The equation of a progressive wave is `y = 8 sin pi (0.2 x - 4t)`. If the x and y are expressed in cm and time in seconds. The value of wavelength and the periodic time will be -

Options

A. 50 cm, 25 sec
B. 0.02 cm, 4 sec
C. 10 cm, 0.5 sec
D. None of these

Answer

To solve the question regarding the progressive wave described by the equation \( y = 8 \sin(\pi (0.2x - 4t)) \), we need to determine the wavelength and the periodic time. Let's break it down step by step.

### Step 1: Identify the wave equation format
The general form of a progressive wave can be expressed as:
\[ y = A \sin(kx - \omega t) \]
where:
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

### Step 2: Compare the given equation with the standard form
From the given equation:
\[ y = 8 \sin(\pi (0.2x - 4t)) \]
we can rewrite it as:
\[ y = 8 \sin(\pi \cdot 0.2x - \pi \cdot 4t) \]

From this, we can identify:
- \( k = \pi \cdot 0.2 \)
- \( \omega = \pi \cdot 4 \)

### Step 3: Calculate the wave number \( k \)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ \pi \cdot 0.2 = \frac{2\pi}{\lambda} \]

### Step 4: Solve for the wavelength \( \lambda \)
To find \( \lambda \), we can rearrange the equation:
\[ \lambda = \frac{2\pi}{\pi \cdot 0.2} \]
This simplifies to:
\[ \lambda = \frac{2}{0.2} = 10 \text{ cm} \]

### Step 5: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of \( \omega \):
\[ \pi \cdot 4 = \frac{2\pi}{T} \]

### Step 6: Solve for the time period \( T \)
Rearranging the equation gives:
\[ T = \frac{2\pi}{4\pi} = \frac{1}{2} \text{ seconds} = 0.5 \text{ seconds} \]

### Final Answers
- Wavelength \( \lambda = 10 \text{ cm} \)
- Periodic time \( T = 0.5 \text{ seconds} \)