The normal modes of a standing wave are all just multiples of the fundamental frequency.
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- Standing Waves and Nodes
Two sine waves travelling along the x-axis in opposite directions add together due to superposition. The resultant displacement (D) in the y-axis is given by the equation:
D (x, t) = 2a sin(kx) cos(ωt)
where a is the amplitude, ω is the angular frequency, t is time, and k is the wave number given by 2π /λ, where λ is the wavelength. This interference will cause a standing wave to form on a string of length L tied at both ends if two boundary conditions are met. For this string, the displacement at each end must be 0. Mathematically:
D (x = 0, t) = 0 and D (x = L, t) = 0
The first boundary condition, x = 0, is already met as sin(0) = 0. The other boundary condition, x= L, will be met if:
2a sin(kL) = 0
This means that sin(kL) must be equal to zero. This is the case if kL is equal to an integer multiple of π:
kL = (2πL) / λ = nπ, n = 1, 2, 3, 4...
Because the string is a fixed length, the only thing that can be varied in the wavelength, λ. Thus, a standing wave can only be form if the wavelength is:
λn = (2πL) / (nπ) = 2L / n
These possible standing waves are called the normal modes of the vibrating string.
For waves like this, the wavelength is related to the frequency by the equation:
f = v / λ
where v is the speed of the wave. Using this, the corresponding frequencies of the normal modes can be found:
fn = v / λn = v / (2L / n) = nv / 2L
The frequency of the first normal mode (n = 1) is:
f1 = v / 2L
This frequency is also known as the fundamental frequency. The frequencies of all subsequent modes can be expressed in term of this frequency:
fn = nf1, n = 1, 2, 3, 4...
These higher frequency modes are also known as harmonics.
- Sine wave generator
- Mechanical wave driver
- String, with something so weigh down the dangling end
- Set up the apparatus like in the video.
- Turn on the sine wave generator and find the first standing wave (n = 1) along the string. Remember the frequency of this mode.
- Find the following modes of the wave. Their frequencies should be integer multiples of the first mode's frequency.