Write the equation of stationary wave and obtain the equations of nodes and anti-nodes by defining them.

Explanation

Equation of stationary wave is,
`y (x,t)= 2 a sin k x cos omega t " "…(1)`
`cos omega t ` represent that each point of string performs SHM and its amplitude is `2a sin kx. `It means, for different value of x on string, amplitudes are different. The points where the amplitude is zero are caled nodes.
The points where the ampitude is maximum are called antinodes.
Nodes:
In equation `(1), 2a sin kx =0`
`therefore sin kx =0`
` therefore kx = n pi` where `n =0, 1,2,3,...,n `
`therefore (2pi)/(lamda) x = n pi`
`therefore x = (n lamda )/(2)` where `n = 0,1,2,3,...,n`
Hence, first node is obtained at `x =0 and ` second at `x = (lamda)/(2),` third at ` x = lamda....` Hence,
distance between two consecutive node is `(lamda)/(2).`
Antinodes:
In equation `(1), |sin kx|=1`
`k x = (n + (1)/(2)) pi` where` n = 0,1,2,3...`
`therefore (2pi )/(lamda ) x = (n + (1)/(2)) pi [ because k = (2pi)/(lamda)]`
`therefore x = (n + (1)/(2) ) (lamda )/(2) ` where `n = 0,1,2,3...`
This represent that fromend of string at distance `(lamda)/(4), (3 lamda)/(4), (5lamda)/(4),...,(2n +1) (lamda)/(4)` first, second, third `.....n^(th)` antinodes are obtained respectively.
Distance between two consecutive antinodes is `(lamda)/(2)` and distance between two consecutive node and antinode is `(lamda)/(4).`