Editing Standing wave (section)

=== Standing wave on an infinite length string ===

To begin, consider a string of infinite length along the ''x''-axis that is free to be stretched [[transverse wave|transversely]] in the ''y'' direction.

For a [[harmonic wave]] traveling to the right along the string, the string's [[displacement (geometry)|displacement]] in the ''y'' direction as a function of position ''x'' and time ''t'' is{{sfn|Halliday|Resnick|Walker|2005|p=432}}
:<math> y_\text{R}(x,t) =  y_\text{max}\sin \left({2\pi x \over \lambda} - \omega t \right). </math>

The displacement in the ''y''-direction for an identical harmonic wave traveling to the left is
:<math> y_\text{L}(x,t) = y_\text{max}\sin \left({2\pi x \over \lambda} + \omega t \right), </math>

where
*''y''<sub>max</sub> is the [[amplitude]] of the displacement of the string for each wave,
*''ω'' is the [[angular frequency]] or equivalently ''2π'' times the [[frequency]] ''f'',
*''λ'' is the [[wavelength]] of the wave.

For identical right- and left-traveling waves on the same string, the total displacement of the string is the sum of ''y''<sub>R</sub> and ''y''<sub>L</sub>,
:<math> y(x,t) = y_\text{R} + y_\text{L} = y_\text{max}\sin \left({2\pi x \over \lambda} - \omega t \right) + y_\text{max}\sin \left({2\pi x \over \lambda} + \omega t \right). </math>

Using the [[Trigonometric identity#Product-to-sum and sum-to-product identities|trigonometric sum-to-product identity]] <math>\sin a + \sin b = 2\sin \left({a+b \over 2}\right)\cos \left({a-b \over 2}\right)</math>,
{{NumBlk|:|<math> y(x,t) = 2y_\text{max}\sin \left({2\pi x \over \lambda} \right) \cos(\omega t). </math>|{{EquationRef|1}}}}

Equation ({{EquationNote|1}}) does not describe a traveling wave. At any position ''x'', ''y''(''x'',''t'') simply oscillates in time with an amplitude that varies in the ''x''-direction as <math>2y_\text{max}\sin \left({2\pi x \over \lambda}\right)</math>.{{sfn|Halliday|Resnick|Walker|2005|p=432}} The animation at the beginning of this article depicts what is happening. As the left-traveling blue wave and right-traveling green wave interfere, they form the standing red wave that does not travel and instead oscillates in place.

Because the string is of infinite length, it has no boundary condition for its displacement at any point along the ''x''-axis. As a result, a standing wave can form at any frequency.

At locations on the ''x''-axis that are ''even'' multiples of a quarter wavelength,
:<math>x = \ldots, -{3\lambda \over 2}, \; -\lambda, \; -{\lambda \over 2}, \; 0, \; {\lambda \over 2}, \; \lambda, \; {3\lambda \over 2}, \ldots </math>

the amplitude is always zero. These locations are called [[node (physics)|nodes]]. At locations on the ''x''-axis that are ''odd'' multiples of a quarter wavelength
:<math>x = \ldots, -{5\lambda \over 4}, \; -{3\lambda \over 4}, \; -{\lambda \over 4}, \; {\lambda \over 4}, \; {3\lambda \over 4}, \; {5\lambda \over 4}, \ldots </math>

the amplitude is maximal, with a value of twice the amplitude of the right- and left-traveling waves that interfere to produce this standing wave pattern. These locations are called [[anti-node]]s.  The distance between two consecutive nodes or anti-nodes is half the wavelength, ''λ''/2.