Interactive physics simulator
Standing Waves
Discover wave resonance, boundary nodes, and acoustic harmonics. Explore how counter-propagating waves superpose to form stationary patterns in guitar strings, wind pipes, and Melde's resonance rig.
Standing Waves & Resonance Simulator
Adjust harmonic modes, pipe boundaries, frequencies, and hanging weights. Observe how boundary reflections trap energy to form stationary nodes.
Live Wave Telemetry
- Wavelength (λ)
- 2.00 m
- Frequency (f)
- 100.0 Hz
- Wave Speed (v)
- 200.0 m/s
- Harmonic State
- Fundamental (n=1)
- Resonance Status
- Locked (100%)
What are Standing Waves?
A standing wave, or stationary wave, is a wave that oscillates in time but whose peak amplitude profile remains locked in a fixed position in space. Unlike traveling waves that transport energy through a medium, standing waves store and trap vibrational energy within stationary patterns.
This phenomenon is the result of superposition. When a wave of wavelength λ travels along a medium (like a plucked string or air column) and meets a boundary, it reflects back in the opposite direction. The incident and reflected waves interfere continuously. If the boundaries constrain the wave to specific length ratios, the two waves superpose constructively at fixed locations to produce antinodes (maximum vibration) and destructively at other locations to form nodes (zero displacement).
Key Components
Standing waves are characterized by specific spatial points:
- Nodes (N): Points along the medium that undergo zero displacement. The incident and reflected waves are permanently 180° out of phase here, causing continuous destructive cancellation.
- Antinodes (A): Points where the wave displacement reaches its maximum amplitude. The overlapping waves arrive in phase here, resulting in constructive superposition.
- Harmonic Series: The set of stable resonant standing wave patterns, each corresponding to an integer multiple of the fundamental frequency.
Boundary Wave Formulas
The relationship between string or pipe length (L) and wavelength (λ) is governed by boundary conditions:
λn = 2L / n (n = 1, 2, 3...)
fn = n · v / (2L) = n · f1
Open-Closed Pipe (Odd Harmonics Only):
λn = 4L / n (n = 1, 3, 5...)
fn = n · v / (4L) = n · f1
Where:
- L is the length of the string or air column (meters)
- λ is the resonant wavelength (meters)
- f is the frequency (Hertz); v is the wave speed (m/s)
- n is the harmonic number (integer mode)
Solved Examples
A steel guitar string of length 0.65 m has a linear mass density μ = 4.0 × 10-3 kg/m. Under a tension of 169 N, calculate (a) the speed of the transverse wave along the string, (b) the fundamental frequency (n = 1), and (c) the frequency of the third harmonic (n = 3).
- Step 1: Calculate the wave speed v along the string using the tension T and linear mass density μ:
v = √(T / μ) = √(169 N / (4.0 × 10-3 kg/m)) = √42250 = 205.55 m/s. - Step 2: Calculate the fundamental wavelength λ1 for a string fixed at both ends (fixed-fixed boundary):
λ1 = 2L = 2 × 0.65 m = 1.30 m. - Step 3: Calculate the fundamental frequency f1 using the wave relation f = v / λ:
f1 = v / λ1 = 205.55 m/s / 1.30 m = 158.12 Hz. - Step 4: Calculate the frequency of the third harmonic f3. On a fixed-fixed string, all integer harmonics exist, so fn = n · f1:
f3 = 3 × f1 = 3 × 158.12 Hz = 474.36 Hz.
Answer: Wave speed = 205.55 m/s, Fundamental frequency (n=1) = 158.12 Hz, Third harmonic (n=3) = 474.36 Hz
An organ pipe of length L = 1.70 m is filled with air where the speed of sound is v = 340 m/s. Calculate the fundamental frequency (first harmonic) and first overtone for:
(a) The pipe open at both ends (Open-Open).
(b) The pipe closed at one end (Open-Closed).
- Step 1 (Open-Open Pipe): For a pipe open at both ends, displacement antinodes form at the open boundaries. The fundamental wavelength is λ1 = 2L = 2 × 1.70 m = 3.40 m.
- Step 2 (Open-Open Frequencies): The fundamental frequency is:
f1 = v / λ1 = 340 m/s / 3.40 m = 100.00 Hz.
The first overtone is the next available harmonic, which is the second harmonic (n = 2):
f2 = 2 × f1 = 2 × 100.00 Hz = 200.00 Hz. - Step 3 (Open-Closed Pipe): For a pipe closed at one end, a displacement node forms at the closed end and an antinode at the open end. The fundamental wavelength is λ1 = 4L = 4 × 1.70 m = 6.80 m.
- Step 4 (Open-Closed Frequencies): The fundamental frequency is:
f1 = v / λ1 = 340 m/s / 6.80 m = 50.00 Hz.
A closed pipe supports only odd harmonics (n = 1, 3, 5...). The first overtone is therefore the third harmonic (n = 3):
f3 = 3 × f1 = 3 × 50.00 Hz = 150.00 Hz.
Answer: (a) Open-Open: Fundamental = 100 Hz, First Overtone = 200 Hz; (b) Open-Closed: Fundamental = 50 Hz, First Overtone = 150 Hz
In a Melde's rig setup, a horizontal string of length L = 1.50 m and linear mass density μ = 3.0 × 10-3 kg/m is attached to a motor vibrating at f = 60 Hz. The string passes over a pulley and is tensioned by a hanging mass M. Calculate the mass M required to set up a standing wave with exactly 3 loops (third harmonic, n = 3). Assume acceleration due to gravity g = 9.8 m/s2.
- Step 1: Determine the required wavelength λ for a standing wave with 3 loops. Each loop is one half-wavelength (L = n · λ/2):
λ = 2L / n = (2 × 1.50 m) / 3 = 1.00 m. - Step 2: Calculate the required wave speed v using the frequency of the oscillator:
v = f · λ = 60 Hz × 1.00 m = 60.00 m/s. - Step 3: Calculate the tension T required to produce this wave speed using the formula v = √(T / μ):
T = v2 · μ = (60.00 m/s)2 × (3.0 × 10-3 kg/m) = 3600 × 0.003 = 10.80 N. - Step 4: Relate the tension to the hanging mass M (T = M · g) and solve for M:
M = T / g = 10.80 N / 9.8 m/s2 = 1.102 kg (approx. 1102 grams).
Answer: Required hanging mass M = 1.10 kg (1102 g) to establish 3 loops
Common Mistakes
- Energy Propagation: Many students confuse standing waves with traveling waves and assume they carry energy from one end to another. Because nodes are permanently still, they act as barriers; energy is trapped between nodes, oscillating between kinetic and potential forms in place.
- Closed Ends of Air Pipes: Assuming that a closed pipe has a displacement antinode. Air molecules cannot move into a solid wall; thus, a closed end must form a displacement node. Conversely, the open end is free to move, forming a displacement antinode.
- Harmonics of Closed Pipes: Forgetting that open-closed pipes cannot support even harmonics (n=2, 4, 6...). The first overtone of an open-closed pipe is the n=3 harmonic (triple the fundamental frequency), not n=2.
Tension & Resonance speed
In Melde's experiment and stringed instruments, the wave velocity depends on the tension T (supplied by tensioners or a hanging mass M · g) and the linear density μ. Tuning the mass directly changes the wave speed, allowing resonance to lock at different integer nodes.
Practice Questions
1. Define a node and an antinode in a standing wave pattern. What is the distance between two consecutive nodes in terms of wavelength?
A **node** is a point along a standing wave where the wave displacement is always zero (complete destructive interference and stillness). An **antinode** is a point where the wave displacement reaches its maximum value (maximum constructive interference and motion).
The distance between two consecutive nodes (or two consecutive antinodes) is exactly half a wavelength (λ/2). The distance between a node and its adjacent antinode is a quarter wavelength (λ/4).
2. Why does an open-closed pipe of a given length sound deeper (lower frequency) than an open-open pipe of the same length?
For a pipe of length L, the fundamental wavelength in an open-open pipe is λ = 2L. In an open-closed pipe, the boundary conditions restrict the wave such that the fundamental wavelength is λ = 4L (double the wavelength). Since frequency is inversely proportional to wavelength (f = v/λ), the open-closed pipe produces a fundamental frequency that is exactly half that of the open-open pipe, resulting in a deeper pitch (one octave lower).
3. How do standing waves form, and why are they called "standing" waves?
Standing waves form when two coherent waves of the same frequency and amplitude travel in opposite directions in the same medium (often due to reflections at boundaries). When they superpose, they create a wave pattern where the nodes and antinodes remain fixed in spatial positions. Because the wave profile oscillates in place and the energy does not propagate along the medium, it appears stationary or "standing".
4. Explain what happens to the standing wave pattern in Melde's experiment when you increase the hanging mass. How does this relate to resonance?
Increasing the hanging mass increases the tension T in the string. According to v = √(T/μ), this increases the wave propagation speed v. Since the oscillator frequency f is fixed, the wavelength increases (λ = v/f). As the wavelength shifts, the string moves out of resonance (the pattern collapses into small, chaotic vibrations) until the tension is tuned to a specific value where the new larger wavelength matches a lower harmonic condition (e.g. n decreases from 3 to 2), triggering a new, stable resonance peak.
FAQ
Frequently Asked Questions
What is a standing wave?
A standing wave (or stationary wave) is a wave pattern that oscillates in time but whose peak amplitude profile does not move in space. It is formed by the superposition of two counter-propagating waves of the same frequency and amplitude.
Do standing waves transmit energy?
No. Unlike traveling waves, standing waves do not transmit net energy through the medium. The energy is trapped and oscillates in place between the fixed node points.
What are the boundary conditions for standing waves on a guitar string?
A guitar string has fixed-fixed boundary conditions. Since the string is clamped at both ends, these ends cannot vibrate and must form displacement nodes. This limits the possible standing wave patterns to wavelengths where L = n · λ/2.
What is the difference between open-open and open-closed pipes?
An open-open pipe is open at both ends, forming displacement antinodes (pressure nodes) at both ends. It supports all integer harmonics (n = 1, 2, 3...). An open-closed pipe is closed at one end, forming a displacement node (pressure antinode) at the closed end. It supports only odd harmonics (n = 1, 3, 5...).
What is Melde's experiment?
Melde's experiment demonstrates standing waves on a tensioned string. It shows how the frequency of a driving oscillator, the length of the string, the linear mass density, and the tension (controlled by a hanging mass) determine the resonant standing wave patterns (harmonics).
How do you calculate the wavelength of the nth harmonic?
For fixed-fixed boundaries or open-open pipes, the wavelength is λn = 2L / n. For open-closed pipes, the wavelength is λn = 4L / n (where n is an odd integer: 1, 3, 5...).
What is resonance in standing waves?
Resonance occurs when the driving frequency of a system matches one of its natural frequencies of vibration. When this happens, the waves reflect back and forth in phase, constructive superposition occurs, and the amplitude of the standing wave increases dramatically.