When two type of progressive ray trains of similar type having the similar amplitude and same time period/ wavelength/ frequency traveling with same velocity along the same straight line in reverse directions superimpose, a new set of rays are formed. These are known as stationary waves or standing waves.
In practice, a stationary wave is created when a wave train is goes at a boundary. The incident and reflected rays then interfere to give a stationary wave.
1. Consider that the two super imposing rays are incident ray y1=asin(ωt+kx) and reflected wave y2=asin(ωt+kx).
Then by rule of superposition y = y1+y2 = a[sin(ωt-kx)+sin(ωt+kx)]
SinC+sinD =2 sin(C+D)/2 cos(C-D)/2)
y = 2acoskx sinωt
2. As this relation satisfies the ray relation; 
. It shows a wave.
3. As it is not of the type f(ax±bt), the ray is not progressive.
4. Amplitude of the ray Asw = 2a coskx
Amplitude in two different cases
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Reflection at open end
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Reflection at closed end
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Asw = 2a coskx
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Asw = 2a sinkx
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Amplitude is maximum when coskx =±1 kx = 0,π,2π
X = 0,λ/2,λ,....nλ/2
where k=2/λ and n=0,1,2,3
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Amplitude is maximum when coskx=±1
kx = 0,π/2,3π/2....... and X = λ/4,3λ/4,....
where k =2π/λ and n=1,2,3,.....
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Amplitude is minimum when coskx = 0
kx = π/2,3π/2......(2n-1)π/2
X = λ/4,3λ/4,....
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Amplitude is minimum when sinkx = 0
kx = π/2,3π/2......(2n-1)π/2
X = 0,λ/2,λ,....nλ/2
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5. Nodes (N): The points where amplitude is minimized are known as nodes.
(i) Distance between two successive nodes is λ/2 .
(ii) Nodes are at fixed rest.
(iii) At point air pressure and density both are very high.
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6. Antinodes (A): The points of maximum amplitude are known as antinodes.
(i) The displacement between two successive antinodes is λ/2 .
(ii) At ends density and air pressure both are very low.
(iii) The distance between a node (N) and adjoining antinode (A) is λ/4 .
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