The equation of a longitudinal standing wave due to superposition of the progressive waves producted by two sources of sound is `s = -20 sin 10 pix sin 100 pit` where `s` is the displacement from mean position measured in `mm, x` is in meters and is in seconds. The specific gravity of the medium is `10^(-3)`. Density of water `= 10^(3)kg//m^(3)`. Find :
(a) Wavelength, frequency and velocity of the progressive waves.
(b) Bulk modulus of the medium and the pressure amplitude.
(c) Minimum distance between pressure antinode and a displacement anotinode.
(d) Intensity at the displacement nodes.

Explanation

To solve the given problem step by step, we will break it down into the four parts as specified in the question.

### Given:
The equation of the standing wave is:
\[ s = -20 \sin(10 \pi x) \sin(100 \pi t) \]
Where:
- \( s \) is the displacement in mm,
- \( x \) is in meters,
- \( t \) is in seconds.

The specific gravity of the medium is \( 10^{-3} \) and the density of water is \( 10^{3} \, \text{kg/m}^3 \).

### (a) Wavelength, Frequency, and Velocity of the Progressive Waves

1. **Frequency Calculation:**
The angular frequency \( \omega \) can be identified from the term \( \sin(100 \pi t) \).
\[ \omega = 100 \pi \]
The frequency \( f \) is given by:
\[ f = \frac{\omega}{2\pi} = \frac{100 \pi}{2\pi} = 50 \, \text{Hz} \]

2. **Wavelength Calculation:**
The wave number \( k \) can be identified from the term \( \sin(10 \pi x) \).
\[ k = 10 \pi \]
The wavelength \( \lambda \) is given by:
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{10\pi} = \frac{1}{5} \, \text{m} = 20 \, \text{cm} \]

3. **Velocity Calculation:**
The velocity \( v \) of the wave is given by:
\[ v = f \cdot \lambda = 50 \cdot \frac{1}{5} = 10 \, \text{m/s} \]

### (b) Bulk Modulus of the Medium and Pressure Amplitude

1. **Bulk Modulus Calculation:**
The bulk modulus \( B \) is given by:
\[ B = \rho v^2 \]
The density \( \rho \) of the medium can be calculated using specific gravity:
\[ \rho = 10^{-3} \cdot 10^{3} = 1 \, \text{kg/m}^3 \]
Now substituting the values:
\[ B = 1 \cdot (10)^2 = 100 \, \text{N/m}^2 \]

2. **Pressure Amplitude Calculation:**
The pressure amplitude \( P_0 \) is given by:
\[ P_0 = B \cdot k \cdot S_0 \]
Where \( S_0 = 20 \, \text{mm} = 20 \times 10^{-3} \, \text{m} \).
\[ P_0 = 100 \cdot 10\pi \cdot (20 \times 10^{-3}) = 25 \, \text{N/m}^2 \]

### (c) Minimum Distance Between Pressure Antinode and Displacement Antinode

The minimum distance between a pressure antinode and a displacement antinode is given by:
\[ \text{Distance} = \frac{\lambda}{4} = \frac{20 \, \text{cm}}{4} = 5 \, \text{cm} \]

### (d) Intensity at the Displacement Nodes

The intensity \( I \) at the displacement nodes is given by:
\[ I = \frac{P_0^2}{2 \rho v} \]
Substituting the values:
\[ I = \frac{(25)^2}{2 \cdot 1 \cdot 10} = \frac{625}{20} = 31.25 \, \text{W/m}^2 \]

### Summary of Results:
- (a) Wavelength = 20 cm, Frequency = 50 Hz, Velocity = 10 m/s
- (b) Bulk Modulus = 100 N/m², Pressure Amplitude = 25 N/m²
- (c) Minimum Distance = 5 cm
- (d) Intensity = 31.25 W/m²