To determine the direction of the velocity of the wave, we analyze the given wave equation:

$z(y, t) = 1.50 \, \text{cm} \, \sin(1250 \, \text{rad/s} \cdot t + 10.0 \, \text{m}^{-1} \cdot y)$

This equation represents a sinusoidal wave. The general form of a wave traveling in one dimension is:

$z(y, t) = A \sin(\omega t \pm k y)$

• $A$: Amplitude of the wave (1.50 cm here).
• $\omega$: Angular frequency ($1250 \, \text{rad/s}$).
• $k$: Wave number ($10.0 \, \text{m}^{-1}$).
• The $+$ sign in $(\omega t + k y)$ indicates the wave is traveling in the negative $y$-direction, while a $-$ sign would indicate the wave is traveling in the positive $y$-direction.

Conclusion:

Since the equation includes a $+$ sign in $(\omega t + k y)$, the wave is traveling in the negative $y$-direction.

Let me know if you would like additional clarification! Here are 5 related questions to explore further:

1. What is the speed of the wave given the angular frequency and wave number?
2. How do we calculate the wavelength of this wave?
3. How can the phase difference between two points on the wave be determined?
4. What is the maximum velocity of a particle in this wave?
5. What happens to the wave if the sign in the equation is reversed?

Tip: The direction of wave travel is determined by the sign of the $\omega t$ and $k y$ terms in the phase. Always check their relative sign carefully!