Dentermine the speed of sound waves in water , and find the wavelength of a wave having a frequency of `242 H_(Z)` . Take `B_(water) = 2xx10^(9) Pa`.

Explanation

To determine the speed of sound waves in water and find the wavelength of a wave with a frequency of 242 Hz, we will follow these steps:

### Step 1: Identify the given values
- Bulk modulus of water, \( B = 2 \times 10^9 \, \text{Pa} \)
- Frequency of the wave, \( f = 242 \, \text{Hz} \)
- Density of water, \( \rho = 10^3 \, \text{kg/m}^3 \)

### Step 2: Use the formula for the speed of sound
The speed of sound in a medium is given by the formula:
\[ v = \sqrt{\frac{B}{\rho}} \]
where \( B \) is the bulk modulus and \( \rho \) is the density of the medium.

### Step 3: Substitute the values into the formula
Substituting the values of \( B \) and \( \rho \):
\[ v = \sqrt{\frac{2 \times 10^9 \, \text{Pa}}{10^3 \, \text{kg/m}^3}} \]

### Step 4: Calculate the speed of sound
Calculating the value inside the square root:
\[ v = \sqrt{2 \times 10^6} \, \text{m/s} \]
Now, calculating the square root:
\[ v \approx 1414 \, \text{m/s} \]

### Step 5: Find the wavelength using the frequency
The wavelength \( \lambda \) can be calculated using the formula:
\[ \lambda = \frac{v}{f} \]
Substituting the values of \( v \) and \( f \):
\[ \lambda = \frac{1414 \, \text{m/s}}{242 \, \text{Hz}} \]

### Step 6: Calculate the wavelength
Calculating the wavelength:
\[ \lambda \approx 5.84 \, \text{m} \]

### Final Results
- Speed of sound in water: \( v \approx 1414 \, \text{m/s} \)
- Wavelength of the wave: \( \lambda \approx 5.84 \, \text{m} \)

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