1. Write the standard form of a progressive wave

A wave traveling in the positive $x$-direction is generally written as

$y = A \sin 2\pi\left(ft - \frac{x}{\lambda} + \phi\right)$

Given:

$Y = 10^{-2} \sin 2\pi\left(160t - 0.5x + \frac{\pi}{4}\right)$

Comparing with the standard form,

$f = 160\ \text{Hz}$

and

$\frac{1}{\lambda} = 0.5$

So,

$\lambda = \frac{1}{0.5} = 2\ \text{m}$

Note: The phase constant term is irrelevant for speed.

2. Use the wave speed formula

$v = f\lambda$

Substitute the values:

$v = 160 \times 2 = 320\ \text{m s}^{-1}$

3. Convert to km h$^{-1}$

We know:

$1\ \text{m s}^{-1} = 3.6\ \text{km h}^{-1}$

Hence,

$v = 320 \times 3.6 = 1152\ \text{km h}^{-1}$

4. Final answer

$\boxed{1152}$

This matches the stored correct answer.