Write the equation for a progressive wave moving along positive X-axis with the following characteristics

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Q-1 Write the equation for a progressive wave moving along positive X-axis with the following characteristics

(i) a = 0.3 m; T = 2 sec., λ = 1m  

(ii) a = 2 x 10-3 m; n = 540 Hz, v = 380 m/sec

Assume that the displacement y = 0 at t = 0 and x = 0.

Ans. (i)  y= 0.3sin2 π (t/2- x)
(ii)  y =0.002sin1080 π (t –x/330)

Q2-Which of the following Are the Solution To the one dimensional wave equation:

(i)\;y=2 sin \,x\;cos \,vt \\ (ii) \;y=5 sin\,2x\;cos \,vt \\ (iii) \; y = x^2- v^2t^2  \\ (i) \; y= 2 x -5t

solution One-dimensional wave equation is

\frac{ \partial^2 y } { \partial t^2} = v^2 \frac { \partial^2 y} { \partial x^2}

(i) differentiate y=2 sin \,x\;cos \,vt (i) twice partially with respect to t and x separately

\frac{\partial^2 y}{\partial t^2}=-v^2y \;\; and \;\;  \frac { \partial^2 y} { \partial x^2} =-y
\therefore \; \; \; \, \:  \frac{\partial^2 y}{\partial t^2}=v^2\frac { \partial^2 y} { \partial x^2}

Here (i) is the solution to the one dimensional wave equation

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(ii) in this case,

\therefore \; \; \; \, \:  \frac{\partial^2 y}{\partial t^2}=-v^2y\; \; and \; \frac { \partial^2 y} { \partial x^2}=-4y
\therefore \; \; \; \, \:  \frac{\partial^2 y}{\partial t^2}=\frac{v^2}{4}\; \frac { \partial^2 y} { \partial x^2}

Here (ii) is not the solution of one dimensional wave equation.

in the same way do for (iii) and (iv) Eq (iii) is not the solution and Eq (iv) is the solution of the wave equation


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