`y_1 = 8 sin (omegat - kx) and y_2 = 6 sin (omegat + kx)` are two waves travelling in a string of area of cross-section s and density rho. These two waves are superimposed to produce a standing wave.
(a) Find the energy of the standing wave between two consecutive nodes.
(b) Find the total amount of energy crossing through a node per second.

Explanation

To solve the problem step by step, we will break it down into two parts as stated in the question.

### Part (a): Find the energy of the standing wave between two consecutive nodes.

1. **Identify the Wave Functions**:
We have two waves given as:
\[ y_1 = 8 \sin(\omega t - kx) \]
\[ y_2 = 6 \sin(\omega t + kx) \]

2. **Calculate the Energy Density**:
The energy density \( U \) of a wave is given by:
\[ U = \frac{1}{2} \rho \omega^2 A^2 \]
where \( A \) is the amplitude of the wave.

3. **Calculate Energy Density for Each Wave**:
- For \( y_1 \):
\[ U_1 = \frac{1}{2} \rho \omega^2 (8)^2 = \frac{1}{2} \rho \omega^2 \cdot 64 = 32 \rho \omega^2 \]
- For \( y_2 \):
\[ U_2 = \frac{1}{2} \rho \omega^2 (6)^2 = \frac{1}{2} \rho \omega^2 \cdot 36 = 18 \rho \omega^2 \]

4. **Total Energy Density**:
The total energy density \( U_{total} \) of the standing wave is the sum of the energy densities of both waves:
\[ U_{total} = U_1 + U_2 = 32 \rho \omega^2 + 18 \rho \omega^2 = 50 \rho \omega^2 \]

5. **Distance Between Two Consecutive Nodes**:
The distance between two consecutive nodes in a standing wave is given by:
\[ \text{Distance} = \frac{\pi}{k} \]

6. **Calculate Energy Between Two Nodes**:
The energy \( E \) between two consecutive nodes can be calculated as:
\[ E = U_{total} \times \text{Volume} \]
The volume \( V \) between two nodes is given by:
\[ V = A \times \text{Distance} = S \times \frac{\pi}{k} \]
Therefore,
\[ E = U_{total} \times S \times \frac{\pi}{k} = 50 \rho \omega^2 \times S \times \frac{\pi}{k} \]

### Part (b): Find the total amount of energy crossing through a node per second.

1. **Power Calculation**:
The power \( P \) is given by the formula:
\[ P = \frac{1}{2} \rho \omega^2 A^2 \times v \]
where \( v \) is the wave speed, which can be expressed as \( v = \frac{\omega}{k} \).

2. **Calculate Power for the Standing Wave**:
The amplitude of the resultant standing wave can be calculated using the principle of superposition. The resultant amplitude \( A \) is:
\[ A = 8 + 6 = 14 \]
Thus, the power becomes:
\[ P = \frac{1}{2} \rho \omega^2 (14)^2 \times \frac{\omega}{k} = \frac{1}{2} \rho \omega^2 \cdot 196 \cdot \frac{\omega}{k} = \frac{98 \rho \omega^3}{k} \]

### Final Answers:
- (a) The energy of the standing wave between two consecutive nodes is:
\[ E = 50 \rho \omega^2 S \frac{\pi}{k} \]
- (b) The total amount of energy crossing through a node per second is:
\[ P = \frac{98 \rho \omega^3}{k} \]