Explain the formulation of stationary waves by analytical method what are nodes and antinodes

Explain the formulation of stationary waves by the analytical method. What are nodes and antinodes? Show that the distance between two successive nodes or antinodes is λ/2. 

1. Consider two simple harmonic progressive waves of equal amplitudes (a) and wavelength (λ) propagating on a long uniform string in opposite directions.
2. The equation of wave travelling along the X-axis in the positive direction is given by,`y_1 = a sin [2pi(nt - x/λ)]`The equation of wave travelling along the X-axis in the negative direction is given by,

   `y_2 = a sin [2pi(nt + x/λ)]`

3. When these waves interfere, the resultant displacement of particles of the string is given by the principle of superposition of waves as y = y1 + y2 
   ∴ y = a sin`[2pi(nt - x/λ)] + a sin [2pi(nt + x/λ)]`
4. By using trigonometry formula,sin C + sin D = 2 sin `((C + D)/2)` cos`((C - D)/2)`y = 2a sin (2πnt) cos `(2pix)/λ`

   y = 2a cos`(2pix)/λ` sin (2πnt) ….(1)

5. Substituting 2a cos `(2pix)/λ` = A in equation (1),y = A sin(2πnt)∴ y = A sinωt ….(∴ ω = 2πn)

   This is the equation of a stationary wave which gives resultant displacement due to two simple harmonic progressive waves.

6. Nodes: The points of a medium, which vibrate with minimum amplitude are called nodes.
   Condition for node: Amplitude is minimum, i.e., A = 0.∴ 2a cos`(2pix)/λ` = 0∴ cos`(2pix)/λ` = 0∴ `(2pix)/λ = pi/2,(3pi)/2,(5pi)/2,.........`∴ x = `λ/4, (3λ)/4, (5λ)/4,`.......i.e., x = (2p – 1)`λ/4` where p = 1, 2, 3,….Distance between two successive nodes:`(3λ)/4 - λ/4 = λ/2;``(5λ)/4 - (3λ)/4 = λ/2`

   ∴ Distance between two successive nodes is `λ/2`

7. Antinodes: The points of a medium, which vibrate with maximum amplitude are called antinodes. Condition for antinode: Amplitude is maximum, i.e., A = ±2a∴ 2a cos `(2pix)/λ = ±2a`∴ cos`(2pix)/λ` = ±1∴ `(2pix)/λ` = 0, π, 2π, 3π, .....∴ x = 0, `λ/2, λ, (3λ)/2,`...........i.e., x = `(λp)/2` where p = 0, 1, 2, 3,…∴ Distance between two successive antinodes:`λ/2 - 0 = λ/2;``λ - λ/2 = λ/2`

   ∴ Distance between two successive antinodes is `λ/2`.

Concept: Stationary Waves

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