electromagnetic waves are characterized by their wavelengths.
  • 14.4.A.3.i Categories of electromagnetic waves include (in order of decreasing wavelength, spanning a range from kilometers to picometers) radio waves, microwaves, infrared, visible, ultraviolet, X-rays, and gamma rays.
  • 14.4.A.3.ii Visible electromagnetic waves are further broken into categories of color, including (in order of decreasing wavelength) red, orange, yellow, green, blue, and violet.
  • 14.4.A.3.iii Visible electromagnetic waves are also called light. Sometimes, electromagnetic waves of all wavelengths are collectively referred to as light or electromagnetic radiation.

Boundary statement: AP Physics 2 expects students to know the ordering of the electromagnetic spectrum (including visible light). However, students will not be expected to define exact wavelength ranges within the electromagnetic spectrum.

• 14.5.A.1 The Doppler effect describes the relationship between the rest frequency of a wave source, the observed frequency of the source, and the relative velocity of the source and the observer.
• 14.5.A.2 A greater relative velocity results in a greater measured difference between the observed and rest frequencies.
  • 14.5.A.2.i For a wave source moving at the same velocity as the observer, the observed frequency is equal to the rest frequency.
  • 14.5.A.2.ii For a wave source moving toward an observer, the observed frequency is greater than the rest frequency.
  • 14.5.A.2.iii For a wave source moving away from an observer, the observed frequency is less than the rest frequency.

Boundary statement: Only qualitative treatments of the Doppler effect are required for AP Physics 2.

• 14.6.A.1 Wave interference is the interaction of two or more wave pulses or waves.
• 14.6.A.2 When two or more wave pulses or waves interact with each other, they travel through each other and overlap rather than bouncing off each other.
• 14.6.A.3 When two or more wave pulses or waves overlap, the resulting displacement can be determined by adding the individual displacements. This is called superposition.
• 14.6.A.4 Wave interference may be constructive or destructive.
  • 14.6.A.4.i When the displacements of the superposed wave pulses or waves are in the same direction, the interaction is called constructive interference.
  • 14.6.A.4.ii When the displacements of the superposed wave pulses or waves are in opposite directions, the interaction is called destructive interference.
  • 14.6.A.4.iii Two or more traveling wave pulses or waves can interact in such a way as to produce amplitude variations in the resultant wave pulse or wave.
• 14.6.A.5 Visual representations of wave pulses or waves are useful in determining the result of two interacting wave pulses or waves.
• 14.6.A.6 Beats arise from the addition of two waves of slightly different frequency.
  • 14.6.A.6.i Waves with different frequencies are sometimes in phase and sometimes out of phase at locations along the waves, causing periodic amplitude changes in the resultant wave.
  • 14.6.A.6.ii The beat frequency is the difference in the frequencies of the two waves.
    • Equation: $\left|f_{\text{beat}}\right| = \left|f_1 - f_2\right|$
  • 14.6.A.6.iii Tuning forks are devices that are commonly used to demonstrate beat frequencies.

• 14.6.B.1 Standing waves can result from interference between two waves that are confined to a region and traveling in opposite directions.
  • 14.6.B.1.i Standing waves have nodes and antinodes. A node is a point on the standing wave where the amplitude is always zero. An antinode is a point on the standing wave where the amplitude is always at maximum.
  • 14.6.B.1.ii The possible wavelengths of a standing wave are determined by the size and boundary conditions of the region to which it is confined.
  • 14.6.B.1.iii Common regions where standing waves can form include pipes with open or closed ends, as well as strings with fixed or loose ends.
• 14.6.B.2 A standing wave with the longest possible wavelength is called the fundamental or first harmonic. The second-longest wavelength is typically called the second harmonic, the third-longest wavelength is called the third harmonic, and so on. However, for a standing wave with a node at one end and an antinode at the other end, only odd harmonics can be established.
• 14.6.B.3 Visual representations of standing waves are useful in determining the relationships between length of the region, wavelength, frequency, wave speed, and harmonic.

• 14.7.A.1 Diffraction is the spreading of a wave around the edges of an obstacle or through an opening.
• 14.7.A.2 Diffraction is most pronounced when the size of the opening is comparable to the wavelength of the wave.
• 14.7.A.3 Diffraction of multiple wavefronts through a single opening leads to observable interference patterns.
• 14.7.A.4 Diffraction is commonly demonstrated by monochromatic light of wavelength $\lambda$ incident on a narrow opening of width $a$ that is a distance $L$ from a screen.
  • 14.7.A.4.i Constructive and destructive interference of multiple wavefronts originating from the opening will result in bright and dark bands on the screen.
  • 14.7.A.4.ii The amount of interference between two wavefronts depends on the path length difference $\Delta D$ of the wavefronts.
  • 14.7.A.4.iii The path length difference $\Delta D$ can be described in terms of the opening width $a$ and the angle $\theta$ between the direction of propagation of the wavefront and the normal to the opening by the equation $\Delta D = a\sin\theta$.
  • 14.7.A.4.iv For small angles, where $\theta < 10°$, the small angle approximation can be used to relate $\lambda$, $a$, and $L$ to $y_{\min}$, the distance from the middle of the central bright fringe to the $m^{\text{th}}$ order of minimum brightness on the screen.
    • Equation: $a\left(\dfrac{y_{\min}}{L}\right) \approx m\lambda$
• 14.7.A.5 The diffraction pattern produced by a wave passing through an opening depends on the shape of the opening.
• 14.7.A.6 Visual representations of single-slit diffraction patterns are useful in determining the physical properties of the slit and the interacting waves.

• 14.8.A.1 The pattern resulting from monochromatic light of wavelength $\lambda$ incident on two slits a distance $d$ apart is caused by a combination of wave diffraction and wave interference.
  • 14.8.A.1.i When only considering wave interference, a double slit creates a pattern of uniformly spaced maxima.
  • 14.8.A.1.ii Constructive and destructive interference of the wavefronts originating from each slit will result in bright and dark bands on the screen.
  • 14.8.A.1.iii The amount of interference between two wavefronts depends on the path length difference $\Delta D$ of the wavefronts.
  • 14.8.A.1.iv The path length difference $\Delta D$ can be described in terms of the slit separation $d$ and the angle $\theta$ between the direction of propagation of the wavefront and the normal to the opening by the equation $\Delta D = d\sin\theta$.
  • 14.8.A.1.v For small angles, where $\theta < 10°$, the small angle approximation can be used to relate $\lambda$, $d$, and $L$ to $y_{\max}$, the distance from the middle of the central bright fringe to the $m^{\text{th}}$ order of maximum brightness on the screen.
    • Equation: $d\left(\dfrac{y_{\max}}{L}\right) \approx m\lambda$
  • 14.8.A.1.vi When considering wave interference and wave diffraction, a double slit creates an interference pattern of maxima and minima superimposed within the envelope created by single-slit diffraction.
• 14.8.A.2 Interference patterns produced by light interacting with a double slit indicate that light has wave properties. The source of this discovery was Young's double-slit experiment.
• 14.8.A.3 Visual representations of double-slit diffraction patterns are useful in determining the physical properties of the slits and the interacting waves.
• 14.8.A.4 A diffraction grating is a collection of evenly spaced parallel slits or openings that produce an interference pattern that is the combination of numerous diffraction patterns superimposed on each other.
• 14.8.A.5 When white light is incident on a diffraction grating, the center maximum is white and the higher-order maxima disperse white light into a rainbow of colors, with the longest-wavelength light (red) appearing farthest from the central maximum.

• 14.9.A.1 When light travels from one medium to another, some of the light is transmitted, some is reflected, and some is absorbed.
• 14.9.A.2 The phase change of a reflected ray depends on the relative indices of refraction of the materials with which the ray interacts.
  • 14.9.A.2.i A phase change of 180 degrees occurs when a light ray is reflected from a medium with a greater index of refraction than the medium through which the ray is traveling.
  • 14.9.A.2.ii No phase change occurs when a light ray is reflected from a medium with a lower index of refraction than the medium through which the ray is traveling.
• 14.9.A.3 The phase of a wave does not change when it is refracted as it passes from one medium into another.
• 14.9.A.4 Thin-film interference occurs when light interacts with a medium whose thickness is comparable to the light's wavelength.
  • 14.9.A.4.i The interactions between the initial reflected light and the light exiting the thin film after being reflected from the second interface exhibit wave interference behavior, resulting in a single wave that is the sum of the two interacting waves.
  • 14.9.A.4.ii The amount of constructive or destructive interference between the two reflected waves depends on the relationship between the thickness of the film, the wavelength of light, any phase shifts, and the angle at which the incident light strikes the film.
• 14.9.A.5 Practical examples of thin-film interference include the color variations seen in soap bubbles and oil films, as well as antireflection coatings.
  • 14.9.A.5.i The spectrum of colors observed in oil films and soap bubbles arises from differences in the thickness of the film.
  • 14.9.A.5.ii Antireflection coatings eliminate reflected light by applying the relationships between indices of refraction, phase shift, and wave interference to create destructive interference of the light reflected from the two surfaces of the coating.
  • 14.9.A.5.iii The simplest antireflection coating has a thickness equal to one-quarter of the wavelength of the light in the coating, and the index of refraction of the coating is greater than that of air and less than that of the surface upon which the coating is applied. This assumes incident light is normal to the surface.

Boundary statement: Quantitative analysis of thin-film interference is limited to waves that are normal to the incident surface.