A string of length l along x-axis is fixed at both ends and is vibrating in second harmonic. If at `t=0`, `y=2.5mm` for incident wave, the equation of standing wave is (`T` is tension and `mu` is linear density)

Options

A. `(2.5mm)sin((2pi)/(l)x)cos(2pisqrt(((T)/(mu l^(2)))t))`
B. `(5mm)sin((pi)/(l)x)cos2pit`
C. `(5mm)sin((2pi)/(l)x)cos(2pisqrt(((T)/(mu l^(2)))t))`
D. `(5mm)cos((2pi)/(l)x)cos(2pisqrt(((T)/(mu l^(2)))t))`

Explanation

To find the equation of the standing wave for a string of length \( l \) vibrating in its second harmonic, we can follow these steps:

### Step 1: Understand the Harmonic Mode
In the second harmonic, the string will have two nodes (at the ends) and one antinode (in the middle). The wavelength \( \lambda \) of the second harmonic is equal to the length of the string \( l \).

### Step 2: Relate Wavelength to String Length
For a string fixed at both ends vibrating in the second harmonic:
\[ L = \lambda \]
Thus, we can express the wavelength as:
\[ \lambda = l \]

### Step 3: Determine Wave Number \( k \)
The wave number \( k \) is given by:
\[ k = \frac{2\pi}{\lambda} \]
Substituting \( \lambda = l \):
\[ k = \frac{2\pi}{l} \]

### Step 4: Determine Angular Frequency \( \omega \)
The angular frequency \( \omega \) can be expressed in terms of tension \( T \) and linear density \( \mu \) as:
\[ \omega = 2\pi f = 2\pi \sqrt{\frac{T}{\mu}} \cdot \frac{1}{l} \]

### Step 5: Write the Equation of the Standing Wave
The general form of the equation for a standing wave on a string fixed at both ends is:
\[ y(x, t) = 2A \sin(kx) \cos(\omega t) \]
Where \( A \) is the amplitude of the wave. Given that at \( t = 0 \), \( y = 2.5 \, \text{mm} \), we can set:
\[ 2A = 2.5 \, \text{mm} \implies A = 1.25 \, \text{mm} \]

### Step 6: Substitute Values into the Equation
Now we can substitute \( A \), \( k \), and \( \omega \) into the standing wave equation:
\[ y(x, t) = 2(1.25 \, \text{mm}) \sin\left(\frac{2\pi}{l} x\right) \cos\left(\sqrt{\frac{T}{\mu}} \cdot \frac{2\pi}{l} t\right) \]

### Final Equation
Thus, the final equation of the standing wave is:
\[ y(x, t) = 2.5 \, \text{mm} \sin\left(\frac{2\pi}{l} x\right) \cos\left(\sqrt{\frac{T}{\mu}} \cdot \frac{2\pi}{l} t\right) \]