1. Write the given wave in standard form

The wave is $y(x,t)=10^{-3}\sin(50t+2x)$

A travelling harmonic wave is generally written as:

• for propagation along positive $x$-axis: $y=A\sin(\omega t-kx)\quad \text{or} \quad y=A\sin(kx-\omega t)$
• for propagation along negative $x$-axis: $y=A\sin(\omega t+kx)$

Since the given equation is $y=A\sin(50t+2x),$ it matches the form $\sin(\omega t+kx)$.

So the wave is propagating along the negative $x$-axis.

2. Identify angular frequency and wave number

Comparing $y=10^{-3}\sin(50t+2x)$ with $y=A\sin(\omega t+kx),$ we get: $\omega=50\ \text{rad s}^{-1}, \qquad k=2\ \text{rad m}^{-1}$

3. Calculate wave speed

Wave speed is $v=\frac{\omega}{k}$

So, $v=\frac{50}{2}=25\ \text{m s}^{-1}$

4. Match with the options

• A: positive $x$-axis, $100\,\text{m s}^{-1}$ ❌
• B: positive $x$-axis, $25\,\text{m s}^{-1}$ ❌
• C: negative $x$-axis, $25\,\text{m s}^{-1}$ ✅
• D: negative $x$-axis, $100\,\text{m s}^{-1}$ ❌

Therefore, the correct option is C.