Four traveling waves are described by the following equations, where all quantities are measured in SI units and y represents the displacement. I: y = 0.12 cos(3x + 2t)II: y = 0.15 sin(6x - 3t)III: y = 0.23 cos(3x + 6t)IV: y = -0.29 sin(1.5x - t)Which of these waves have the same speed?

waves i and iv have the same speed

Explanation:

The speed of a traveling wave the speed of the wave is given by

           v = λ f

There is also the speed of oscillation of the particle given by

            v_osc = dy / dt

The wave equation has the form

           y = A sin (kx –wt)

Let's look for wavelength and frequency

         k = 2π /λ

         λ = 2π / k

         w = 2π f

         f = w / 2π

We substitute in the wave speed equation

          v = 2π /k  w/ 2π

          v = w / k

Let's look for the speed in each wave

i) y = 0.12 cos (3x + 2t)

    k = 3 m-1

    w = 2 rad / s

Let's use the trigonometric relationship

    cos (α -π/2) = cos α cos π/2 + sin α sin π/2

    cos(α -π/2) = sin α  

     y = 0.12 sin ( 3x + 2t - π/2)

     v₁ = 2/3

     v₁ = 0.667 m / s

ii) y = 0.15 sin (6x-3t)

     v₂ = 3/6

     v₂ = 0.5 m / s

iii) y = 0.23 cos (3x + 6t)

    y= 0.23 sin(3x +6t -π/2)

    v₃ = -6 / 3

    v₃ = -2 m / s

iv) y = -0.29 sin (1.5x-t)

    v₄ = 1 / 1.5

    v₄ = 0.667 m / s

From these results we see that waves i and iv have the same speed, but differentiate range of motion

it is called kinetic energy