Editing Standing wave (section)

=== Standing wave on a string with two fixed ends ===

Next, consider a string with fixed ends at {{nowrap|''x'' {{=}} 0}} and {{nowrap|''x'' {{=}} ''L''}}. The string will have some damping as it is stretched by traveling waves, but assume the damping is very small. Suppose that at the {{nowrap|''x'' {{=}} 0}} fixed end a sinusoidal force is applied that drives the string up and down in the y-direction with a small amplitude at some frequency ''f''. In this situation, the driving force produces a right-traveling wave. That wave [[reflection (physics)|reflects]] off the right fixed end and travels back to the left, reflects again off the left fixed end and travels back to the right, and so on. Eventually, a steady state is reached where the string has identical right- and left-traveling waves as in the infinite-length case and the power dissipated by damping in the string equals the power supplied by the driving force so the waves have constant amplitude.

Equation ({{EquationNote|1}}) still describes the standing wave pattern that can form on this string, but now Equation ({{EquationNote|1}}) is subject to [[boundary condition]]s where {{nowrap|''y'' {{=}} 0}} at {{nowrap|''x'' {{=}} 0}} and {{nowrap|''x'' {{=}} ''L''}} because the string is fixed at {{nowrap|''x'' {{=}} ''L''}} and because we assume the driving force at the fixed {{nowrap|''x'' {{=}} 0}} end has small amplitude. Checking the values of ''y'' at the two ends,
:<math> y(0,t) = 0, </math>
:<math> y(L,t) = 2y_\text{max}\sin \left({2\pi L \over \lambda} \right) \cos(\omega t) = 0. </math>

[[File:Standing waves on a string.gif|thumb|200px|upright|Standing waves in a string&nbsp;– the [[fundamental frequency|fundamental]] mode and the first 5 [[harmonic]]s.]]

This boundary condition is in the form of [[Wave equation#The Sturm–Liouville formulation|the Sturm–Liouville formulation]]. The latter boundary condition is satisfied when <math> \sin \left({2\pi L \over \lambda} \right) = 0 </math>. ''L'' is given, so the boundary condition restricts the wavelength of the standing waves to{{sfn|Halliday|Resnick|Walker|2005|p=434}}
{{NumBlk|:|<math> \lambda = \frac{2L}{n}, </math>|{{EquationRef|2}}}}
:<math>n = 1, 2, 3, \ldots </math>

Waves can only form standing waves on this string if they have a wavelength that satisfies this relationship with ''L''. If waves travel with speed ''v'' along the string, then equivalently the frequency of the standing waves is restricted to{{sfn|Halliday|Resnick|Walker|2005|p=434}}{{sfn|Serway|Faughn|1992|p=472}}
:<math> f = \frac{v}{\lambda} = \frac{nv}{2L}. </math>

The standing wave with {{nowrap|''n'' {{=}} 1}} oscillates at the [[fundamental frequency]] and has a wavelength that is twice the length of the string. Higher integer values of ''n'' correspond to modes of oscillation called [[harmonic]]s or [[overtone]]s. Any standing wave on the string will have ''n'' + 1 nodes including the fixed ends and ''n'' anti-nodes.

To compare this example's nodes to the description of nodes for standing waves in the infinite length string, Equation ({{EquationNote|2}}) can be rewritten as
:<math> \lambda = \frac{4L}{n}, </math>
:<math> n = 2, 4, 6, \ldots </math>

In this variation of the expression for the wavelength, ''n'' must be even. Cross multiplying we see that because ''L'' is a node, it is an ''even'' multiple of a quarter wavelength,
:<math> L = \frac{n\lambda}{4}, </math>
:<math> n = 2, 4, 6, \ldots </math>

This example demonstrates a type of [[resonance]] and the frequencies that produce standing waves can be referred to as ''resonant frequencies''.{{sfn|Halliday|Resnick|Walker|2005|p=434}}{{sfn|Serway|Faughn|1992|p=475-476}}<ref>{{cite AV media | date = May 21, 2014 | title = String Resonance | url = http://digitalsoundandmusic.com/video/?tutorial=oZ38Y0K8e-Y | access-date = August 22, 2020 | publisher = Digital Sound & Music | id = YouTube Video ID: oZ38Y0K8e-Y}}</ref>