Editing Wave interference (section)

=== Real-valued wave functions ===
The above can be demonstrated in one dimension by deriving the formula for the sum of two waves.  The equation for the amplitude of a [[sinusoid]]al wave traveling to the right along the x-axis is
<math display="block">W_1(x,t) = A\cos(kx - \omega t)</math>
where <math>A</math> is the peak amplitude, <math>k = 2\pi/\lambda</math> is the [[wavenumber]] and <math>\omega = 2\pi f</math> is the [[angular frequency]] of the wave.  Suppose a second wave of the same frequency and amplitude but with a different phase is also traveling to the right
<math display="block">W_2(x,t) = A\cos(kx - \omega t + \varphi)</math>
where <math>\varphi</math> is the phase difference between the waves in [[radian]]s.  The two waves will [[Superposition principle|superpose]] and add: the sum of the two waves is
<math display="block">W_1 + W_2 = A[\cos(kx - \omega t) + \cos(kx - \omega t + \varphi)].</math>
Using the [[trigonometric identity]] for the sum of two cosines: <math display="inline">\cos a + \cos b = 2\cos\left({a-b \over 2}\right)\cos\left({a+b \over 2}\right),</math> this can be written
<math display="block">W_1 + W_2 = 2A\cos\left({\varphi \over 2}\right)\cos\left(kx - \omega t + {\varphi \over 2}\right).</math>
This represents a wave at the original frequency, traveling to the right like its components, whose amplitude is proportional to the cosine of <math>\varphi/2</math>.
* ''Constructive interference'': If the phase difference is an even multiple of {{pi}}: <math>\varphi = \ldots,-4\pi, -2\pi, 0, 2\pi, 4\pi,\ldots</math> then <math>\left|\cos(\varphi/2)\right| = 1</math>, so the sum of the two waves is a wave with twice the amplitude <math display="block">W_1 + W_2 = 2A\cos(kx - \omega t)</math>
* ''Destructive interference'': If the phase difference is an odd multiple of {{pi}}: <math>\varphi = \ldots,-3\pi,\, -\pi,\, \pi,\, 3\pi,\, 5\pi,\ldots</math> then <math>\cos(\varphi/2) = 0\,</math>, so the sum of the two waves is zero <math display="block">W_1 + W_2 = 0</math>