How does the wave y = sin ( x -a) for a = 0 , `a = (pi)/(4) , a = (pi)/(2) , a = (3pi)/(2)` and ` a = pi` look like ?

Explanation

From the above picture we observe that y = sin (x - a) for a = 0 .
`a = (pi)/(4) , a = (pi)/(2) , a = (3pi)/(2)` and `a = pi` , the function y = sin (x - a) shifts towards right . Further , we can take a = vt and v `= (pi)/(4)` , and sketching for different times t = 0s , t = 1 s , t = 2s etc . , we once again observe that y = sin ( x - vt) moves towards the right . If y = sin (x + vt) then the travelling (or progressive) wave moves towards the left . thus , any arbitrary function of type y = f (x - vt) characterising the wave must move towards right and similarly , any arbitrary function of type y = f (x + vt) characterizing the wave must move towards left .