Consider the wave `y = (5 mm) sin[1 cm^(-1) x - (60 s^(-1)) t]`. Find (a) the ampitude, (b) the angular wave number, ( c ) the wavelength, (d) the frequency, (e) the time period and (f) the wave velocity.

Explanation

To solve the given wave equation \( y = (5 \, \text{mm}) \sin[1 \, \text{cm}^{-1} \, x - (60 \, \text{s}^{-1}) \, t] \), we will find the following parameters step by step:

### Step 1: Identify the Amplitude
The amplitude \( A \) is the coefficient of the sine function in the wave equation.

**Solution:**
From the equation, we see that the amplitude \( A = 5 \, \text{mm} \).

### Step 2: Determine the Angular Wave Number
The angular wave number \( k \) is the coefficient of \( x \) in the sine function.

**Solution:**
From the equation, we have \( k = 1 \, \text{cm}^{-1} \).

### Step 3: Calculate the Wavelength
The wavelength \( \lambda \) can be calculated using the formula:
\[ \lambda = \frac{2\pi}{k} \]

**Solution:**
Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{1 \, \text{cm}^{-1}} = 2\pi \, \text{cm} \]

### Step 4: Find the Frequency
The angular frequency \( \omega \) is the coefficient of \( t \) in the sine function. The frequency \( f \) can be calculated using the formula:
\[ f = \frac{\omega}{2\pi} \]

**Solution:**
From the equation, \( \omega = 60 \, \text{s}^{-1} \):
\[ f = \frac{60}{2\pi} = \frac{30}{\pi} \, \text{Hz} \]

### Step 5: Determine the Time Period
The time period \( T \) is the reciprocal of the frequency:
\[ T = \frac{1}{f} \]

**Solution:**
Substituting the value of \( f \):
\[ T = \frac{1}{\frac{30}{\pi}} = \frac{\pi}{30} \, \text{s} \]

### Step 6: Calculate the Wave Velocity
The wave velocity \( v \) can be calculated using the formula:
\[ v = f \lambda \]

**Solution:**
Substituting the values of \( f \) and \( \lambda \):
\[ v = \left(\frac{30}{\pi}\right) \times (2\pi) = 60 \, \text{cm/s} \]

### Summary of Results
(a) Amplitude \( A = 5 \, \text{mm} \)
(b) Angular wave number \( k = 1 \, \text{cm}^{-1} \)
(c) Wavelength \( \lambda = 2\pi \, \text{cm} \)
(d) Frequency \( f = \frac{30}{\pi} \, \text{Hz} \)
(e) Time period \( T = \frac{\pi}{30} \, \text{s} \)
(f) Wave velocity \( v = 60 \, \text{cm/s} \)