Editing Eigenfunction (section)

===Vibrating strings===
[[File:Standing wave.gif|thumb|270px|The shape of a standing wave in a string fixed at its boundaries is an example of an eigenfunction of a differential operator. The admissible eigenvalues are governed by the length of the string and determine the frequency of oscillation.]]
Let {{math|''h''(''x'', ''t'')}} denote the transverse displacement of a stressed elastic chord, such as the [[vibrating string]]s of a [[string instrument]], as a function of the position {{mvar|x}} along the string and of time {{mvar|t}}. Applying the laws of mechanics to [[infinitesimal]] portions of the string, the function {{mvar|h}} satisfies the [[partial differential equation]]
<math display="block" alt="the second partial derivative of h with respect to t equals c squared times the second partial derivative of h with respect to x">\frac{\partial^2 h}{\partial t^2} = c^2\frac{\partial^2 h}{\partial x^2},</math>
which is called the (one-dimensional) [[wave equation]]. Here {{mvar|c}} is a constant speed that depends on the tension and mass of the string.

This problem is amenable to the method of [[separation of variables]]. If we assume that {{math|''h''(''x'', ''t'')}} can be written as the product of the form {{math|''X''(''x'')''T''(''t'')}}, we can form a pair of ordinary differential equations:
<math display="block" alt="d square big X over d x squared equals negative of omega over c quantity squared times big X, and d squared big T over d t squared equals negative omega squared times T">\frac{d^2}{dx^2}X=-\frac{\omega^2}{c^2}X, \qquad \frac{d^2}{dt^2}T = -\omega^2 T.</math>

Each of these is an eigenvalue equation with eigenvalues <math display="inline">-\frac{\omega^2}{c^2}</math> and {{math|−''ω''<sup>2</sup>}}, respectively. For any values of {{mvar|ω}} and {{mvar|c}}, the equations are satisfied by the functions
<math display="block">X(x) = \sin\left(\frac{\omega x}{c} + \varphi\right), \qquad T(t) = \sin(\omega t + \psi),</math>
where the phase angles {{mvar|φ}} and {{mvar|ψ}} are arbitrary real constants.

If we impose boundary conditions, for example that the ends of the string are fixed at {{math|1=''x'' = 0}} and {{math|1=''x'' = ''L''}}, namely {{math|1=''X''(0) = ''X''(''L'') = 0}}, and that {{math|1=''T''(0) = 0}}, we constrain the eigenvalues. For these boundary conditions, {{math|1=sin(''φ'') = 0}} and {{math|1=sin(''ψ'') = 0}}, so the phase angles {{math|1=''φ'' = ''ψ'' = 0}}, and
<math display="block" alt="sine of omega divided by c quantity equals 0">\sin\left(\frac{\omega L}{c}\right) = 0.</math>

This last boundary condition constrains {{mvar|ω}} to take a value {{math|1=''ω<sub>n</sub>'' = {{sfrac|''ncπ''|''L''}}}}, where {{mvar|n}} is any integer. Thus, the clamped string supports a family of standing waves of the form
<math display="block">h(x,t) = \sin\left(\frac{n\pi x}{L} \right) \sin(\omega_n t).</math>

In the example of a string instrument, the frequency {{math|''ω<sub>n</sub>''}} is the frequency of the {{mvar|n}}-th [[harmonic]], which is called the {{math|(''n'' − 1)}}-th [[overtone]].