The following four waves are sent along strings with the same linear densities (x is in meters and t is in seconds). Rank the waves according to (a) their wave speed and (b) the tension in the strings along which they travel, greatest first:
1 $$y_1 = (3 mm) sin(x - 3t)$$, (1) $$y_2 = (6 mm) sin(2x - t)$$, (3) $$y_3 = (1 mm) sin(4x - t)$$, (4) $$y_4 = (2 mm) sin(x - 2t)$$,

Explanation:

y_1 = (3 mm) sin(x - 3t)

comparing it with standard wave equation

y = A sin( ωt-kx )

we see  

ω = -3 , k = -1

velocity = ω / k

= 3

y_2 = (6 mm) sin(2x - t)

we see  

ω = -1 , k = -2

velocity = ω / k

= .5

y_3 = (1 mm) sin(4x - t)

we see  

ω = -1 , k = -4

velocity = ω / k

= .25

y_4 = (2 mm) sin(x - 2t)

we see  

ω = -2 , k = -1

velocity = ω / k

= 2

So greatest velocity to lowest velocity

y_1 = (3 mm) sin(x - 3t) ,  y_4 = (2 mm) sin(x - 2t) ,y_2 = (6 mm) sin(2x - t) ,  y_3 = (1 mm) sin(4x - t)

b )

Given the mass per unit length of wire the same , velocity is proportional to

√ T , where T is tension

so in respect of tension in the wire same order will exist for highest to lowest tension .