Standing waves form when waves reflect and interfere so that fixed points called nodes and high-motion points called antinodes appear. This cheat sheet covers standing waves on strings fixed at both ends, pipes open at both ends, and pipes closed at one end. Students need these patterns because many sound and vibration problems depend on choosing the correct harmonic model before using formulas.
Worked examples help connect diagrams, boundary conditions, wavelengths, and frequencies.
The central relationship is , where is wave speed, is frequency, and is wavelength. For strings fixed at both ends and open pipes, allowed wavelengths follow and frequencies follow . For pipes closed at one end, only odd harmonics occur, with and for odd .
In air at room temperature, many problems use for sound.
Key Facts
- For any wave, the speed equation is , so frequency can be found with .
- A string fixed at both ends has nodes at both ends, and its allowed wavelengths are for .
- The harmonic frequencies for a string fixed at both ends are , where is the harmonic number.
- An open pipe has antinodes at both ends, and its allowed wavelengths are for .
- The harmonic frequencies for an open pipe are , using the sound speed in air for .
- A pipe closed at one end has a node at the closed end and an antinode at the open end.
- A pipe closed at one end supports only odd harmonics, so and .
- The fundamental frequency is the lowest allowed frequency, so it is for strings and open pipes, but for a closed pipe.
Vocabulary
- Standing wave
- A wave pattern formed by interference in which nodes and antinodes stay in fixed positions.
- Node
- A point in a standing wave where the medium has zero displacement.
- Antinode
- A point in a standing wave where the medium has maximum displacement.
- Harmonic
- An allowed standing-wave frequency that fits the boundary conditions of the string or pipe.
- Fundamental frequency
- The lowest allowed frequency of a standing wave system, also called the first harmonic.
- Boundary condition
- A required behavior at the end of a string or pipe, such as a node at a fixed end or an antinode at an open end.
Common Mistakes to Avoid
- Using the open-pipe formula for a closed pipe is wrong because a closed pipe has one node and one antinode, so its fundamental wavelength is , not .
- Including even harmonics for a closed pipe is wrong because a pipe closed at one end supports only .
- Forgetting that is the physical length of the string or pipe is wrong because may be , , , or depending on the system and harmonic.
- Using the speed of sound for a vibrating string is wrong because a string wave speed depends on the string tension and mass per length, not the air temperature.
- Treating nodes and antinodes as interchangeable is wrong because fixed string ends and closed pipe ends are nodes, while open pipe ends are antinodes.
Practice Questions
- 1 A string fixed at both ends is long and has wave speed . Find the fundamental frequency and the third harmonic frequency .
- 2 An open pipe is long. Using , find and .
- 3 A pipe closed at one end is long. Using , find the first three allowed harmonic frequencies.
- 4 Explain why a closed pipe does not produce a second harmonic, even though an open pipe of the same length can.
Understanding Standing Waves on Strings and in Pipes Worked Examples
The most important step in a standing wave problem is to translate the picture into pieces of a wavelength. A segment from one node to the next node is half a wavelength. A segment from a node to the nearest antinode is one quarter of a wavelength.
Count these segments along the length before doing any calculation. This method works even when the drawing looks unfamiliar.
The lowest pattern has the fewest sections that can fit the boundary conditions. Each higher pattern fits one more section into the same length, so its wavelength becomes shorter and its frequency rises.
For a vibrating string, the wave speed is not usually the speed of sound. It depends on the tension in the string and on its mass per unit length. A tighter string carries waves faster, which raises every resonant frequency.
A heavier string of the same length and tension carries waves more slowly, which lowers the pitch. This explains why guitar players tune a string by changing its tension. Pressing a string against a fret shortens the vibrating part of the string.
A shorter length has a higher fundamental frequency. In lab questions, check whether the given speed belongs to the string itself or to sound in air after the string makes the instrument body vibrate.
Pipes need a different kind of physical thinking. Sound in a pipe is a pattern of air pressure changes and air motion. At an open end, air can move freely, so the motion is greatest there.
At a closed end, the air cannot move through the wall, so its motion is zero. The air motion pattern is often easier to draw than the pressure pattern. Pressure behaves in the opposite way at the ends.
This can cause confusion if a textbook diagram labels pressure nodes instead of displacement nodes. Real open pipes act slightly longer than their measured length because some vibrating air extends just beyond each opening. This end correction is small, but it matters in precise experiments and musical instruments.
Worked examples often contain errors from labels rather than arithmetic. Write down the pipe type or string type first. Then identify the harmonic from the number of loops or from the stated mode.
For a closed pipe, be careful with naming. The next allowed resonance after the fundamental is commonly called the third harmonic, even though it is the second resonance that occurs. Keep length units consistent with wave speed units.
If speed is in meters per second, convert centimeters to meters before finding frequency. Finally, test whether the answer makes sense.
Higher modes must have higher frequencies, and a closed pipe with the same length has a lower fundamental than an open pipe. These quick checks catch many wrong model choices.