For the first standing wave, notice that L is equal to 1( λ). For the second standing
wave, L is equal to 2( λ), and for the third, L = 3( λ). A pattern is established: A
standing wave can only form when the length of the string is a multiple of λ.
L = n( λ)Solving this for the wavelength, we get
These are called the harmonic (or resonant) wavelengths, and the integer n is
known as the harmonic number.
Since we typically have control over the frequency of the waves we create, it’s
more helpful to figure out the frequencies that generate a standing wave. Because λf
= v, and because v is fixed by the physical characteristics of the string, the special
λ’s found above correspond to equally special frequencies. From fn = v/λn, we get