For brass, bulk modulus and modulus of rigidity are `10.72 xx 10^(10)Nm^(-2)` and `3.6 xx 10^(10)Nm^(-2)`. If the density of brass is `8.4 xx 10^(3) kgm^(-3)`, find the speed of longitudinal and transverse waves in it.

Answer

To find the speed of longitudinal and transverse waves in brass, we can use the following formulas:

1. **Speed of Longitudinal Waves (v_L)**:
\[ v_L = \sqrt{\frac{B}{\rho}} \]
where \( B \) is the bulk modulus and \( \rho \) is the density.

2. **Speed of Transverse Waves (v_T)**:
\[ v_T = \sqrt{\frac{G}{\rho}} \]
where \( G \) is the modulus of rigidity.

### Given Data:
- Bulk Modulus (B) = \( 10.72 \times 10^{10} \, \text{N/m}^2 \)
- Modulus of Rigidity (G) = \( 3.6 \times 10^{10} \, \text{N/m}^2 \)
- Density (\( \rho \)) = \( 8.4 \times 10^{3} \, \text{kg/m}^3 \)

### Step 1: Calculate the Speed of Longitudinal Waves
Using the formula for the speed of longitudinal waves:
\[ v_L = \sqrt{\frac{B}{\rho}} = \sqrt{\frac{10.72 \times 10^{10}}{8.4 \times 10^{3}}} \]

Calculating the denominator:
\[ 8.4 \times 10^{3} = 8400 \, \text{kg/m}^3 \]

Now substituting the values:
\[ v_L = \sqrt{\frac{10.72 \times 10^{10}}{8400}} = \sqrt{1.2738 \times 10^{7}} \approx 3565.5 \, \text{m/s} \]

### Step 2: Calculate the Speed of Transverse Waves
Using the formula for the speed of transverse waves:
\[ v_T = \sqrt{\frac{G}{\rho}} = \sqrt{\frac{3.6 \times 10^{10}}{8.4 \times 10^{3}}} \]

Calculating the denominator:
\[ 8.4 \times 10^{3} = 8400 \, \text{kg/m}^3 \]

Now substituting the values:
\[ v_T = \sqrt{\frac{3.6 \times 10^{10}}{8400}} = \sqrt{4.2857 \times 10^{6}} \approx 2070.0 \, \text{m/s} \]

### Final Answers:
- Speed of Longitudinal Waves (v_L) ≈ **3565.5 m/s**
- Speed of Transverse Waves (v_T) ≈ **2070.0 m/s**