A string of length L fixed at both ends vibrates in its fundamental mode at a frequency v and a maximum amplitude A. (a) Find the wavelength and the wave number k. (b) Take the origin at one end of the string and the X-axis along the string. Take the Y-axis along the direction of the displacement. Take t = 0 at the instant when the middle point of the string passes through its mean position and is going towards the positive y-direction. Write the equation describing the standing wave.

Explanation

To solve the problem step by step, we will break it down into two parts as requested: (a) finding the wavelength and wave number, and (b) writing the equation for the standing wave.

### Part (a): Finding Wavelength and Wave Number

1. **Understanding the Fundamental Mode**:
In the fundamental mode of vibration for a string fixed at both ends, the length of the string (L) is equal to half the wavelength (λ). This is because there is one complete wave (one antinode and two nodes) fitting into the length of the string.

\[ L = \frac{\lambda}{2} \]

2. **Solving for Wavelength (λ)**:
Rearranging the equation gives us:

\[ \lambda = 2L \]

So, the wavelength of the wave is \( \lambda = 2L \).

3. **Finding the Wave Number (k)**:
The wave number (k) is defined as:

\[ k = \frac{2\pi}{\lambda} \]

Substituting the value of λ we found:

\[ k = \frac{2\pi}{2L} = \frac{\pi}{L} \]

Thus, the wave number is \( k = \frac{\pi}{L} \).

### Part (b): Writing the Equation for the Standing Wave

1. **General Form of the Standing Wave**:
The general equation for a standing wave can be written as:

\[ y(x, t) = A \sin(kx) \cos(\omega t + \phi) \]

where:
- \( A \) is the maximum amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

2. **Setting the Initial Conditions**:
We are told to take the origin at one end of the string (x = 0) and that at \( t = 0 \), the middle point of the string (x = L/2) is passing through its mean position and moving in the positive y-direction. This means that at \( t = 0 \):

\[ y\left(\frac{L}{2}, 0\right) = 0 \]

3. **Substituting Values**:
Substitute \( k = \frac{\pi}{L} \) into the standing wave equation:

\[ y\left(x, t\right) = A \sin\left(\frac{\pi}{L} x\right) \cos(\omega t + \phi) \]

4. **Finding the Phase Constant (φ)**:
Since at \( t = 0 \) the middle point (x = L/2) is at the mean position (y = 0):

\[ 0 = A \sin\left(\frac{\pi}{L} \cdot \frac{L}{2}\right) \cos(\phi) \]

The sine term becomes:

\[ \sin\left(\frac{\pi}{2}\right) = 1 \]

Therefore, we have:

\[ 0 = A \cdot 1 \cdot \cos(\phi) \]

Since \( A \neq 0 \), it follows that:

\[ \cos(\phi) = 0 \implies \phi = \frac{\pi}{2} \]

5. **Final Equation**:
Now substituting \( \phi \) back into the equation:

\[ y(x, t) = A \sin\left(\frac{\pi}{L} x\right) \cos\left(\omega t + \frac{\pi}{2}\right) \]

We can express \( \cos\left(\omega t + \frac{\pi}{2}\right) \) as \( -\sin(\omega t) \):

\[ y(x, t) = -A \sin\left(\frac{\pi}{L} x\right) \sin(\omega t) \]

Thus, the final equation for the standing wave is:

\[ y(x, t) = A \sin\left(\frac{\pi}{L} x\right) \cos(2\pi vt) \]