A transverse harmonic wave is described by `y(x, t) = 0.05 sin pi(4t - 0.03 x)` (x is in metre). What will be the instantaneous phase difference between two points separated by 20 cm ?

Options

A. `(pi)/(200)` rad
B. `(pi)/(800)` rad
C. `(6 pi)/(1000)` rad
D. `(pi)/(670)` rad

Explanation

To find the instantaneous phase difference between two points separated by 20 cm in the given transverse harmonic wave described by the equation \( y(x, t) = 0.05 \sin(\pi(4t - 0.03x)) \), we can follow these steps:

### Step 1: Identify the wave equation parameters
The general form of a transverse harmonic wave is given by:
\[ y(x, t) = A \sin(\omega t - kx) \]
From the given equation, we can identify:
- \( \omega = 4\pi \) (angular frequency)
- \( k = 0.03\pi \) (wave number)

### Step 2: Calculate the wavelength (\( \lambda \))
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ 0.03\pi = \frac{2\pi}{\lambda} \]
Solving for \( \lambda \):
\[ \lambda = \frac{2\pi}{0.03\pi} = \frac{2}{0.03} = \frac{200}{3} \text{ m} \]

### Step 3: Determine the phase difference formula
The phase difference \( \Delta \phi \) between two points separated by a distance \( \Delta x \) is given by:
\[ \Delta \phi = k \Delta x \]

### Step 4: Convert the distance to meters
The distance between the two points is given as 20 cm. We need to convert this to meters:
\[ \Delta x = 20 \text{ cm} = 0.2 \text{ m} \]

### Step 5: Calculate the phase difference
Now we can substitute \( k \) and \( \Delta x \) into the phase difference formula:
\[ \Delta \phi = k \Delta x = (0.03\pi)(0.2) \]
Calculating this gives:
\[ \Delta \phi = 0.03\pi \times 0.2 = 0.006\pi \]

### Step 6: Final Result
Thus, the instantaneous phase difference between the two points separated by 20 cm is:
\[ \Delta \phi = 0.006\pi \text{ radians} \]

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