A plane progressive wave propagating through air has an amplitude of `10^(-3)m` and a frequency of 480 Hz. Calculate the energy density and intensity of wave. Given velocity of wave `=340 ms^(-1)` and density of air `= 1.29"kg m"^(-3)`.

Answer

To solve the problem of calculating the energy density and intensity of a plane progressive wave propagating through air, we can follow these steps:

### Step 1: Write down the given values
- Amplitude (A) = \(10^{-3} \, \text{m}\)
- Frequency (f) = \(480 \, \text{Hz}\)
- Velocity of wave (v) = \(340 \, \text{m/s}\)
- Density of air (ρ) = \(1.29 \, \text{kg/m}^3\)

### Step 2: Calculate the angular frequency (ω)
The angular frequency (ω) can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2 \times 3.14 \times 480 \approx 3015.84 \, \text{rad/s} \]

### Step 3: Calculate the wave number (k)
The wave number (k) can be calculated using the formula:
\[ k = \frac{\omega}{v} \]
Substituting the values of ω and v:
\[ k = \frac{3015.84}{340} \approx 8.86 \, \text{rad/m} \]

### Step 4: Calculate the energy density (E)
The energy density (E) of the wave can be calculated using the formula:
\[ E = \frac{1}{2} \rho A^2 \omega^2 \]
Substituting the values:
\[ E = \frac{1}{2} \times 1.29 \times (10^{-3})^2 \times (3015.84)^2 \]
Calculating this step-by-step:
1. Calculate \( (10^{-3})^2 = 10^{-6} \)
2. Calculate \( (3015.84)^2 \approx 9095488.4 \)
3. Now substitute:
\[ E = \frac{1}{2} \times 1.29 \times 10^{-6} \times 9095488.4 \approx 5.86 \, \text{J/m}^3 \]

### Step 5: Calculate the intensity (I)
The intensity (I) of the wave can be calculated using the formula:
\[ I = E \cdot v \]
Substituting the values of E and v:
\[ I = 5.86 \times 340 \approx 1992.4 \, \text{W/m}^2 \]

### Final Results
- Energy Density (E) = \(5.86 \, \text{J/m}^3\)
- Intensity (I) = \(1992.4 \, \text{W/m}^2\)

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