Question 1

What are the standing wave formulas for a string and a pipe?

Accepted Answer

A string fixed at both ends and a pipe open at both ends have the same pattern: wavelengths λ = 2L ÷ n and frequencies f = n·v ÷ (2L) for n = 1, 2, 3, … (OpenStax University Physics, Volume 1, sections 16.6 and 17.4). A pipe closed at one end has a node at the closed end and an antinode at the open end, so λ = 4L ÷ n and f = n·v ÷ (4L) with only odd n = 1, 3, 5, …. Here v is the wave speed, L the length and n the harmonic number; n = 1 is the fundamental. A closed pipe therefore sounds an octave below an open pipe of the same length, and it lacks the even harmonics.

Question 2

How do I find the speed of the wave on a string?

Accepted Answer

The speed is v = √(T ÷ μ), with the tension T in newtons and the linear density μ in kilograms per meter (1 lbf = 4.448 N; 1 lb/ft = 1.488 kg/m). OpenStax Example 16.7: a 2.00 m string with μ = 0.006 kg/m under the weight of a 2.00 kg mass (19.6 N) has v = 57.15 m/s and resonates at 14.29, 28.58 and 42.87 Hz in its first three modes. Example 16.5 gives a guitar string with μ = 3.09 × 10⁻⁴ kg/m under 56.40 N: 427.23 m/s. Choose “Tension” to find the tension that a given note needs: a string with n = 6 at 2.00 kHz, μ = 0.0065 kg/m and L = 1.50 m needs 6500 N.

Question 3

Which speed of sound does the pipe use?

Accepted Answer

The speed of sound in dry air at the temperature you give, v = √(γRT ÷ M) with γ = 1.4 and M = 0.02897 kg/mol, the same value as on our speed of sound calculator: 343.2 m/s (1,126 ft/s) at 20°C (68°F). Pitch follows the air: the same pipe plays higher when it is warm, about 0.17% per degree Celsius near room temperature. You can also type any speed, for another gas or for sound in water. OpenStax Example 17.6 asks for a pipe closed at one end that sounds 128 Hz at 22.0°C: L = v ÷ (4f) = 0.672 m with the book’s 344 m/s (0.673 m with 344.4 m/s), and its fourth overtone, the ninth harmonic, is 1.15 kHz.

Question 4

How long must a flute or an organ pipe be for a given note?

Accepted Answer

Pick “Length”, the pipe type and the frequency. A flute is open at both ends, so L = v ÷ (2f): for middle C, 262 Hz, at 20°C the length is 0.655 m (25.8 in). An organ pipe closed at one end needs only half that length for the same note: L = v ÷ (4f). OpenStax asks the same for an oboe modeled as an open pipe: 110 Hz at 343 m/s needs 1.56 m. Real instruments have finger holes, tapering bores and a mouthpiece, so this gives the ideal cylindrical length, a starting point rather than a blueprint.

Question 5

What is the end correction, and should I use it?

Accepted Answer

The antinode at an open end is not exactly at the opening but a little beyond it, so the pipe acts slightly longer than it is. OpenStax gives the correction for a thin-walled tube as about 0.6 times the radius, added to the length; it is added once for each open end, so a pipe open at both ends gains 1.2 times its radius and a pipe closed at one end gains 0.6 times (per UConn physics notes). It matters for wide, short pipes: a 1 in radius adds about 0.6 in to a closed pipe. Leave the radius blank for the ideal formulas, as most textbook problems do.

Question 6

What is the difference between a harmonic and an overtone?

Accepted Answer

A harmonic is a multiple of the fundamental frequency, counted from the fundamental itself: the first harmonic is the fundamental (n = 1), the second harmonic is twice that, and so on. An overtone is any resonance above the fundamental, counted from the first one above it. For a string or an open pipe the first overtone is the second harmonic; for a pipe closed at one end it is the third harmonic, because the even harmonics are missing (OpenStax section 17.5). The table lists both.

Standing wave calculator