Question

A stationary wave is set up in a string fixed at both ends. Which of the following cannot represent stationary wave equation

(a)  (b)  (c) (d) 

A point on a string at has an initial displacement of . If the wave travels to the right, then the corresponding equation is
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Answer 1

Given:

A stationary wave is set up in a string fixed at both ends.

To find:

Equation for the stationary wave.

Calculation:

Stationary waves are formed by superimposition of two waves having same amplitude and frequency but travelling in opposite directions towards each other.

Let 1st wave be y1:

$y1 = a \sin( \omega t - kx)$

Let 2nd wave be y2:

$y2 = a \sin( \omega t + kx)$

Superimposition of the waves:

$\therefore \: y = y1 + y2$

$= > \: y = a \sin( \omega t - kx) + a \sin( \omega t + kx)$

$= > \: y = a \bigg \{ \sin( \omega t - kx) + \sin( \omega t + kx) \bigg \}$

$= > \: y = a \bigg \{ 2 \sin( \omega t) \cos(kx) \bigg \}$

$= > \: y = 2a \bigg \{ \sin( \omega t) \cos(kx) \bigg \}$

So , this is the equation of the standing wave having maximum amplitude as 2a and minimum amplitude as 0.

So , final answer is :

$\boxed{ \bold{ \red{ y = 2a \bigg \{ \sin( \omega t) \cos(kx) \bigg \}}}}$

Answer 2

$\huge{\mathbb{\red{ANSWER}}}$

Let us consider a progressive wave of amplitude a and wavelength λ travelling in the direction of X axis. This is the equation of a stationary wave. ∴ A = + 2a.

$\huge{\mathbb{\red{Thanks}}}$