Use the wave equation to find the speed of a wave given by `y(x,t) =(3.00 mm) sin [(3.00 m^(-1)) x-(8.00 s^(-1))t])`.

Answer

To find the speed of the wave given by the equation \( y(x,t) = (3.00 \, \text{mm}) \sin [(3.00 \, \text{m}^{-1}) x - (8.00 \, \text{s}^{-1}) t] \), we can follow these steps:

### Step 1: Identify the wave parameters
From the wave equation, we can identify the wave number \( k \) and angular frequency \( \omega \):
- The wave number \( k \) is the coefficient of \( x \), which is \( 3.00 \, \text{m}^{-1} \).
- The angular frequency \( \omega \) is the coefficient of \( t \), which is \( 8.00 \, \text{s}^{-1} \).

### Step 2: Use the relationship between wave speed, wave number, and angular frequency
The speed of the wave \( v \) can be calculated using the relationship:
\[ \omega = v \cdot k \]
Rearranging this equation gives:
\[ v = \frac{\omega}{k} \]

### Step 3: Substitute the values of \( \omega \) and \( k \)
Now, substituting the values we identified:
\[ v = \frac{8.00 \, \text{s}^{-1}}{3.00 \, \text{m}^{-1}} \]

### Step 4: Calculate the speed
Now, performing the division:
\[ v = \frac{8.00}{3.00} \approx 2.67 \, \text{m/s} \]

### Final Answer
The speed of the wave is approximately \( 2.67 \, \text{m/s} \).

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