Standing Waves

AP Physics 2· 12 min read

1. Formation of Standing Waves via Superposition★★☆☆☆⏱ 3 min

A standing wave pattern emerges when two identical traveling waves of equal frequency and amplitude move in exactly opposite directions through the same medium. The waves continuously interfere with each other, producing a fixed pattern of oscillation that does not appear to travel across space.

📘 Definition

Standing Wave

A non-propagating wave pattern formed by the perfect constructive and destructive interference of two counter-propagating identical traveling waves, with no net energy transfer across the medium.

y_{total} = 2A \sin(kx) \cos(\omega t)

📐 Worked Example

Two traveling waves on a string are described by $y_{1} = 0.02 sin (5 x - 20 t)$ m and $y_{2} = 0.02 sin (5 x + 20 t)$ m. Find the amplitude of the resulting standing wave at $x = 0.314$ m.

1
Use the position-dependent amplitude formula for superposed counter-propagating waves:
$A(x) = 2A \sin(kx)$
2
Substitute given values, noting $k = 5$ and $A = 0.02$ m:
$A(x) = 2(0.02) \sin(5 * 0.314) = 0.04 * \sin(\pi/2) = 0.04 \text{ m}$

✓ Quick check

Test your understanding of standing wave formation

At what position in the example above would the oscillation amplitude be 0 m?
- x=0 m
- x=0.1 m
- x=0.314 m
- x=1 m
Reveal answer
x=0 m —
At x=0, $sin (0) = 0$ , so the amplitude is zero, forming a node.

2. Nodes, Antinodes and Boundary Conditions★★☆☆☆⏱ 3 min

Nodes are points of zero oscillation, while antinodes are points of maximum oscillation. The positions of nodes and antinodes are completely determined by the boundary conditions of the medium, which define what motion is allowed at each end of the string or air column.

📘 Definition

Node / Antinode

N / A

Node: Zero amplitude point from total destructive interference. Antinode: Maximum amplitude point from total constructive interference.

Boundary Type	Allowed Motion	Node/Antinode at Boundary
Fixed string end	Zero displacement	Node
Free string end	Maximum displacement	Antinode
Closed air column end	Zero air motion	Node
Open air column end	Maximum air motion	Antinode

📐 Worked Example

A string of length 1.2 m is fixed at both ends. What is the distance between adjacent nodes on the fundamental standing wave?

1
For the fundamental mode, exactly half a wavelength spans the full length of the string, with nodes only at the two ends:
$L = \frac{\lambda_1}{2}$
2
Adjacent nodes are always separated by $λ /2$ , which equals the full length of the string for the fundamental mode:
$d_{node} = 1.2 \text{ m}$

3. Harmonic Series for Strings and Air Columns★★★☆☆⏱ 4 min

f_n = \frac{nv}{2L}, \quad n = 1, 2, 3... \quad (\text{Fixed-fixed / Open-open})

f_n = \frac{nv}{4L}, \quad n = 1, 3, 5... \quad (\text{Closed-open})

📐 Worked Example

A 0.8 m long closed-open air column has a sound wave speed of 343 m/s. Calculate the 3rd harmonic frequency.

1
Use the closed-open air column harmonic formula, with n=3 for the 3rd odd harmonic:
$f_3 = \frac{3v}{4L}$
2
Substitute the given values to compute the result:
$f_3 = \frac{3 * 343}{4 * 0.8} \approx 322 \text{ Hz}$

✓ Quick check

Verify your harmonic rule knowledge

Which of the following systems cannot produce a 2nd harmonic?
- 1m fixed-fixed string
- 1m open-open pipe
- 1m closed-open pipe
- 1m stretched wire
Reveal answer
1m closed-open pipe —
Closed-open systems only support odd harmonics, so n=2 is impossible.

4. Standing Wave Resonance★★★☆☆⏱ 2 min

Resonance occurs when an external driving force applies energy to the medium at a frequency that exactly matches one of the system's natural harmonic frequencies. This produces a stable, large-amplitude standing wave pattern that persists as long as energy is supplied to offset damping losses.

📐 Worked Example

A 2m long open-open pipe has a sound speed of 340 m/s. Find the lowest driving frequency that will produce a standing wave.

1
The lowest resonant frequency is the fundamental, n=1 for open-open systems:
$f_1 = \frac{v}{2L}$
2
Substitute given values to find the result:
$f_1 = \frac{340}{2*2} = 85 \text{ Hz}$

5. Common Pitfalls

Wrong move:

Applying the 2L denominator formula to closed-open air columns

Why:

Closed-open systems have a quarter-wavelength fundamental, not a half-wavelength fundamental

Correct move:

Always use 4L as the denominator for closed-open pipes, and only allow odd n values

Wrong move:

Counting nodes to find harmonic number and getting n off by 1

Why:

For a fixed-fixed string, the nth harmonic has n+1 total nodes, not n nodes

Correct move:

Count the number of half-wavelength segments across the medium to get n directly

Wrong move:

Treating open air pipe ends as nodes instead of antinodes

Why:

Open air ends allow maximum longitudinal air motion, so they are antinodes by definition

Correct move:

Map all boundary conditions to node/antinode first before doing any frequency calculations

Wrong move:

Claiming standing waves transfer net energy across the medium

Why:

Counter-propagating waves carry equal energy in opposite directions, leading to zero net energy flow

Correct move:

Only traveling waves have net energy transport through the medium

Wrong move:

Calculating distance between adjacent antinodes as full $λ$

Why:

Adjacent nodes and adjacent antinodes are always separated by $λ /2$ , not a full wavelength

Correct move:

Remember consecutive points of identical phase on a standing wave are always $λ /2$ apart

6. Quick Reference Cheatsheet

System Type	Boundary Conditions	Fundamental Wavelength	Harmonic Formula	Allowed n values
Fixed-fixed string	Both ends nodes	2L	$f_{n} = n v / (2 L)$	1, 2, 3...
Open-open pipe	Both ends antinodes	2L	$f_{n} = n v / (2 L)$	1, 2, 3...
Closed-open pipe	One end node, one end antinode	4L	$f_{n} = n v / (4 L)$	1, 3, 5...

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · Paper 1
Closed-open pipe harmonic MCQ
2024 · Paper 2
String standing wave FRQ
2022 · Paper 1
Resonance frequency calculation

What's Next

Mastering standing waves is critical for scoring on both multiple choice and free response questions in the AP Physics 2 waves and optics unit, as this topic frequently appears alongside Doppler effect and wave interference questions. The harmonic rules you learned here also extend to other wave systems, including standing sound waves in musical instruments and standing electromagnetic waves in laser cavities. Next, you will apply your understanding of superposition and standing waves to solve problems involving thin film interference, another high-frequency exam topic. You will also build on these concepts to analyze diffraction patterns formed when light passes through single and double slits, connecting physical wave behavior to measurable optical effects.