In the wave equation

y = 0.5 sin \(\frac{2 \pi}{\lambda}\) (400t - x)m

the velocity of the wave will be:

Concept:

Wave Equation:

The general form of a wave equation is y = A sin(kx - ωt), where:

A = Amplitude of the wave

k = Wave number (k = 2π / λ)

ω = Angular frequency (ω = 2πf)

t = Time

x = Position

The wave velocity v can be calculated using the relation:

v = ω / k

Calculation:

Given the wave equation: y = 0.5 sin(2π / λ (400t - x)) m, we can identify the following:

ω = 2π × 400 = 800π rad/s

k = 2π / λ

The velocity of the wave is given by:

v = ω / k = (800π) / (2π / λ) = 400λ

Since the equation is in the standard wave form, we can conclude that the velocity of the wave is 400 m/s.

∴ The velocity of the wave is 400 m/s, which corresponds to Option 3.