Question
Derive an expression for equation of stationary wave on a stretched string.
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Answer
When two progressive waves having the same amplitude, wavelength and speed propagate in opposite directions through the same region of a medium, their superposition under certain conditions creates a stationary interference pattern called a stationary wave.Consider two simple harmonic progressive waves, of the same amplitude A, wavelength A and frequency n = ω/2π, travelling on a string stretched along the x-axis in opposite directions. They may be represented by
$y_1 = A \sin (ωt – kx) (along the + x-axis) and … (1)$
$y_2 = A \sin (ωt + kx) (along the – x-axis) …. (2)$
where k = 2π/λ is the propagation constant.
By the superposition principle, the resultant displacement of the particle of the medium at the point at which the two waves arrive simultaneously is the algebraic sum
$y = y_1 + y_2 = A [\sin (ωt – kx) + \sin (ωt + kx)]$
Using the trigonometrical identity,
$\sin C+\sin D =2 \sin \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$
$y=2 A \sin \omega t \cos (-k x)$
$=2 A \sin \omega t \cos kx [\because \cos (- k x)=\cos ( kx )]$
$=2 A \cos kx \sin \omega t \ldots(3)$
$\therefore y = R \sin \omega t, \ldots(4)$
$\text { where } R =2 A \cos k x \ldots \ldots$
Equation (4) is the equation of a stationary wave.
$y_1 = A \sin (ωt – kx) (along the + x-axis) and … (1)$
$y_2 = A \sin (ωt + kx) (along the – x-axis) …. (2)$
where k = 2π/λ is the propagation constant.
By the superposition principle, the resultant displacement of the particle of the medium at the point at which the two waves arrive simultaneously is the algebraic sum
$y = y_1 + y_2 = A [\sin (ωt – kx) + \sin (ωt + kx)]$
Using the trigonometrical identity,
$\sin C+\sin D =2 \sin \left(\frac{C+D}{2}\right) \cos \left(\frac{C-D}{2}\right)$
$y=2 A \sin \omega t \cos (-k x)$
$=2 A \sin \omega t \cos kx [\because \cos (- k x)=\cos ( kx )]$
$=2 A \cos kx \sin \omega t \ldots(3)$
$\therefore y = R \sin \omega t, \ldots(4)$
$\text { where } R =2 A \cos k x \ldots \ldots$
Equation (4) is the equation of a stationary wave.
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