To solve the problem step by step, we will analyze the wave equation given and extract the required parameters.
Given wave equation:
\[ y = (5 \, \text{mm}) \sin\left(1 \, \text{cm}^{-1} \, x - (60 \, \text{s}^{-1}) \, t\right) \]
### Step 1: Find the Amplitude
The amplitude \( a \) is the coefficient of the sine function in the wave equation.
**Solution:**
\[ a = 5 \, \text{mm} \]
### Step 2: Find the Angular Wave Number
The angular wave number \( k \) is the coefficient of \( x \) in the sine function.
**Solution:**
\[ k = 1 \, \text{cm}^{-1} \]
### Step 3: Find the Wavelength
The wavelength \( \lambda \) is related to the angular wave number \( k \) by the formula:
\[ \lambda = \frac{2\pi}{k} \]
**Solution:**
\[ \lambda = \frac{2\pi}{1 \, \text{cm}^{-1}} = 2\pi \, \text{cm} \]
### Step 4: Find the Frequency
The frequency \( f \) can be calculated from the angular frequency \( \omega \) using the formula:
\[ f = \frac{\omega}{2\pi} \]
where \( \omega = 60 \, \text{s}^{-1} \).
**Solution:**
\[ f = \frac{60}{2\pi} \approx 9.55 \, \text{Hz} \]
### Step 5: Find the Time Period
The time period \( T \) is the reciprocal of the frequency:
\[ T = \frac{1}{f} \]
**Solution:**
\[ T = \frac{1}{\frac{60}{2\pi}} = \frac{2\pi}{60} = \frac{\pi}{30} \, \text{s} \]
### Step 6: Find the Wave Velocity
The wave velocity \( v \) can be calculated using the formula:
\[ v = f \cdot \lambda \]
**Solution:**
\[ v = \left(\frac{60}{2\pi}\right) \cdot (2\pi) = 60 \, \text{cm/s} \]
### Summary of Results:
(a) Amplitude: \( 5 \, \text{mm} \)
(b) Angular wave number: \( 1 \, \text{cm}^{-1} \)
(c) Wavelength: \( 2\pi \, \text{cm} \)
(d) Frequency: \( \approx 9.55 \, \text{Hz} \)
(e) Time period: \( \frac{\pi}{30} \, \text{s} \)
(f) Wave velocity: \( 60 \, \text{cm/s} \)
