Consider two simple harmonic progressive waves, oif the same amplitude A, wavelength `lambda` and frequency `n=omega//2pi`, travelling along the x-axis in opposite directions. They can be represented by
`y_(1)=A sin (omegat-kx)" "` (along the +xaxis)`" "`....(1)
and `y_(2)=A sin (omegat+kx)" "` (along the -xaxis)`" "`....(2)
where `k=2pi//lambda` is the propagation constant.
By the superposition principle, the resultant displacement of the particle of the medium at the point atwhich the two waves arrive simultaneously is the algebraic sum
`y=y_(1)+y_(2)=A [sin(omegat-kx)+sin(omegat+kx)]" "`.....(3)
Now, `sin C+sin D=2 sin ((C+D)/(2))cos((C-D)/(2))`
`:. y=2A sin omegat cos (-kx)=2A sin omegat cos kx" "`......(4)
[`:' cos (-kx)=cos(kx)`]
`=2A cos kx sin omegat" "`......(5)
where `R=2A cos kx`.
The above equation shows that the resultant wave is a simple harmonic wave of amplitude |R| and having the same period as that of the individual waves. The absence of the terms`-+kx` in the sine function and `omegat` in the cosine function implies that the resultant wave does not propagate along the positive or negative x-axis.Hence, it is called a stationary wave. In this case there is no energy transport.
The absence of the term in t in the cosine function shows that each particle vibrates with a fixed amplitude |R| that varies periodically, only with x.
Nodes and antinodes : The points at which the particles of the midum are always at rest are called the nodes.
At nodes, R=0
`:. cos (2pix)/(lambda)=0" "(":'" A ne 0"and "k=(2pi)/(lambda))" "`....(6)
`:. (2pix)/(lambda)=(pi)/(2),(3pi)/(2),(5pi)/(2),(7pi)/(2)`,.....
`:. x=(lambda)/(4),(3lambda)/(4),(5lambda)/(4),(7lambda)/(4),....,(2p+1)(lambda)/(4),.....," "` where p=0,1,2,....
Therefore, the distance between successive nodes is
`[2(p+1)+1](lambda)/(4)-(2p+1)(lambda)/(4)=(lambda)/(2)" "`....(7)
The points at which the particles of the medium vibrate with the maximum amplitude are called the antinodes.
At antinodes, `R=-+2A`
`:. cos(2pix)/(lambda)=-+1" "`.....(8)
`:. (2pix)/(lambda)=0,pi,2pi,3pi`,....
`:. x=0,(lambda)/(2),lambda,(3lambda)/(2),.....,(plambda)/(2),...," "` where p=0,1,2,...
Therefore, the distance between successive antinodes is
`((p+1).lambda)/(2)-(plambda)/(2)=(lambda)/(2)" "`......(9)
`:.` Distance between successive nodes=distance between successive antinodes `=lambda//2`.
Thus, the nodes and antinodes occur alternately and are equally spaced.
