The phase difference between the vibrations of two medium particles due to the transmission of a wave is `2pi//3`.The distance between the particles is 15 cm.Determine the wavelength of the wave.

Answer

To solve the problem, we need to find the wavelength of the wave given the phase difference and the distance between two medium particles. Let's break it down step by step.

### Step 1: Understand the relationship between phase difference, distance, and wavelength
The phase difference (\( \Delta \phi \)) between two particles in a wave can be expressed using the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
where:
- \( \Delta \phi \) is the phase difference,
- \( \lambda \) is the wavelength,
- \( \Delta x \) is the distance between the two particles.

### Step 2: Rearrange the formula to find the wavelength
We can rearrange the formula to solve for the wavelength (\( \lambda \)):
\[ \lambda = \frac{2\pi}{\Delta \phi} \Delta x \]

### Step 3: Substitute the known values
We know:
- The phase difference \( \Delta \phi = \frac{2\pi}{3} \)
- The distance \( \Delta x = 15 \, \text{cm} = 0.15 \, \text{m} \) (converting centimeters to meters for standard SI units)

Now, substituting these values into the rearranged formula:
\[ \lambda = \frac{2\pi}{\frac{2\pi}{3}} \times 0.15 \]

### Step 4: Simplify the expression
Calculating the fraction:
\[ \lambda = \frac{2\pi \times 3}{2\pi} \times 0.15 \]
The \( 2\pi \) cancels out:
\[ \lambda = 3 \times 0.15 \]
\[ \lambda = 0.45 \, \text{m} \]

### Step 5: Convert to centimeters
Since the question asks for the wavelength in centimeters:
\[ \lambda = 0.45 \, \text{m} = 45 \, \text{cm} \]

### Final Answer:
The wavelength of the wave is \( 45 \, \text{cm} \).

---