y(x, t) equation of a lingitudinal wave is given as
`y = 10^(-2)2(pi)[1000t+(50)/(17)x]` (All Si units
If density of the gas is `10^(-3)kg//m^(3)`, find the pressure amplitube

Options

A. 200.62`N//m^(2)`
B. 421.24`N//m^(2)`
C. 100.26 `N//m^(2)`
D. 21.36 `N//m^(2)`

Answer

To find the pressure amplitude of the longitudinal wave given by the equation \( y(x, t) = 10^{-2} \cdot 2\pi \left(1000t + \frac{50}{17}x\right) \) with a density of the gas \( \rho = 10^{-3} \, \text{kg/m}^3 \), we can follow these steps:

### Step 1: Identify the wave speed
The wave speed \( v \) for a longitudinal wave can be determined from the wave equation. The general form of a longitudinal wave can be expressed as:
\[ y(x, t) = A \sin(kx - \omega t) \]
In our case, we can identify the angular frequency \( \omega \) and the wave number \( k \) from the equation given.

From the equation:
\[ y = 10^{-2} \cdot 2\pi \left(1000t + \frac{50}{17}x\right) \]
we can see that:
- The coefficient of \( t \) (which is \( 1000 \)) corresponds to \( \omega \).
- The coefficient of \( x \) (which is \( \frac{50}{17} \)) corresponds to \( k \).

### Step 2: Calculate the wave speed
The wave speed \( v \) is given by the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ \omega = 1000 \, \text{rad/s}, \quad k = \frac{50}{17} \, \text{rad/m} \]
Calculating \( v \):
\[ v = \frac{1000}{\frac{50}{17}} = 1000 \cdot \frac{17}{50} = 340 \, \text{m/s} \]

### Step 3: Relate pressure amplitude to wave speed and density
The relationship between the pressure amplitude \( P \) and wave speed \( v \) is given by:
\[ v = \sqrt{\frac{P}{\rho}} \]
where \( P \) is the pressure amplitude and \( \rho \) is the density of the medium.

### Step 4: Solve for pressure amplitude
Rearranging the equation to solve for \( P \):
\[ P = \rho v^2 \]
Substituting the values:
\[ \rho = 10^{-3} \, \text{kg/m}^3, \quad v = 340 \, \text{m/s} \]
Calculating \( P \):
\[ P = 10^{-3} \cdot (340)^2 = 10^{-3} \cdot 115600 = 115.6 \, \text{N/m}^2 \]

### Step 5: Conclusion
Thus, the pressure amplitude of the longitudinal wave is approximately:
\[ P \approx 115.6 \, \text{N/m}^2 \]