Source: PHYS 116 Final Exam Notes, University of Washington
Difficulty: Intermediate | Prerequisites: Superposition (Ch. 16.1), standing waves on a string (Ch. 16.2–16.3)
Big Picture
This set of topics applies everything from the first three sections to sound in air. Standing sound waves in tubes explain how wind instruments work. Interference from two sources explains why you hear loud and quiet spots in a room with two speakers. Beats explain the pulsing sound when two notes are slightly out of tune. If you are comfortable with nodes, antinodes, and the superposition principle, these sections are largely a matter of applying familiar rules to new geometries.
Sound waves form standing waves inside tubes, with the boundary conditions (open or closed ends) determining which harmonics are allowed. When two speakers emit the same frequency, whether you hear a loud or quiet spot depends on the path length difference between you and each speaker. When two frequencies are close but not identical, you hear a pulsing loudness called beats, at a rate equal to the difference between the two frequencies.
Open-open tube (open tube)
A tube open at both ends, such as a flute. Both ends are pressure nodes (displacement antinodes). All integer harmonics are allowed: m = 1, 2, 3, ...
Open-closed tube (closed tube)
A tube open at one end and closed at the other. The open end is a pressure node; the closed end is a pressure antinode. Only odd harmonics are allowed: m = 1, 3, 5, ...
Pressure node
A point in a sound wave where the pressure stays at atmospheric pressure (no variation). Open ends of tubes are pressure nodes.
Pressure antinode
A point where the pressure variation is at its maximum, alternating between peak compression and peak rarefaction. Closed ends of tubes are pressure antinodes.
Path length difference (Δd)
The difference in distance from two sources to a given point: Δd = d₂ - d₁. This quantity determines whether the waves arrive in phase or out of phase at that point.
In phase
Two waves are in phase when their crests and troughs are aligned. This happens when the path length difference is a whole number of wavelengths. The result is constructive interference.
Out of phase
Two waves are out of phase when the crest of one aligns with the trough of the other. This happens when the path length difference is a half-integer number of wavelengths. The result is destructive interference.
Beat frequency (f_beat)
The rate at which the loudness of combined sound oscillates when two tones of slightly different frequency are played together. Equal to the absolute value of the difference: f_beat = |f₁ - f₂|. In simple terms, it is how many "pulses" of loud-soft-loud you hear per second.
Pitch
The perceived frequency of a sound, determined by the fundamental frequency. Your brain identifies the fundamental as the pitch even when many harmonics are present.
Timbre (tone quality)
The quality of a sound that lets you distinguish between instruments playing the same note at the same loudness. Determined by the mix of higher harmonics present. Think of it as the "colour" of a sound.
Sound is a longitudinal pressure wave. Standing sound waves in tubes follow the same logic as standing waves on strings, but the boundary conditions are different.
An open end of a tube is a pressure node (the pressure stays at atmospheric). A closed end is a pressure antinode (pressure varies maximally).
Open-open (or closed-closed) tube:
Both ends share the same type of boundary condition. The allowed wavelengths and frequencies are the same as a string fixed at both ends.
All harmonics are present: m = 1, 2, 3, ...
Adjacent harmonics are separated by the fundamental frequency: Δf = f₁.
Open-closed tube:
The two ends have different boundary conditions (one node, one antinode). This changes the allowed patterns.
Only odd harmonics are present: m = 1, 3, 5, ...
The tube is effectively "longer" in terms of the wavelength it supports: λ₁ = 4L instead of 2L.
Most real sounds are a mix of many frequencies. When you play middle C on a guitar, the fundamental is 262 Hz, but standing waves also form at 524, 786, 1048 Hz, and so on (the full harmonic series).
Your brain identifies the fundamental frequency as the pitch. Middle C sounds like middle C regardless of the instrument.
Your brain uses the pattern of higher harmonics to identify timbre, which is how you tell a guitar from a trumpet playing the same note.
The higher harmonics do not change the period of the sound wave. They change only its shape. A trumpet's middle C and a guitar's middle C both have a period of 3.82 ms (corresponding to 262 Hz), but the waveform shapes are entirely different.
When two speakers emit identical sound waves (same frequency, wavelength, and amplitude), whether a given point is loud or quiet depends on the path length difference.
Path length difference: Δd = d₂ - d₁, where d₁ and d₂ are the distances from each speaker to the point.
Constructive interference (loud spot):
Occurs when Δd = mλ, where m = 0, 1, 2, 3, ...
The waves arrive in phase. The combined amplitude is doubled.
Destructive interference (quiet spot):
Occurs when Δd = (m + ½)λ, where m = 0, 1, 2, 3, ...
The waves arrive out of phase. The combined amplitude is zero (complete cancellation).
Spherical waves:
The same path length difference rules apply to spherical waves spreading out from two point sources.
At any point, count how many wavelengths separate it from each source. If the difference is a whole number, you get constructive interference. If it is a half-integer, you get destructive interference.
When two sound waves with slightly different frequencies f₁ and f₂ are played together, the result is a tone whose loudness oscillates.
The oscillation frequency (the perceived pitch) is the average: f_osc = ½(f₁ + f₂). This is the frequency your ear hears.
The rate of the loud-soft-loud pulsing is the beat frequency: f_beat = |f₁ - f₂|.
The beat frequency is always slower than the oscillation frequency.
Practical use: if you hear 3 beats per second and one tone is 610 Hz, the other tone is either 607 Hz or 613 Hz. You cannot tell which from the beat frequency alone.
Open-open (or closed-closed) tube:
\lambda_m = \frac{2L}{m} \qquad f_m = m \left( \frac{v}{2L} \right) = m f_1 \qquad m = 1, 2, 3, \ldotsOpen-closed tube:
\lambda_m = \frac{4L}{m} \qquad f_m = m \left( \frac{v}{4L} \right) = m f_1 \qquad m = 1, 3, 5, \ldots \text{ (odd only)}Path length difference (two-source interference):
Constructive (loud): Δd = mλ, where m = 0, 1, 2, ...
Destructive (quiet): Δd = (m + ½)λ, where m = 0, 1, 2, ...
Beat frequency:
f_{\text{beat}} = |f_1 - f_2|Oscillation frequency (perceived pitch of beats):
f_{\text{osc}} = \frac{1}{2}(f_1 + f_2)Speed of sound relation:
v = \lambda fWind instruments work by creating standing sound waves in air columns. A flute (open-open) supports all harmonics. A clarinet (effectively open-closed) supports only odd harmonics, which gives it a distinctly different tone quality from a flute even when playing the same note. Piano tuners use beats to tune strings: they adjust the tension until the beat frequency between two strings drops to zero, meaning the strings are at the same frequency.
Students often confuse displacement nodes with pressure nodes in sound waves. In a tube, the open end is a pressure node (pressure stays constant) but a displacement antinode (air moves freely). The closed end is the opposite.
A common error with open-closed tubes is using the open-open formula. Remember: open-closed uses 4L in the denominator and allows only odd harmonics.
Students sometimes think the beat frequency is the pitch they hear. The pitch is the oscillation frequency (the average). The beat frequency is just the rate of the loudness modulation.
When given two consecutive harmonics of an open-open tube, students sometimes assume they are f₁ and f₂. Check first: for an open-open tube, adjacent harmonics differ by exactly f₁. If 400 Hz and 300 Hz are adjacent harmonics, the fundamental is 100 Hz, not 300 Hz.
⚠️ Know the difference between open-open and open-closed tube formulas. You will be tested on which harmonics are allowed in each case.
⚠️ Path length difference problems are a staple. Expect to be given two speakers, a listener position, a frequency, and the speed of sound, and asked whether the spot is loud or quiet.
⚠️ The worked example from the notes (Example 16.11: two speakers 3.0 m apart, 170 Hz, listener 4 m away, v = 340 m/s) is a classic exam template. Know the steps: find d₁ and d₂, compute Δd, find λ, express Δd as a multiple of λ.
⚠️ Beat frequency problems: given f_beat and one frequency, state both possible values for the other frequency. The exam may ask you to explain why there are two possible answers.
Fill in the blank: An open end of a tube is a pressure ______. (node)
True or false: An open-closed tube supports all harmonics. (False, only odd harmonics.)
Fill in the blank: Constructive interference from two sources occurs when the path length difference is ______ wavelengths. (a whole number of)
True or false: The beat frequency tells you the pitch you hear. (False, the pitch is the oscillation frequency, which is the average of the two.)
Fill in the blank: If f₁ = 440 Hz and f₂ = 444 Hz, the beat frequency is ______ Hz. (4)
Q: A tube open at both ends supports standing waves with harmonics at 300 Hz and 400 Hz, with no harmonics in between. What is the fundamental frequency?
A: For an open-open tube, adjacent harmonics differ by f₁. The difference is 400 - 300 = 100 Hz, so f₁ = 100 Hz. (The 300 Hz harmonic is the third, and 400 Hz is the fourth.)
Q: Two speakers are 3.0 m apart and play identical tones at 170 Hz. Sam stands directly in front of one speaker at a distance of 4.0 m. The speed of sound is 340 m/s. Is Sam at a loud spot or a quiet spot?
A: First find the distances. d₁ = 4.0 m (directly in front). d₂ = √(3.0² + 4.0²) = √25 = 5.0 m. Path length difference: Δd = 5.0 - 4.0 = 1.0 m. Next find the wavelength: λ = v/f = 340/170 = 2.0 m. Express Δd in wavelengths: 1.0/2.0 = 0.5 = ½λ. A half-wavelength path difference means destructive interference. Sam is at a quiet spot.
Q: You hear 3 beats per second when two sound tones are played. One tone is 610 Hz. What are the possible frequencies of the other tone?
A: f_beat = |f₁ - f₂| = 3, so f₂ = 610 ± 3. The other tone is either 607 Hz or 613 Hz.
Q: A clarinet (open-closed tube) has a length of 0.60 m. Assuming the speed of sound is 340 m/s, what is its fundamental frequency? What is the frequency of its next allowed harmonic?
A: For an open-closed tube, f₁ = v/(4L) = 340/(4 × 0.60) = 340/2.4 ≈ 142 Hz. Only odd harmonics are allowed, so the next harmonic is m = 3: f₃ = 3 × 142 ≈ 425 Hz.
Q: Why do a guitar and a trumpet playing the same note at the same volume sound different?
A: They have different timbres. Both produce the same fundamental frequency (same pitch) and can play at the same loudness, but the relative strengths of their higher harmonics differ. Your brain uses this harmonic "recipe" to distinguish the two instruments.
Standing sound waves extend the string-based standing wave formulas to air columns, so everything from Ch. 16.2-16.3 carries forward. Two-source interference connects to optics later in the course, where the same path length difference logic applies to light passing through double slits (Young's experiment). Beats connect to any scenario where two close frequencies are present, including radar and sonar applications.
Standing sound waves, open tube, closed tube, open-open tube, open-closed tube, pressure node, pressure antinode, longitudinal wave, harmonics in tubes, odd harmonics, path length difference, constructive interference two sources, destructive interference two sources, in phase, out of phase, loud spot, quiet spot, beats, beat frequency, oscillation frequency, pitch, timbre, tone quality, fundamental frequency, v = λf, PHYS 116, general physics waves, University of Washington