Theme C — Wave Behaviour
C.4.3 Boundary Conditions for Standing Waves — IB Physics HL Revision Notes & Topical Questions
5 min read 9 parts Lesson 29 of 37 1 quiz
Curriculum Equation Key Wave Speed & Doppler Effect FormulasFast Exam Solution
v = f × λ | f_obs = f × (v ± v_o) / (v ∓ v_s) | T = 1 / f
Lesson summary
C.4.3 Boundary Conditions for Standing Waves — IB PHYSICS SL notes: Decide whether a boundary imposes a node or an antinode and predict the allowed standing-wave patterns for strings and pipes.Boundary conditions — constraints at the edges of the medium that decide where nodes an…
- Allowed patterns
- Unsustained waves
- Number of nodes
- Possible boundaries
- String fixed at both ends
- Natural frequencies
Part 1 of 7Why boundaries matter
Boundary conditions
- Boundary conditions constraints at the edges of the medium that decide where nodes and antinodes must lie
Allowed patterns
- Allowed patterns they decide which wavelengths, and so which frequencies, are possible
Media
- Media standing waves form on strings and in pipes when a progressive wave superposes with its reflection
Unsustained waves
- Unsustained waves waves that fail the conditions are not sustained, because the reflected wave does not reinforce the incident wave
Number of nodes
- Number of nodes it depends on the frequency of the progressive waves and on the boundary conditions
Part 2 of 7Strings
1Possible boundaries
- Possible boundaries fixed at both ends, free at both ends, or one end fixed and one free
2Fixed end
- Fixed end the string cannot move there, so there must be a node
- Examples are a string clamped at a wall or the bridge of a guitar.
- The reflected wave is in antiphase (a phase change of π) and cancels the incident wave at the end.
3Free end
- Free end the end moves freely, so an antinode forms and the wave reflects in phase without inversion
- An example is a ring sliding on a rod.
4String fixed at both ends
- String fixed at both ends a node lies at each end, so an integer number of half-wavelengths fits the length, L = nλ/2
5Natural frequencies
- Natural frequencies at these specific frequencies standing waves with different numbers of nodes and antinodes form
6What sets the frequency
- What sets the frequency the tension, adjusted by tuning pegs, and the mass per unit length, which is why guitar strings have different thicknesses
Part 3 of 7Pipes
Possible boundaries
- Possible boundaries closed at both ends, open at both ends, or closed at one end and open at the other
Closed end
- Closed end it is a displacement node because air cannot move through the stopper or wall
- The reflected wave is in antiphase with the incident wave and they cancel.
Open end
- Open end it is a displacement antinode because air is free to move
- The reflected wave is in phase and the amplitudes add.
End correction
- End correction in reality the antinode lies slightly beyond the pipe, but this is ignored unless stated
Pipe open at both ends
- Pipe open at both ends antinodes at both ends
Pipe closed at one end
- Pipe closed at one end a node at the closed end and an antinode at the open end
Length relation
- Length relation a pipe closed at one end has L = λ/4 for its lowest pattern, not λ/2
Part 4 of 7Reflection and phase
1String reflection
- String reflection a wave reflected from a fixed end changes phase by π, and there is no change at a free end
2Pressure in a pipe
- Pressure in a pipe an open end reflects a compression as a rarefaction, and a closed end reflects a compression as a compression
- The open end is a pressure node and the closed end is a pressure antinode.
- This is the opposite of the displacement picture.
3Two strings joined
- Two strings joined a thicker string acts like a nearly fixed end for the thinner string and inverts the reflected pulse
- A thinner string behaves more like a free end.
Part 5 of 7Finding the allowed patterns
1
Sketch the medium.
2
Put a node or an antinode at each end according to the boundary type.
3
Draw the simplest pattern that fits between the ends.
4
Draw the patterns with additional nodes.
5
Count how many half-wavelengths or quarter-wavelengths fit and write L as a multiple of them.
6
Find the wavelength, then the frequency from f = v/λ.
Speed
- Speed use about 340 m s−1 for sound in air at room temperature and the given speed for a string
Estimate check
- Estimate check a longer string or pipe must give a lower note
Part 6 of 7Resonance in a closed pipe
Driven pipe
- Driven pipe a pipe closed at one end driven by a loudspeaker at the open end resonates strongly where the closed end is a node and the open end an antinode
Shortening
- Shortening each resonant frequency rises in proportion
Slide whistle
- Slide whistle it changes pitch continuously by varying the length of the air column
Part 7 of 7More from the specification
More from the specification
Boundary resonance scale
- Boundary resonance scale fixed or open physical terminal boundaries determine harmonic standing wavelengths observable across laboratory experimental scale.
Free and Forced Oscillations
- An oscillating system undergoes free oscillations at its natural resonant frequency when displaced and released,
- whereas forced oscillations occur when a periodic driving force continuously transfers energy to the oscillator.
Damping in Harmonic Systems
- Damping forces oppose motion in an oscillating system, removing mechanical energy and causing the amplitude of harmonic oscillations to decay over time through light, critical,
- or heavy damping.
Resonance and Standing Wave Harmonics
- When the driving frequency matches the natural frequency of an oscillator, resonance occurs, yielding peak energy transfer;
- standing waves form from the superposition of opposing travelling waves, establishing discrete boundary-dependent frequencies known as harmonics.
Free and Forced Oscillations
- An oscillating system undergoes free oscillations at its natural resonant frequency when displaced and released,
- whereas forced oscillations occur when a periodic driving force continuously transfers energy to the oscillator.
Damping in Harmonic Systems
- Damping forces oppose motion in an oscillating system, removing mechanical energy and causing the amplitude of harmonic oscillations to decay over time through light, critical,
- or heavy damping.
Resonance and Standing Wave Harmonics
- When the driving frequency matches the natural frequency of an oscillator, resonance occurs, yielding peak energy transfer;
- standing waves form from the superposition of opposing travelling waves, establishing discrete boundary-dependent frequencies known as harmonics.
Forced oscillation
- Oscillation caused by a periodic driving force
Heavy damping
- Damping greater than critical, giving a slow return without oscillation
Resistive force
- Friction and drag opposing motion.
Damping
- Energy loss to friction that reduces amplitude.
Oscillation
- Repeated back-and-forth motion.
Harmonic
- Resonant mode of a stationary wave, with frequency a whole-number multiple of the fundamental where allowed.
Equilibrium
- State with zero net force.
Stationary
- Not moving.
Force
- A push or pull measured in newtons.
Driving frequency
- The frequency of the periodic external force
Equilibrium position
- Position where the net force is zero.
Key terms
| Term | Meaning |
|---|---|
| Boundary condition | Constraint at an edge fixing whether there is a node or antinode. |
| Fixed end | Point that cannot move, so a node. |
| Free end | Point free to move, so an antinode. |
| Open pipe end | Displacement antinode. |
| Closed pipe end | Displacement node. |
Avoid these mistakes
1
Placing an antinode at a closed end of a pipe — the displacement there is zero.
2
Forgetting that reflection from a fixed end inverts the wave.
3
Mixing up pressure nodes and displacement nodes.
4
Using L = λ/2 for a pipe closed at one end instead of L = λ/4.
5
Thinking boundary conditions are optional — only waves that fit them persist.
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