The vibrations of a string fixed at both ends are represented by `y = 16 sin (pi x/15) cos 96 pi t` where x and y are in cm and t in seconds. Then the phase difference between the points at x = 13 cm and X = 16 cm in radian is

Options

A. `pi//5`
B. `pi` rad
C. 0
D. `2pi//5`

Answer

To find the phase difference between the points at \( x = 13 \, \text{cm} \) and \( x = 16 \, \text{cm} \) for the given wave equation \( y = 16 \sin\left(\frac{\pi x}{15}\right) \cos(96 \pi t) \), we can follow these steps:

### Step 1: Identify the phase of the wave
The wave function can be expressed in the form:
\[ y = A \sin(kx) \cos(\omega t) \]
where:
- \( k = \frac{\pi}{15} \) (the wave number),
- \( \omega = 96\pi \) (the angular frequency).

The phase of the wave at any point \( x \) is given by \( \phi = kx \).

### Step 2: Calculate the phase at \( x = 13 \, \text{cm} \)
Substituting \( x = 13 \) into the phase equation:
\[ \phi_{13} = k \cdot 13 = \frac{\pi}{15} \cdot 13 = \frac{13\pi}{15} \]

### Step 3: Calculate the phase at \( x = 16 \, \text{cm} \)
Now substituting \( x = 16 \):
\[ \phi_{16} = k \cdot 16 = \frac{\pi}{15} \cdot 16 = \frac{16\pi}{15} \]

### Step 4: Find the phase difference
The phase difference \( \Delta \phi \) between the two points is given by:
\[ \Delta \phi = \phi_{16} - \phi_{13} \]
Substituting the values we calculated:
\[ \Delta \phi = \frac{16\pi}{15} - \frac{13\pi}{15} = \frac{3\pi}{15} = \frac{\pi}{5} \]

### Final Answer
Thus, the phase difference between the points at \( x = 13 \, \text{cm} \) and \( x = 16 \, \text{cm} \) is:
\[ \Delta \phi = \frac{\pi}{5} \, \text{radians} \]

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