Unit 14: Waves, Sound, and Physical Optics

This unit covers how waves behave, from sound traveling through air to light interfering through slits. It includes wave properties, sound and the Doppler effect, standing waves in strings and pipes, beats, and the interference and diffraction of light.

14.1 Wave Properties

A wave carries energy through a medium or through space without carrying matter along with it. Each piece of the medium oscillates around its rest position while the disturbance moves on. Frequency f is the number of oscillations per second, period T = 1/f is the time for one oscillation, and wavelength λ is the distance the disturbance travels in one period.

Wave speed, frequency, and wavelength are tied together by v = fλ. The frequency is set by the source, so when a wave crosses into a new medium the frequency stays the same while the speed and wavelength change to fit.

Worked example. A wave on a string has frequency 250 Hz and wavelength 0.80 m, so its speed is v = fλ = 250 × 0.80 = 200 m/s. It then enters a thicker section of the string where the speed drops to 150 m/s. The frequency stays 250 Hz, and the new wavelength is λ = v/f = 150/250 = 0.60 m. Check: 250 × 0.60 = 150, which matches the new speed.

14.2 Sound

Sound is a longitudinal mechanical wave. Air molecules oscillate back and forth along the direction the wave travels, forming compressions (crowded regions) and rarefactions (spread-out regions). The speed of sound in air is about 343 m/s at 20°C, roughly a million times slower than light.

Intensity is power per unit area. The ear responds to an enormous range of intensities, so loudness is measured on the logarithmic decibel scale. Every increase of 10 dB means ten times the intensity. A whisper and a jet engine differ by a factor of about a trillion in intensity but only about 120 dB on the scale.

The Doppler effect shifts the observed frequency when source and observer move relative to each other. The AP equation sheet gives f′ = f₀(v + v_O)/(v − v_S). Take v_O as positive when the observer moves toward the source, and v_S as positive when the source moves toward the observer. Either approaching motion raises the observed frequency; moving away flips the sign and lowers it. The wave speed in the medium does not change.

14.3 Standing Waves on Strings

When a wave reflects back on itself, the incident and reflected waves interfere to form a standing wave, a pattern that does not travel. A node is a point where the medium never moves. An antinode is a point where it oscillates with maximum amplitude.

A string fixed at both ends can only support standing waves with a node at each end, which forces the length to hold a whole number of half-wavelengths: L = nλ/2 for n = 1, 2, 3, .... The allowed frequencies are f_n = nv/(2L). The n = 1 mode is the fundamental. The higher modes are harmonics, and each harmonic is a whole-number multiple of the fundamental. The first mode above the fundamental is called the first overtone, so on a string the second harmonic is the first overtone.

Worked example. A 0.65 m guitar string carries waves at 390 m/s. The fundamental is f₁ = v/(2L) = 390/(2 × 0.65) = 390/1.30 = 300 Hz. The third harmonic is three times that, f₃ = 3 × 300 = 900 Hz. As a check, the fundamental wavelength is λ₁ = 2L = 1.30 m, and 300 × 1.30 = 390 m/s, the given wave speed.

14.4 Standing Waves in Pipes

Air columns in pipes resonate the same way strings do, but the boundary conditions depend on whether each end is open or closed. An open end is a displacement antinode, because the air there moves freely. A closed end is a displacement node, because the air cannot move against the wall.

Worked example. A pipe of length 0.60 m is closed at one end. With v = 343 m/s, the fundamental is f₁ = v/(4L) = 343/(4 × 0.60) = 343/2.40 ≈ 143 Hz. The next allowed mode is the third harmonic, which is the first overtone, f₃ = 3 × 143 ≈ 429 Hz. A closed pipe has no second harmonic, no fourth harmonic, and no other even multiples.

14.5 Beats

When two waves of slightly different frequency overlap, the interference alternates between constructive and destructive, and the listener hears the loudness swell and fade. The beat frequency is the difference, f_beat = |f₁ − f₂|. Two tuning forks at 440 Hz and 444 Hz produce 4 beats per second, heard as a slow pulsing on top of the tone.

14.6 Interference of Waves

When two waves meet, their displacements add point by point. Constructive interference happens when the waves arrive in step, crest on crest, and the amplitudes add. Destructive interference happens when they arrive exactly out of step, crest on trough, and the amplitudes cancel. In terms of path difference, the extra distance one wave travels:

14.7 Double-Slit Interference

Light passing through two narrow slits spreads by diffraction, and the two waves interfere on a distant screen. Bright fringes appear where the path difference is a whole number of wavelengths: d sinθ = mλ, m = 0, ±1, ±2, .... Here d is the slit separation and θ is the angle from the central maximum. Dark fringes sit halfway between the bright ones, at d sinθ = (m + 1/2)λ.

Worked example. Slits 0.25 mm apart are lit by 600 nm light. For the second bright fringe (m = 2): sinθ = mλ/d = 2 × 6.0×10⁻⁷/(2.5×10⁻⁴) = 4.8×10⁻³, so θ ≈ 0.275°. On a screen 1.5 m away, the fringe sits y = L tanθ ≈ 1.5 × 4.8×10⁻³ = 7.2×10⁻³ m, or 7.2 mm, from the center. (At these small angles tanθ ≈ sinθ.)

14.8 Single-Slit Diffraction

A single slit still spreads the light, and different parts of the slit interfere with each other. The result is a broad bright central maximum flanked by dimmer fringes. The dark minima sit at a sinθ = mλ, m = ±1, ±2, ..., where a is the slit width. The central maximum stretches from the m = −1 minimum to the m = +1 minimum, so narrowing the slit pushes the first minima outward and the central maximum gets wider.

14.9 Diffraction Gratings

A diffraction grating is thousands of evenly spaced slits. The principal maxima follow the same equation as the double slit, d sinθ = mλ, with d now the spacing between adjacent slits. With so many slits contributing, the maxima are much sharper, narrower, and brighter than a double slit's, with faint secondary maxima in between. That sharpness is why gratings are used to measure wavelengths precisely.

14.10 Thin-Film Interference

A thin film, like a soap bubble, reflects light from its front and back surfaces, and the two reflections interfere. Reflection from a higher-index medium flips the wave by half a cycle, a 180° phase reversal. Reflection from a lower-index medium causes no reversal. For a soap film in air, the front (air-to-film) reflection reverses while the back (film-to-air) reflection does not, so whether the interference is constructive or destructive depends on the film thickness compared with the wavelength. That is why the colors shift as the film thins.

Confusions That Cost Points

Practice Questions

Original questions written for this guide in the style of the AP exam. Answers and explanations are on the next page, so complete the questions before checking them.

Answer Key

1. B. The frequency is set by the source and stays 120 Hz in the new medium, so λ = v/f = 180/120 = 1.5 m. C keeps the old wavelength, treating λ as the fixed quantity when it is f that is fixed. D computes (240/180) × 2.0 = 2.7 m, inverting the ratio. A computes f/v = 0.67, the reciprocal of the wavelength.

2. A. For a string fixed at both ends, f_n = nf₁, so f₁ = 900/3 = 300 Hz. B halves the frequency, as if the third harmonic were twice the fundamental. C doubles instead of dividing by 3, which no harmonic series supports. D uses the ratio 2/3 instead of 1/3.

3. B. The fundamental is f₁ = v/(4L) = 343/(4 × 0.50) = 343/2.0 = 171.5 Hz. A closed pipe supports only odd harmonics, so the first overtone is the third harmonic: f₃ = 3 × 171.5 = 514.5 ≈ 515 Hz. A answers the fundamental instead of the overtone. C computes 2 × 171.5, applying the open-pipe series with its even harmonics to a closed pipe. D computes 4 × 171.5, using n = 4 as if every harmonic existed.

4. C. Bright fringes from two slits sit where the path difference is a whole number of wavelengths: d sinθ = mλ. D is the single-slit minima condition, a different setup giving dark fringes. A gives the dark fringes of the double slit. B adds a factor of 2 that appears in no interference condition.

5. D. In f′ = f₀(v + v_O)/(v − v_S), a source moving toward the observer makes the denominator smaller than v, so the observed frequency is higher than the emitted one. A makes the sign error, treating approach like recession. B confuses wave speed, which is unchanged, with frequency. C gets the frequency right but wrongly claims the speed of sound changes too.

6. C. The front reflection goes from air (lower n) to film (higher n), so it undergoes the 180° reversal. The back reflection goes from film to air, higher to lower n, so it does not. D ignores the direction rule entirely. A denies any reversal. B assigns the reversal to the wrong surface.

7. D. The beat frequency is the difference: |440 − 444| = 4 Hz, heard as four loudness pulses per second. A gives 442 Hz, the average, which is the perceived pitch of the combined tone, not the beat. B halves the difference for no reason the beat formula supports. C adds the frequencies.

8. A. The first minima sit at sinθ = ±λ/a, so a smaller slit width a pushes the minima to larger angles and the central maximum between them gets wider. B inverts the relationship. C confuses width with brightness: the pattern does dim, but the width changes too. D misplaces the central maximum, which stays centered at θ = 0.

One-Page Recall Check

Say each answer out loud before you look back, and mark the ones you cannot finish. Anything you cannot say out loud yet belongs in your flashcard deck. In Rycal, add those items to the Waves, Sound, and Physical Optics deck and let spaced review bring them back over the next few days.

• State v = fλ and explain which of the three quantities is fixed by the source.
• Distinguish a transverse wave from a longitudinal wave and give an example of each.
• Give the speed of sound in air at 20°C.
• Explain what the decibel scale measures and what an increase of 10 dB means for intensity.
• Write the Doppler expression from the equation sheet and state the sign rule for approaching versus receding motion.
• Define node and antinode in a standing wave.
• Give the allowed frequencies for a string fixed at both ends.
• Give the harmonic series for an open-open pipe and for a pipe closed at one end.
• Explain how displacement nodes and pressure nodes relate in a sound standing wave.
• State the beat frequency rule and use it on 440 Hz and 444 Hz.
• State the path-difference conditions for constructive and destructive interference.
• Write the double-slit bright-fringe condition and the single-slit minima condition, and say which is which.
• Explain how a diffraction grating pattern differs from a double-slit pattern with the same slit spacing.
• Explain phase reversal in thin-film interference, including which surface of a soap film in air reverses.

Key terms for this unit

Wave, Frequency, Period, Wavelength, Transverse wave, Longitudinal wave, Sound, Intensity, Decibel, Doppler effect, Standing wave, Node, Antinode, Fundamental, Harmonic, Overtone, Beat frequency, Constructive interference, Destructive interference, Path difference, Double-slit interference, Single-slit diffraction, Diffraction grating, Thin-film interference, Phase reversal.

About this guide. Written for Rycal and aligned to the College Board AP Physics 2 course framework, Unit 14. All questions and explanations are original Rycal writing. Rycal is independent and is not affiliated with or endorsed by the College Board.