A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m−1 x). What is the direction of the velocity of the wave?

To determine the direction of the velocity of the wave described by the equation $y(x, t) = 2.50 \, \text{cm} \cos(2,500 \, \text{rad/s} \, t - 15.0 \, \text{m}^{-1} \, x)$, we will analyze the wave equation in detail.

1. ### Break Down the Problem

We need to identify the parameters in the wave equation to understand its propagation direction.

2. ### Relevant Concepts

A transverse wave in the form of $y(x, t) = A \cos(\omega t - kx)$ describes:

• $A$: Amplitude
• $\omega$: Angular frequency
• $k$: Wave number

Given:

• $\omega = 2,500 \, \text{rad/s}$
• $k = 15.0 \, \text{m}^{-1}$

3. ### Analysis and Detail

In the wave equation $y(x, t)$:

• The term $\omega t - kx$ indicates the wave's phase.
• The sign before $kx$ determines the direction of propagation.

4. ### Verify and Summarize

Since the term for $x$ in the cosine function has a negative sign (-), it indicates that the wave is traveling in the positive x-direction.

Final Answer

The direction of the velocity of the wave is in the positive x-direction.