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Standing waves

\|{\vec {v}}\|={\sqrt {\frac {\|{\vec {T}}\|}{\mu }}}

Wave speed is equal to the square root of tension divided by the linear density of the string.

μ = m/L

Linear density of the string is equal to the mass divided by the length of the string.

λ_max = 2L

The fundamental wavelength is equal to two times the length of the string.

Variables

λ: wavelength (m)
λ_max: fundamental wavelength (m)
μ: linear density (g/m)
v: wave speed (m/s)
F: force (N)
m: mass (kg)
L: length of the string (m)
l: meters (m)

Definition of terms

Tension (F): (not frequency) in the string (t is used for time in these equations). Units: newtons (N)

Linear density (μ): of the string, Greek mu. Units: grams per meter (g/m)

Velocity (v) of the wave (m/s)

Mass (m): Units: grams (g). (We would use kilograms but they are too big for most strings).

Length of the string (L): Units: meters (m)

Fundamental frequency: the frequency when the wavelength is the longest allowed, this gives us the lowest sound that we can get from the system.

In a string, the length of the string is half of the largest wavelength that can create a standing wave, called its fundamental wavelength.