Normal Modes of Standing Waves

The normal modes of a standing wave are all just multiples of the fundamental frequency.

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• Standing Waves and Nodes
• Harmonics
• Frequency

Theory:

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:

 λ_n = (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:

 f_n = v / λ_n = 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:

 f_n = nf₁,     n = 1, 2, 3, 4...

These higher frequency modes are also known as harmonics. 

Apparatus:

• Sine wave generator
• Mechanical wave driver
• String, with something so weigh down the dangling end
• Pulley

Procedure

1. Set up the apparatus like in the video. 
2. Turn on the sine wave generator and find the first standing wave (n = 1) along the string. Remember the frequency of this mode.
3. Find the following modes of the wave. Their frequencies should be integer multiples of the first mode's frequency.