A plane longitudinal wave having angular frequency `omega` = 500 rad /sec is travelling in positive x-direction in a medium of density `rho` =1 kg/m and bulk modulus `4 xx 10^4 N//m^2` . The loudness at a point in the medium is observed to be 20 dB. Assuming at x = 0 initial phase of the medium particles to be zero, find the equation of the wave

Options

A. `y = 2 xx 10^(-9) sin (500 t - (5x)/(2) )`
B. `y = 3 xx 10^(-9)sin (500 t + (5x)/(2))`
C. `y = 3 xx 10^(-9)sin (500 t + (5x)/(2))`
D. `y = 2 xx 10^(-9) sin (5000 t - (5x)/(2))`

Answer

To find the equation of the wave, we will follow these steps:

### Step 1: Identify the given parameters
- Angular frequency, \( \omega = 500 \, \text{rad/s} \)
- Density of the medium, \( \rho = 1 \, \text{kg/m}^3 \)
- Bulk modulus, \( B = 4 \times 10^4 \, \text{N/m}^2 \)
- Loudness, \( L = 20 \, \text{dB} \)

### Step 2: Calculate the velocity of the wave
The velocity \( v \) of the wave in a medium can be calculated using the formula:
\[ v = \sqrt{\frac{B}{\rho}} \]
Substituting the values:
\[ v = \sqrt{\frac{4 \times 10^4}{1}} = \sqrt{4 \times 10^4} = 200 \, \text{m/s} \]

### Step 3: Convert loudness to intensity
The loudness in decibels is related to intensity by the formula:
\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]
Where \( I_0 \) is the reference intensity, typically \( 10^{-12} \, \text{W/m}^2 \). Rearranging gives:
\[ \frac{I}{I_0} = 10^{\frac{L}{10}} \implies I = I_0 \cdot 10^{\frac{L}{10}} = 10^{-12} \cdot 10^{2} = 10^{-10} \, \text{W/m}^2 \]

### Step 4: Relate intensity to amplitude
The intensity \( I \) of a wave is given by:
\[ I = \frac{\omega^2 A^2 \rho}{2} \]
Where \( A \) is the amplitude. Rearranging for \( A^2 \):
\[ A^2 = \frac{2I}{\omega^2 \rho} \]
Substituting the known values:
\[ A^2 = \frac{2 \times 10^{-10}}{(500)^2 \times 1} = \frac{2 \times 10^{-10}}{250000} = 8 \times 10^{-15} \]
Thus,
\[ A = \sqrt{8 \times 10^{-15}} = 2 \times 10^{-7} \, \text{m} \]

### Step 5: Calculate the wave number \( k \)
The wave number \( k \) is given by:
\[ k = \frac{\omega}{v} \]
Substituting the values:
\[ k = \frac{500}{200} = 2.5 \, \text{rad/m} \]

### Step 6: Write the equation of the wave
The general equation for a longitudinal wave traveling in the positive x-direction is:
\[ y = A \sin(\omega t - kx) \]
Substituting the values of \( A \), \( \omega \), and \( k \):
\[ y = 2 \times 10^{-7} \sin(500t - 2.5x) \]

### Final Answer
The equation of the wave is:
\[ y = 2 \times 10^{-7} \sin(500t - 2.5x) \]

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