Wave overtones

For resonance in a taut string, the first harmonic is determined for a wave form with one antinode and two nodes. That is, the two ends of the string are nodes because they do not vibrate while the middle of the string is an antinode because it experiences the greatest change in amplitude. This means that one half of a full wavelength is represented by the length of the resonating structure.

The frequency of the first harmonic is equal to wave speed divided by twice the length of the string. (Recall that wave speed is equal to wavelength times frequency.)

F₁ = v/2L

The wavelength of the first harmonic is equal to double the length of the string.

λ₁ = 2L

The "nth" wavelength is equal to the fundamental wavelength divided by n.

λ_n = λ₁/n

Harmonics for a taut string*

	Harmonic number	Overtone number	F =	λ =
F₁	First harmonic	---	F₁ = v/2L	λ₁ = 2L
F₂	Second harmonic	First overtone	F₂ = 2F₁	λ₂ =λ₁/2
F₃	Third harmonic	Second overtone	F₃ = 3F₁	λ₃ = λ₁/3
F_n	Nth harmonic	(Nth - 1) overtone	F_n = nF₁	λ_n = λ₁/n

* or any wave system with two identical ends, such as a pipe with two open or closed ends. In the case of a pipe with two open ends, there are two antinodes at the ends of the pipe and a single node in the middle of the pipe, but the mathematics work out identically.

Definition of terms

Frequency (F): Units: (1/s), hertz (Hz)

Fundamental frequency, first harmonic (F)₁: The lowest frequency (longest wavelength) allowed for the system.

Length of string (L): (or pipe, etc.) Units: meters (m).

Wavelength (λ): Units: meters (m).

The first overtone is the first allowed harmonic above the fundamental frequency (F₁).

In the case of a system with two different ends (as in the case of a tube open at one end), the closed end is a node and the open end is an antinode. The first resonant frequency has only a quarter of a wave in the tube. This means that the first harmonic is characterized by a wavelength four times the length of the tube.

F₁ = v/4L

The wavelength of the first harmonic is equal to four times the length of the string.

λ₁ = 4L

The "nth" wavelength is equal to the fundamental wavelength divided by n.

λ_n = λ₁/n

Note that "n" must be odd in this case as only odd harmonics will resonate in this situation.

Harmonics for a system with two different ends*

	Harmonic number	Overtone number	F =	λ =
F₁	First harmonic	---	F₁ = v/4L	λ₁ = 4L
F₂	Third harmonic	First overtone	F₂ = 3F₁	λ₂ = λ₁/3
F₃	Fifth harmonic	Second overtone	F₃ = 5F₁	λ₃ = λ₁/5
F_n	Nth harmonic†	(Nth - 1)/2 overtone	F_(n-1)/2 = nF₁	λ_n = λ₁/n

* such as a pipe with one end open and one end closed
†In this case only the odd harmonics resonate, so n is an odd integer.

V_s: velocity of sound

dependent on qualities of the medium transmitting the sound, (the air) such as its density, temperature, and “springiness.” A complicated equation, we concentrate only on temperature.
increases as temperature increases (molecules move faster.)
is higher for liquids and solids than for gasses (molecules are closer together.)
for “room air” is 340 meters per second (m/s).
Speed of sound is 343 meters per second at 20 degrees C. Based on the material sound is passing through and the temperature, the speed of sound changes.

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