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

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.