A standing wave is formed by two harmonic waves, `y_1 = A sin (kx-omegat) and y_2 = A sin (kx + omegat)` travelling on a string in opposite directions. Mass density is 'ρ' and area of cross section is S. Find the total mechanical energy between two adjacent nodes on the string.

Explanation

To solve the problem of finding the total mechanical energy between two adjacent nodes on a string where a standing wave is formed by two harmonic waves, we will follow these steps:

### Step 1: Understand the standing wave formation
The two waves are given as:
- \( y_1 = A \sin(kx - \omega t) \)
- \( y_2 = A \sin(kx + \omega t) \)

These waves travel in opposite directions and combine to form a standing wave.

### Step 2: Determine the distance between two adjacent nodes
The distance between two adjacent nodes in a standing wave is given by:
\[ \text{Distance between nodes} = \frac{\lambda}{2} \]
We know that the wavelength \( \lambda \) is related to the wave number \( k \) by:
\[ \lambda = \frac{2\pi}{k} \]
Thus, the distance between two nodes becomes:
\[ \text{Distance} = \frac{1}{2} \cdot \frac{2\pi}{k} = \frac{\pi}{k} \]

### Step 3: Calculate the volume between two nodes
The volume \( V \) between two adjacent nodes can be calculated using the cross-sectional area \( S \) and the length between the nodes:
\[ V = S \cdot \text{Length} = S \cdot \frac{\pi}{k} \]

### Step 4: Determine the energy density of the wave
The energy density \( u \) in a wave is given by:
\[ u = \frac{1}{2} \rho v^2 + \frac{1}{2} T \left( \frac{\partial y}{\partial x} \right)^2 \]
For a standing wave, the maximum value of the displacement \( \frac{\partial y}{\partial x} \) is related to the amplitude \( A \) and the wave number \( k \):
\[ \frac{\partial y}{\partial x} = A k \cos(\omega t) \]
Thus, the energy density can be simplified to:
\[ u = \frac{1}{2} \rho v^2 + \frac{1}{2} T (A k \cos(\omega t))^2 \]
Where \( v \) is the wave speed given by \( v = \sqrt{\frac{T}{\rho}} \).

### Step 5: Calculate the total mechanical energy between two nodes
The total mechanical energy \( E \) between two adjacent nodes is given by the product of energy density and volume:
\[ E = u \cdot V \]
Substituting the expressions we derived:
\[ E = u \cdot \left(S \cdot \frac{\pi}{k}\right) \]

### Final Expression
The total mechanical energy can be expressed as:
\[ E = \left(\frac{1}{2} T (A k)^2 \cdot \frac{\pi}{k} \cdot S\right) \]