Editing Standing wave (section)

=== Standing wave on a string with one fixed end ===

[[File:Transient to standing wave.gif|thumb|upright=1.2|[[Transient (oscillation)|Transient]] analysis of a damped [[traveling wave]] reflecting at a boundary]]

Next, consider the same string of length ''L'', but this time it is only fixed at {{nowrap|''x'' {{=}} 0}}. At {{nowrap|''x'' {{=}} ''L''}}, the string is free to move in the ''y'' direction. For example, the string might be tied at {{nowrap|''x'' {{=}} ''L''}} to a ring that can slide freely up and down a pole. The string again has small damping and is driven by a small driving force at {{nowrap|''x'' {{=}} 0}}.

In this case, Equation ({{EquationNote|1}}) still describes the standing wave pattern that can form on the string, and the string has the same boundary condition of {{nowrap|''y'' {{=}} 0}} at {{nowrap|''x'' {{=}} 0}}. However, at {{nowrap|''x'' {{=}} ''L''}} where the string can move freely there should be an anti-node with maximal amplitude of ''y''. Equivalently, this boundary condition of the "free end" can be stated as {{nowrap|''∂y/∂x'' {{=}} 0}} at {{nowrap|''x'' {{=}} ''L''}}, which is in the form of [[Wave equation#The Sturm–Liouville formulation|the Sturm–Liouville formulation]]. The intuition for this boundary condition {{nowrap|''∂y/∂x'' {{=}} 0}} at {{nowrap|''x'' {{=}} ''L''}} is that the motion of the "free end" will follow that of the point to its left.

Reviewing Equation ({{EquationNote|1}}), for {{nowrap|''x'' {{=}} ''L''}} the largest amplitude of ''y'' occurs when {{nowrap|''∂y/∂x'' {{=}} 0}}, or
:<math> \cos \left({2\pi L \over \lambda}\right) = 0. </math>

This leads to a different set of wavelengths than in the two-fixed-ends example. Here, the wavelength of the standing waves is restricted to
:<math> \lambda = \frac{4L}{n}, </math>
:<math> n = 1, 3, 5, \ldots </math>

Equivalently, the frequency is restricted to
:<math> f = \frac{nv}{4L}. </math>

In this example ''n'' only takes odd values. Because ''L'' is an anti-node, it is an ''odd'' multiple of a quarter wavelength. Thus the fundamental mode in this example only has one quarter of a complete sine cycle–zero at {{nowrap|''x'' {{=}} 0}} and the first peak at {{nowrap|''x'' {{=}} ''L''}}–the first harmonic has three quarters of a complete sine cycle, and so on.

This example also demonstrates a type of resonance and the frequencies that produce standing waves are called ''resonant frequencies''.