In the wave equation
y=0.5 sin (2 π)/λ(400 t-x) m
the velocity of the wave will be:

Solution

Physics sir · 4 steps

Match it to y = A sin k(vt - x): whatever multiplies t inside the bracket is the wave speed.

1. Standard form. A travelling wave can be written as y = A sin 2π/λ(vt - x), where v is the wave speed. Here k = 2π/λ has already been taken outside the bracket.
2. Read off v. Compare with y = 0.5 sin 2π/λ(400t - x). The coefficient of x inside the bracket is 1, so the coefficient of t is the speed: v = 400 m s⁻¹.
3. Cross-check. Opening the bracket gives ω = (2π × 400)/λ and k = 2π/λ. Dividing them, λ cancels out.

   v = ω/k = (2π × 400/λ)/(2π/λ) = 400 m s⁻¹

4. So. The wave travels at 400 m s⁻¹, which is option (c).

The idea to keep. Wave speed is always ω/k. When the equation is written as sin k(vt - x) with x having coefficient 1 in the bracket, the number beside t is the speed itself.

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