Standing Wave Resonant Frequency Calculator

Standing waves form when two waves of the same frequency travel in opposite directions. Resonant frequencies occur when the length fits a whole number of half-wavelengths (or quarter for closed pipes):

String or open pipe (both ends same): f_n = nv/(2L) where n = 1, 2, 3, 4, 5…

Closed pipe (one end open, one closed): f_n = nv/(4L) where n = 1, 3, 5, 7… (odd harmonics only)

Where:

• f_n = frequency of nth harmonic
• v = wave speed (m/s)
• L = length of string or pipe (m)
• n = harmonic number

Wave speeds (typical):

• String: v = √(T/μ) where T = tension, μ = mass per unit length
• Air at 20°C: v ≈ 343 m/s
• Water: v ≈ 1480 m/s

Musical implications:

The fundamental sets the pitch. The strength of the higher harmonics sets the timbre, which is why a flute and a violin playing the same note sound nothing alike:

• Flute: open at both ends, so all harmonics are available, but the upper ones are weak. That near-pure spectrum is what makes the tone sound breathy and simple.
• Clarinet: a cylindrical bore closed at the reed, so only odd harmonics survive. Hence the hollow tone.
• Violin: a bowed string, rich in every harmonic. Complex and warm.

The odd-harmonic rule has a consequence players feel before they ever learn the physics. Overblow a flute and it jumps an octave, to n=2. Overblow a clarinet and there is no n=2 to jump to, so it leaps to n=3, an octave plus a fifth. That is why clarinet fingering above the break looks nothing like the fingering below it: an octave instrument needs twelve fingerings before the register key takes over, while a clarinet needs nineteen. The saxophone and the oboe are reed instruments too, but their bores are conical rather than cylindrical, which restores the even harmonics and lets them overblow at the octave like a flute.

One caveat on the pipe formulas: a real pipe resonates as if it were slightly longer than it measures, because the air just outside an open end moves with the column. The usual correction is about 0.6 times the bore radius per open end. On a wide, short pipe this matters. A 20 cm pipe with a 2 cm bore is acoustically about 21.2 cm long, dropping the predicted fundamental by roughly 6%. Enter the corrected length if you are matching a measured frequency.

Standing waves also produce:

• Chladni patterns in vibrating plates
• The resonances of concert halls and instrument bodies
• Laser cavity modes (standing light waves between mirrors)