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Questions
Derive an expression for the equation of stationary wave on a stretched string.
Explain the formulation of stationary waves by the analytical method.
Solution
Consider two simple harmonic progressive waves of equal amplitudes (A) and wavelength (λ) propagating on a long uniform string in opposite directions (remember 2π/λ = k and 2πn = ω).
The equation of wave travelling along the x-axis in the positive direction is
`"y"_1 = a sin { 2π (nt - x/λ) } ...(1)`
The equation of wave travelling along the x-axis in the negative direction is
`"y"_2 = a sin { 2π (nt + x/λ) } ...(2)`
When these waves interfere, the resultant displacement of particles of string is given by the principle of superposition of waves as
y = y1 +y2
`y = a sin {2π (nt - x/λ)} + a sin {2π (nt + x/λ)}`
By using,
`sin "C"+sin"D"=2sin(("C"+"D")/2)cos(("C"-"D")/2)`, we get
y = 2a sin (2πnt) cos `( 2πx)/λ`
y = 2a cos `(2πx)/λ` sin (2πnt) or, ...(3)
Using 2a cos `(2πx)/λ` = A in equation 3, we get
y = A sin (2πnt)
As ω = 2πn, we get, y = A sin ωt.
This is the equation of a stationary wave, which gives resultant displacement due to two simple harmonic progressive waves. It may be noted that the terms in position x and time t appear separately and not as a combination 2π (nt ± x/λ).
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