Use the wave equation to find the speed of a wave given in terms of the general function h(x,t):
`y(x,t) =(4.00 mm) sin[(22.0 m^(-1)) x+(8.00 s^(-1)t]` .

Answer

To find the speed of the wave given by the function \( y(x,t) = (4.00 \, \text{mm}) \sin[(22.0 \, \text{m}^{-1}) x + (8.00 \, \text{s}^{-1}) t] \), we can follow these steps:

### Step 1: Identify the wave parameters
The wave function is given in the form:
\[ y(x,t) = A \sin(kx + \omega t) \]
where:
- \( A = 4.00 \, \text{mm} \) (amplitude)
- \( k = 22.0 \, \text{m}^{-1} \) (wave number)
- \( \omega = 8.00 \, \text{s}^{-1} \) (angular frequency)

### Step 2: Write the relationship for wave speed
The speed \( v \) of a wave can be calculated using the relationship:
\[ v = \frac{\omega}{k} \]
where:
- \( \omega \) is the angular frequency
- \( k \) is the wave number

### Step 3: Substitute the values of \( \omega \) and \( k \)
Substituting the values we identified:
\[ v = \frac{8.00 \, \text{s}^{-1}}{22.0 \, \text{m}^{-1}} \]

### Step 4: Calculate the speed
Now, perform the division:
\[ v = \frac{8.00}{22.0} \]
\[ v \approx 0.3636 \, \text{m/s} \]

### Step 5: Round the answer
Rounding to three significant figures, we get:
\[ v \approx 0.364 \, \text{m/s} \]

### Final Answer
The speed of the wave is approximately \( 0.364 \, \text{m/s} \).

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