Theory
A mechanical wave on a string can be simplified to a set of linear differential equations given the following assumptions, some of which have already been explicitly stated:
The amplitude of vibrations is low, i.e., the vertical displacement at any point along the string is small relative to its length, implying that tension should not significantly deviate from the string’s rest tension.
The radius of the string is small enough and its tension high enough such that inharmonicity due to stiffness can be considered negligible.
The string is restricted to planar motion and torsional components do not play a significant role.
Using these assumptions, the differential equation modeling vertical displacement as a function of space and time led to the one-dimensional wave equation:
This equation shows that the second time derivative of vertical displacement relates to the second spacial derivative by a squared term, a term that is equal to the square root of the string's tension divided by its bulk modulus.
Solutions to this differential equation are also subject to the following boundary conditions:
1.
2.
3.
Where L denotes the string's total length. Applying these conditions, the solution becomes:
For which the Fourier coefficients are determined by:
In our case of a string being vertically displaced at some location , this coefficient equation becomes:
Where h denotes the vertical displacement of the string. Many steps have been omitted from the mathematical development for the sake of brevity. With this equation for the successive Fourier coefficients, an analytical method for comparing experiment to predicted model was formed. This quantitative approach gave opportunity to test vibrations adhering to the linear case (constant tension in the string in space and time).