๐Standing Waves
What are Standing Waves?
Standing waves form when two identical waves travel in opposite directions and superpose. The result appears to "stand still" โ some points never move (nodes) while others oscillate maximally (antinodes).
Nodes: points of zero amplitude (destructive interference always occurs here) Antinodes: points of maximum amplitude (constructive interference always occurs here)
Standing waves on a string fixed at both ends:
Harmonics: ฮป_n = 2L/n, where n = 1, 2, 3, ...
f_n = nv/(2L) = n ร fโ
fโ is the fundamental frequency (1st harmonic). Higher harmonics are integer multiples.
Standing Waves in Tubes
Standing sound waves in air columns:
Closed tube (closed at both ends or one closed): different harmonic series Open tube (open at both ends): - Antinodes at both open ends - ฮป_n = 2L/n, f_n = nv/(2L) โ all harmonics
Closed at one end: - Node at closed end, antinode at open end - ฮป_n = 4L/n (odd n only!), f_n = nv/(4L) โ only odd harmonics
This is why a clarinet (closed at one end) sounds different from a flute (open at both ends)!
โ๏ธ Worked Example
Problem: A guitar string is 0.65 m long. The wave speed in the string is 400 m/s. Find the fundamental frequency and first two harmonics.
๐ Key Equations
Standing waves
f_n = n(v)/(2L) (open string/tube, n=1,2,3,\ldots)f_n = n(v)/(4L) (closed one end, n=1,3,5,\ldots)ฮป_n = (2L)/(n)โ ๏ธ Common Mistakes
Misconception: Nodes are points where the wave amplitude is largest.
โ Correct thinking: Nodes are points of ZERO amplitude โ they never move. Antinodes are the points of maximum amplitude.
Why: The word "node" comes from the Latin for "knot" โ a fixed point. Antinodes are the opposite: maximum oscillation.
Misconception: A tube closed at one end supports the same harmonics as an open tube.
โ Correct thinking: A tube closed at one end only supports odd harmonics (n = 1, 3, 5, ...) and uses f_n = nv/(4L). An open tube supports all harmonics.
Why: A closed end forces a node there; an open end forces an antinode. The geometry only fits odd multiples of a quarter-wavelength into the closed tube.
Misconception: Higher harmonics are always louder or more prominent than the fundamental.
โ Correct thinking: The fundamental (1st harmonic) is typically the dominant frequency and determines the perceived pitch. Higher harmonics add timbre (tone colour) but are usually less intense.
Why: Instruments are designed so the fundamental is strongest. Harmonics shape the sound quality (why a violin and a flute sound different at the same pitch).
๐ Practice Problems
Try these problems. Check your answer when ready.
A string of length 0.5 m has wave speed 200 m/s. What is the fundamental frequency?
f_1 = v/(2L)The fundamental frequency of a string is 120 Hz. What are the frequencies of the 2nd and 3rd harmonics?
How many nodes and antinodes are in the 3rd harmonic of a string fixed at both ends?
An organ pipe closed at one end is 0.85 m long. If v_sound = 340 m/s, find the fundamental frequency and the next allowed harmonic.
f_n = nv/(4L), n = 1, 3, 5, \ldotsA string's length is doubled while keeping tension and linear density the same. How does the fundamental frequency change?
A pipe open at both ends has a fundamental of 330 Hz. An identical pipe is closed at one end. What is its fundamental, and which of its harmonics (if any) matches the open pipe's 2nd harmonic?
Finished reading through this lesson?