Consider two harmonic progressive waves (formed by string ) that have the same amplitude and same velocity but move in opposite directions . Then the displacment of the first wave (incide wave ) is `y_(1)=Asin (kx-omegat)` (waves move toward right) and the displacement of the second wave (reflected wave) is `y_(2)=asin(kx+omegat)` (wave moves toward left) is both will interfere with each other by the principle of superposition, the net displacement is `y=y_(1)+y_(2)`
Substituing equation (1) and equation) in equation(3) we get `y=Asin(kx-omegat)+Asin(kz+omegat)`
Using trigonometric identity we rewrite equation (4) as `y(x,t)=2Acos(omegat)sin(kx)`
This represent a stationary wave or standing wave which means that his wave does not move either forward or backyard wheres progressive or travelling waves will move forward of backward further the displacement of the particle in equation (5) can be written in more compct from `y(x,t)A. cos(omegat)`
where `A=2asin (kx)` implying that the particuler element of the string executes simple harmonic motion with amplitude equals to A. The maxmium of this amplitutde occours at positions for which `sin(kx)=1Rightarrowkx=(pi)/(2),(3pi)/(2),(5pi)/(2)..=mpi`
Where m takes half integar of half integral values . The position of maximum amplitude is known as antitode Expressing wave number in terms of wavelength, we can represent the anti-nodal position as `x_(m)=(2m+1/2)(lamda)/(2),where,m=0,1,2,..`
For m=0 we have max at `x_(0)=(lamda)/(2)`
For m=1 we have max at `x_(1)=(3lamda)/(4)`
For m=2 we have max at `x_(2)=(5lamda)/(4)` and so on.
The distance between two successive antitides can be computed by `x_(m)-x_(m-1)=((2m+1)/2)(lamda)/(2)-(((2m+1)+1)/(2))(lamda)/(2)=(lamda)/(2)`
when n takes integer or integral values note that elements at these points do not vibrate (not move), and point are called nodes. The nth nodal positions is given by, `x_(n)=n(lamda)/(2)where,n=0,1,2...`
For n=0, we have minimum at `x_(0)=0`
For n=1, we have minimum at `x_(1)=(lamda)/(2)`
For n=2, we have minimum at `x_(2)=lamda and` so on
The distance betwwen any rwo successive nodes can be calculated as `x_(n)-x_(n-1)=n(lamda)/(2)-(n-1)(lamda)/(2)=(lamda)/(2)` .
