The equation of a transverse wave travelling along a string is y(x, t)=4.0 sin [20 × 10⁻³ x+600 t] mm, where x is in mm and t is in second. The velocity of the wave is :

Solution

Physics sir · 4 steps

Speed is ω/k, but watch the units of k, and let the plus sign tell you the direction.

1. Read off constants. Compare with y = A sin(kx + ω t). Here k = 20 × 10⁻³ mm⁻¹ = 0.02 mm⁻¹ and ω = 600 s⁻¹.
2. Fix the units. Since x is in mm, k is per millimetre. One metre is 1000 mm, so k = 0.02 × 1000 = 20 m⁻¹.

   v = ω/k = (600 s⁻¹)/(20 m⁻¹) = 30 m s⁻¹

3. Find the direction. When kx and ω t have the same sign, the wave moves towards -x. With opposite signs, as in kx - ω t, it moves towards +x.
4. So. The wave moves at 30 m s⁻¹ along the negative x direction, so its velocity is -30 m s⁻¹, option (d).

The idea to keep. For y = A sin(kx ± ω t), the speed is ω/k in matching units. A plus sign means the wave travels towards -x, and a minus sign means it travels towards +x.

Answer from the NTA final answer key. Solution by PrepWiser.