Consider the wave `y = (10 mm) sin[(5picm^(-1))x-(60pis^(-1))t + (pi)/(4)]`. Find (a) the amplitude (b) the wave number (c) the wavelength (d) the frequency f the time period and (f) the wave velcity (g) phase constant of `SHM` of particle at `x = 0`.

Answer

To solve the problem step by step, we will analyze the wave equation given and extract the required parameters.

Given wave equation:
\[ y = (10 \, \text{mm}) \sin\left[(5\pi \, \text{cm}^{-1})x - (60\pi \, \text{s}^{-1})t + \frac{\pi}{4}\right] \]

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

**Solution:**
\[ A = 10 \, \text{mm} \]

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

**Solution:**
\[ k = 5\pi \, \text{cm}^{-1} \]

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

**Solution:**
\[ \lambda = \frac{2\pi}{5\pi} = \frac{2}{5} \, \text{cm} = 0.4 \, \text{cm} \]

### Step 4: Identify the Angular Frequency
The angular frequency \( \omega \) is the coefficient of \( t \) in the argument of the sine function.

**Solution:**
\[ \omega = 60\pi \, \text{s}^{-1} \]

### Step 5: Calculate the Frequency
The frequency \( f \) can be calculated using the relationship:
\[ \omega = 2\pi f \]

**Solution:**
\[ f = \frac{\omega}{2\pi} = \frac{60\pi}{2\pi} = 30 \, \text{Hz} \]

### Step 6: Calculate the Time Period
The time period \( T \) is the reciprocal of the frequency.

**Solution:**
\[ T = \frac{1}{f} = \frac{1}{30} \, \text{s} \]

### Step 7: Calculate the Wave Velocity
The wave velocity \( v \) can be calculated using the formula:
\[ v = \frac{\omega}{k} \]

**Solution:**
\[ v = \frac{60\pi}{5\pi} = 12 \, \text{cm/s} \]

### Step 8: Identify the Phase Constant of SHM at \( x = 0 \)
The phase constant \( \phi \) is the constant term in the argument of the sine function.

**Solution:**
At \( x = 0 \):
\[ \phi = \frac{\pi}{4} \]

### Summary of Results:
(a) Amplitude \( A = 10 \, \text{mm} \)
(b) Wave number \( k = 5\pi \, \text{cm}^{-1} \)
(c) Wavelength \( \lambda = 0.4 \, \text{cm} \)
(d) Frequency \( f = 30 \, \text{Hz} \)
(e) Time period \( T = \frac{1}{30} \, \text{s} \)
(f) Wave velocity \( v = 12 \, \text{cm/s} \)
(g) Phase constant at \( x = 0 \) \( \phi = \frac{\pi}{4} \)