The harmonic frequencies of standing waves on a medium, such as a vibrating string or an air column in a musical instrument, are related to the mass and tension of the medium.
When a medium is under tension, such as a stretched string or an air column in a tube, it has a natural frequency of vibration. This natural frequency is determined by the properties of the medium, including its mass per unit length (linear density) and the tension applied to it.
For a string under tension:
The fundamental (first harmonic) frequency (f1) of a vibrating string is inversely proportional to the length of the string (L) and directly proportional to the square root of the tension (T) and the linear density (μ) of the string. Mathematically, it can be expressed as:
f1 = (1/2L) * sqrt(T/μ)
The harmonic frequencies of higher modes (2nd harmonic, 3rd harmonic, etc.) are integer multiples of the fundamental frequency. The nth harmonic frequency (fn) can be calculated as:
fn = n * f1
where n represents the harmonic number.
For an air column in a tube:
In a closed-end tube (e.g., a tube closed at one end), only odd-numbered harmonics are present. The fundamental frequency (f1) is determined by the length of the tube (L) and the speed of sound in the medium (v):
f1 = v / (4L)
The higher harmonics in a closed-end tube can be calculated as:
fn = (2n - 1) * f1
In an open-end tube (e.g., a tube open at both ends), both odd and even-numbered harmonics are present. The fundamental frequency (f1) is still determined by the length of the tube (L) and the speed of sound (v):
f1 = v / (2L)
The higher harmonics in an open-end tube can be calculated as:
fn = n * f1
In summary, the harmonic frequencies of standing waves in a medium are influenced by the mass per unit length (linear density) and the tension applied to the medium. Changing either the mass or tension can alter the fundamental frequency and the subsequent harmonics of the standing wave.