Some points on a vibrating string barely move while their neighbours move strongly. The pattern comes from waves travelling in opposite directions. The ends of the string decide which patterns can keep fitting.
We will build the patterns from superposition, distinguish sound pressure from particle displacement, and use a resonance-tube measurement to infer wavelength, speed and an end correction.
What you will learn to do
- Construct a standing wave from counterpropagating components.
- Distinguish permanent nodes from instantaneously zero displacement.
- Infer fixed-string modes and tension from node counts.
- State displacement and pressure conditions at open and closed acoustic ends.
- Distinguish harmonic, overtone and allowed-mode numbers.
- Infer wave speed from successive resonance lengths and audit the uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Standing waves, pipe modes and resonance measurements Learning goals: - Construct a standing wave from counterpropagating components. - Distinguish permanent nodes from instantaneously zero displacement. - Infer fixed-string modes and tension from node counts. - State displacement and pressure conditions at open and closed acoustic ends. - Distinguish harmonic, overtone and allowed-mode numbers. - Infer wave speed from successive resonance lengths and audit the uncertainty. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
Can two nonzero displacements from different waves add to zero at one point?
Compare your reasoning
Yes. Superposition adds signed displacements. Opposite contributions can cancel, even though each travelling wave is present.
Revisit the explanation →01 / Standing waves, pipe modes and resonance measurements
Two travelling waves can produce a pattern that stays in place
Take two sinusoidal waves of equal amplitude, frequency and wavelength travelling in opposite directions on the same ideal string. Adding A sin(kx − ωt) and A sin(kx + ωt) gives 2A sin(kx) cos(ωt). The position dependence and time dependence now appear as separate factors.
At each fixed x, the string oscillates harmonically with amplitude |2A sin(kx)|. The pattern does not translate along the string. Some positions never move and some have the largest amplitude. The two constituent waves still propagate at their usual speed even though the combined node pattern is stationary.
The ideal standing wave has no net time-averaged power flow along the string because its equal travelling components carry equal average powers in opposite directions. This does not mean the string has no energy or that local energy never moves. Kinetic and elastic energy change during the cycle.
Unequal counterpropagating amplitudes generally leave a travelling component and incomplete nodes. A reflection with losses may therefore produce a partly standing pattern. The exact fixed nodes described here belong to the ideal equal-component model.
The maximum standing-wave displacement is 2A, where A is the amplitude of each travelling component.
STOP & THINK
In an ideal standing wave formed from two equal opposite travelling waves, which statement is correct?
- Neither component propagates
- The nodes stay fixed although the component waves propagate
- The medium has zero energy
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Standing waves, pipe modes and resonance measurements Focus: Two travelling waves can produce a pattern that stays in place Learning goals: - Construct a standing wave from counterpropagating components. - Distinguish permanent nodes from instantaneously zero displacement. - Infer fixed-string modes and tension from node counts. - State displacement and pressure conditions at open and closed acoustic ends. - Distinguish harmonic, overtone and allowed-mode numbers. - Infer wave speed from successive resonance lengths and audit the uncertainty. Equations in this section (LaTeX): y(x,t)=2A\sin(kx)\cos(\omega t) Meaning: The maximum standing-wave displacement is 2A, where A is the amplitude of each travelling component. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
02 / Standing waves, pipe modes and resonance measurements
Nodes and antinodes describe amplitudes, not a single snapshot
A node is a position where displacement is zero at every time. For the standing wave above, nodes satisfy sin(kx) = 0, so adjacent nodes are separated by λ/2. An antinode is halfway between adjacent nodes, where the displacement amplitude is greatest. A node and its nearest antinode are separated by λ/4.
At some instants cos(ωt) = 0 and the entire ideal string passes through its equilibrium line. That does not make every point a node. A node is defined over time; at that instant the antinodes are moving fastest while the nodes remain stationary.
All non-node points within one loop between neighbouring nodes oscillate in phase. Adjacent loops oscillate in opposite phase because sin(kx) changes sign across a node. Amplitudes within a loop vary with position, even though the time dependence is shared.
Count loops rather than isolated peaks in a drawing. A standing-wave snapshot can show positive and negative lobes, but a lobe’s sign reverses after half a period. The number and positions of nodes remain the same.
Locate the nodes in a three-loop string mode
- One extreme
- 0
- Half a period later
- 0
WORKED EXAMPLE
Recover wavelength from the right pair of features
The distance from a node to the nearest antinode of a standing wave is 0.12 m. The constituent travelling-wave speed is 24 m/s. Find wavelength and frequency.
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Topic: Standing waves, pipe modes and resonance measurements Focus: Nodes and antinodes describe amplitudes, not a single snapshot Learning goals: - Construct a standing wave from counterpropagating components. - Distinguish permanent nodes from instantaneously zero displacement. - Infer fixed-string modes and tension from node counts. - State displacement and pressure conditions at open and closed acoustic ends. - Distinguish harmonic, overtone and allowed-mode numbers. - Infer wave speed from successive resonance lengths and audit the uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
03 / Standing waves, pipe modes and resonance measurements
Fixed ends select the wavelengths that fit
A string fixed at x = 0 and x = L must have zero displacement at both endpoints. The form sin(kx) automatically satisfies the first boundary. The second requires sin(kL) = 0, so kL = nπ with positive integer n.
Consequently L contains n half-wavelengths: λ_n = 2L/n. The corresponding frequencies are f_n = nc/(2L). The first mode is the fundamental or first harmonic; the second mode is the second harmonic and first overtone. The nth fixed-string mode has n loops, n antinodes and n − 1 internal nodes.
The fundamental frequency depends on both the allowed geometry and the medium speed. For an ideal uniform string, substituting c = √(F_T/μ) gives f₁ = (1/2L)√(F_T/μ). Changing length, tension and linear density are different ways to change the resonances.
A standing mode can be weakly excited if the driving point is near one of its nodes or the forcing does not couple to its shape. An allowed frequency is therefore not a guarantee of a large observed response. Damping, driving location and driving frequency all influence the amplitude.
These are fixed–fixed string modes under a uniform-speed model.
WORKED EXAMPLE
Infer the mode from nodes and determine the tension
A uniform string of length 0.80 m and linear density 0.0050 kg/m vibrates at 75 Hz with two internal nodes. Find its mode number and tension.
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Topic: Standing waves, pipe modes and resonance measurements
Focus: Fixed ends select the wavelengths that fit
Learning goals:
- Construct a standing wave from counterpropagating components.
- Distinguish permanent nodes from instantaneously zero displacement.
- Infer fixed-string modes and tension from node counts.
- State displacement and pressure conditions at open and closed acoustic ends.
- Distinguish harmonic, overtone and allowed-mode numbers.
- Infer wave speed from successive resonance lengths and audit the uncertainty.
Equations in this section (LaTeX):
\lambda_n=\frac{2L}{n},\qquad f_n=\frac{nc}{2L},\qquad n=1,2,3,\ldots
Meaning: These are fixed–fixed string modes under a uniform-speed model.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.04 / Standing waves, pipe modes and resonance measurements
Boundary conditions explain reflection signs
At an ideal fixed string end, the endpoint cannot move. The incident and reflected displacements must cancel there, so the reflected displacement pulse is inverted. This is a displacement phase reversal of π for a sinusoidal component.
At an ideal freely moving, massless endpoint with no transverse force, the string slope at the end is zero. The incident and reflected displacement pulses reinforce there, giving no inversion and an antinode. Real endpoint masses, springs or losses produce more complicated reflections.
A string with one fixed end and one ideal free end therefore needs a displacement node at one end and antinode at the other. Its lowest pattern contains a quarter wavelength; the next contains three quarters. This geometry foreshadows the odd-harmonic sequence for an air column closed at one end.
Reflection statements must name the variable. An acoustic pressure reflection can have a different sign from the associated particle-displacement reflection. Memorizing an open-end sign without specifying pressure or displacement is a common source of mistakes.
STOP & THINK
Why must an ideal fixed string end reflect a displacement pulse with inversion?
- The endpoint displacement must remain zero
- The tension must vanish
- The wave speed must become negative
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Standing waves, pipe modes and resonance measurements Focus: Boundary conditions explain reflection signs Learning goals: - Construct a standing wave from counterpropagating components. - Distinguish permanent nodes from instantaneously zero displacement. - Infer fixed-string modes and tension from node counts. - State displacement and pressure conditions at open and closed acoustic ends. - Distinguish harmonic, overtone and allowed-mode numbers. - Infer wave speed from successive resonance lengths and audit the uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Standing waves, pipe modes and resonance measurements
Sound has pressure and displacement patterns with different nodes
In a sound wave, neighbouring fluid elements compress or expand a small region. Pressure variation is related to the spatial derivative of longitudinal displacement, not simply to the displacement itself. In a standing acoustic wave, displacement nodes coincide with pressure antinodes, and displacement antinodes with pressure nodes.
At a rigid closed end, air cannot move through the wall, so longitudinal displacement has a node. Pressure variation can be large there. At an ideal open end, the pressure stays approximately at atmospheric pressure, so pressure variation has a node and displacement has an antinode near the opening.
The open-end boundary is approximate because air outside the tube also participates. The effective displacement antinode lies slightly outside the geometric opening. An end correction accounts for this in a simple model; its magnitude depends on radius and the opening geometry.
Draw either a displacement pattern or a pressure pattern and label it explicitly. A picture with nodes but no named variable is incomplete. Switching between the two pictures without changing node labels can reverse the inferred pipe type.
| End | Displacement | Pressure variation |
|---|---|---|
| Rigid closed end | Node | Antinode |
| Open end, idealized | Antinode near opening | Node near opening |
STOP & THINK
At a rigid closed end of an ideal air column, which combination applies?
- Displacement antinode and pressure node
- Displacement node and pressure antinode
- Both displacement and pressure nodes
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Standing waves, pipe modes and resonance measurements Focus: Sound has pressure and displacement patterns with different nodes Learning goals: - Construct a standing wave from counterpropagating components. - Distinguish permanent nodes from instantaneously zero displacement. - Infer fixed-string modes and tension from node counts. - State displacement and pressure conditions at open and closed acoustic ends. - Distinguish harmonic, overtone and allowed-mode numbers. - Infer wave speed from successive resonance lengths and audit the uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Standing waves, pipe modes and resonance measurements
Fit the appropriate half-waves or quarter-waves
An open–open pipe has displacement antinodes near both ends. The fundamental fits half a wavelength into its effective length, giving f_n = nc/(2L_eff) for every positive integer n. Its harmonic sequence has the same integer pattern as a fixed–fixed string, although the endpoint displacement conditions differ.
A closed–open pipe has a displacement node at one end and antinode at the other. Its effective length fits an odd number of quarter-wavelengths. The frequencies are f_j = (2j − 1)c/(4L_eff), where j = 1, 2, 3, … labels the allowed modes.
The second allowed closed–open mode is the third harmonic and the first overtone. Mode number, harmonic number and overtone number are related but are not interchangeable. Saying second harmonic for that mode would describe a frequency absent from the ideal odd-only sequence.
Real instruments can depart from these simple uniform-tube models because of holes, varying cross-section, end geometry, temperature gradients and nonlinear excitation. Use the ideal formulas when their assumptions are stated, and treat effective length consistently in comparisons.
All positive integer harmonics occur in the first ideal model; only odd harmonics occur in the second.
WORKED EXAMPLE
Name the overtone before choosing its frequency
An ideal closed–open air column has effective length 0.50 m. Sound speed is 340 m/s. Find its fundamental and first two overtones.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Standing waves, pipe modes and resonance measurements
Focus: Fit the appropriate half-waves or quarter-waves
Learning goals:
- Construct a standing wave from counterpropagating components.
- Distinguish permanent nodes from instantaneously zero displacement.
- Infer fixed-string modes and tension from node counts.
- State displacement and pressure conditions at open and closed acoustic ends.
- Distinguish harmonic, overtone and allowed-mode numbers.
- Infer wave speed from successive resonance lengths and audit the uncertainty.
Equations in this section (LaTeX):
f_n^{\mathrm{open-open}}=\frac{nc}{2L_{\mathrm{eff}}},\qquad f_j^{\mathrm{closed-open}}=\frac{(2j-1)c}{4L_{\mathrm{eff}}}
Meaning: All positive integer harmonics occur in the first ideal model; only odd harmonics occur in the second.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.07 / Standing waves, pipe modes and resonance measurements
Use successive resonances to eliminate an unknown end correction
In a resonance-tube experiment, a sound source of fixed frequency drives an air column open at the top and effectively closed by a water surface below. Changing the water level changes the air-column length. A strong response occurs near lengths compatible with the source frequency and the column’s boundary conditions.
For the first resonance, L₁ + e = λ/4, where e is the open-end correction. The next resonance satisfies L₂ + e = 3λ/4. If the same correction is adequate at both lengths, subtraction eliminates it: L₂ − L₁ = λ/2. Thus c = 2f(L₂ − L₁).
The method depends on identifying consecutive resonances. Skipping one doubles the length difference and can give a speed estimate twice the correct value. Recording several resonances and checking equal spacing is stronger evidence than relying on only two loudness peaks.
Measure air-column length from the water surface to the same physical opening each time, keep the source frequency stable and avoid changing the opening geometry. The sound speed depends on the gas state, especially temperature, so record conditions if comparing measurements. The correction cancels only to the extent that it remains the same.
L₁ and L₂ are consecutive first and second closed–open resonant lengths. The speed relation also works for any adjacent pair under a common correction.
WORKED EXAMPLE
Infer sound speed and test the end correction
A source of frequency 400 Hz gives the first two air-column resonances at 0.205 m and 0.630 m in a narrow resonance tube. Treat the end correction as the same for both. Find wavelength, sound speed and the correction.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Standing waves, pipe modes and resonance measurements Focus: Use successive resonances to eliminate an unknown end correction Learning goals: - Construct a standing wave from counterpropagating components. - Distinguish permanent nodes from instantaneously zero displacement. - Infer fixed-string modes and tension from node counts. - State displacement and pressure conditions at open and closed acoustic ends. - Distinguish harmonic, overtone and allowed-mode numbers. - Infer wave speed from successive resonance lengths and audit the uncertainty. Equations in this section (LaTeX): \lambda=2(L_2-L_1),\qquad c=2f(L_2-L_1),\qquad e=\frac\lambda4-L_1 Meaning: L₁ and L₂ are consecutive first and second closed–open resonant lengths. The speed relation also works for any adjacent pair under a common correction. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
08 / Standing waves, pipe modes and resonance measurements
Use a graph to separate a slope from an offset
With several closed–open resonances, index the allowed modes by j = 1, 2, 3, … . The measured lengths satisfy L_j = (λ/2)j − λ/4 − e. A graph of L_j against j is approximately straight, with slope λ/2. The end correction affects the intercept, not the slope, in the common-correction approximation.
A best-fit slope can reduce sensitivity to one uncertain peak. It does not remove a mistaken mode assignment or a systematic change of end conditions. Residuals that curve or jump can reveal that the simple model or indexing is failing.
For independent uncertainty estimates, the standard uncertainty in L₂ − L₁ is the square root of the sum of squared length uncertainties. For worst-case bounds, add their absolute bounds instead. State which convention is being used rather than mixing them.
Because c = 2fΔL, small worst-case relative bounds add as δc/c ≈ δf/f + δ(ΔL)/ΔL. A shared ruler zero offset cancels from the difference, while a ruler scale-factor error does not. This is an example of how experimental design can remove a specific systematic error without removing all uncertainty.
WORKED EXAMPLE
Find a defensible uncertainty bound
A resonance measurement has ΔL = 0.400 m. Each of the two measured lengths has an independent stated worst-case reading bound of 1 mm, and the source frequency bound is 0.5%. Estimate the worst-case relative bound in the inferred sound speed.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Standing waves, pipe modes and resonance measurements Focus: Use a graph to separate a slope from an offset Learning goals: - Construct a standing wave from counterpropagating components. - Distinguish permanent nodes from instantaneously zero displacement. - Infer fixed-string modes and tension from node counts. - State displacement and pressure conditions at open and closed acoustic ends. - Distinguish harmonic, overtone and allowed-mode numbers. - Infer wave speed from successive resonance lengths and audit the uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
09 / Standing waves, pipe modes and resonance measurements
Use several pieces of evidence before naming a mode
A frequency alone does not determine a mode number unless the wave speed, length and boundary type are known. A spacing between neighbouring resonances can be more informative than an isolated peak. For both ideal open–open and closed–open pipes, adjacent allowed frequencies are separated by c/(2L_eff), but their lowest frequencies differ.
A missing observed peak does not prove a mode is forbidden. The driver may couple weakly to that mode, or the detector may lie near a node of the variable it measures. Boundary conditions establish the allowed modes; the experiment determines which of them are strongly excited and detected.
For exam problems, begin by naming the variable, drawing the endpoint conditions and counting half- or quarter-wavelength intervals. Only then assign harmonic and overtone labels. For experiments, also check damping, end corrections and mode indexing before reporting a physical parameter.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Standing waves, pipe modes and resonance measurements Focus: Use several pieces of evidence before naming a mode Learning goals: - Construct a standing wave from counterpropagating components. - Distinguish permanent nodes from instantaneously zero displacement. - Infer fixed-string modes and tension from node counts. - State displacement and pressure conditions at open and closed acoustic ends. - Distinguish harmonic, overtone and allowed-mode numbers. - Infer wave speed from successive resonance lengths and audit the uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- Adjacent nodes are λ/2 apart; a nearest node–antinode pair is λ/4 apart.
- An instant when the whole string crosses equilibrium does not make every point a node.
- Fixed–fixed strings and ideal open–open pipes have all integer harmonics.
- Closed–open pipes have only odd harmonics in the ideal uniform-tube model.
- Pressure and displacement nodes in a standing sound wave occur at different positions.
- Consecutive resonance-length differences remove a common end correction.
- A slope can isolate wavelength while an intercept contains an offset.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A fixed–fixed string has wave speed 60 m/s and length 1.2 m. What is its second-harmonic frequency?
Answer in Hz.
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SELF CHECK 2
Consecutive closed–open resonance lengths differ by 0.30 m for a 500 Hz source. What sound speed follows under a common end correction?
Answer in m/s.
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SELF CHECK 3
What is the second overtone of an ideal closed–open pipe?
- Second harmonic
- Third harmonic
- Fifth harmonic
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SELF CHECK 4
At an instant the entire standing-wave string lies on its equilibrium line. Are all points nodes?
- Yes
- No; a node remains at zero displacement at every time
- Only the points moving fastest are nodes
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SELF CHECK 5
Which error cancels when two lengths measured on the same ruler are subtracted?
- A common additive zero offset
- A multiplicative scale-factor error
- Every reading error
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
A pipe open at both endsIntermediate · Free sample · Hints and guided reasoningCancel the open-end correctionIntermediate · Hints and guided reasoningFit the longest standing wave on a string.Intermediate · Hints and guided reasoningThe next resonance skips a harmonicIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
At an ideal open end of an air column, is there a pressure node or a displacement node? Explain the distinction.
Compare your reasoning
There is approximately a pressure-variation node and a displacement antinode. Air can move at the opening while its pressure stays close to atmospheric pressure. The two kinds of node are not interchangeable.
Revisit the explanation →A few days later: change the situation
Why do an ideal fixed–fixed string and an ideal closed–open pipe have different allowed harmonic patterns?
Compare your reasoning
They impose different conditions at their ends. The string fits an integer number of half wavelengths, while the pipe fits an odd number of quarter wavelengths. Boundary conditions select the modes.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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