A transverse sine wave of amplitude 10 cm and wavelength 200 cm travels from left to right along a long horizontal stretched, string with a speed of 100 cm/s. Take the origin at left end of the string. At time t = 0 the left end of the string is at the origin and is moving downward. Then the equation of the wave will be ( in CGS system )

Options

A. `y = 10 sin (0.01 pi xx - pit)`
B. `y = 10 sin (pit- 0.01 pi xx)`
C. `y = 10 sin (0.02 pi xx- 0.01 pi t)`
D. `y = 10 sin (pi t - 0.02 pi xx)`

Answer

To find the equation of the transverse sine wave, we can follow these steps:

### Step 1: Identify the given parameters
- Amplitude \( A = 10 \) cm
- Wavelength \( \lambda = 200 \) cm
- Speed of the wave \( v = 100 \) cm/s

### Step 2: Write the general form of the wave equation
The general form of a transverse wave traveling in the positive x-direction is given by:
\[ y(x, t) = A \sin(kx - \omega t) \]
where:
- \( A \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

### Step 3: Calculate the wave number \( k \)
The wave number \( k \) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( \lambda \):
\[ k = \frac{2\pi}{200} = \frac{\pi}{100} \text{ cm}^{-1} \]

### Step 4: Calculate the frequency \( f \)
The frequency \( f \) can be calculated using the formula:
\[ f = \frac{v}{\lambda} \]
Substituting the values of \( v \) and \( \lambda \):
\[ f = \frac{100}{200} = 0.5 \text{ Hz} \]

### Step 5: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is given by:
\[ \omega = 2\pi f \]
Substituting the value of \( f \):
\[ \omega = 2\pi \times 0.5 = \pi \text{ rad/s} \]

### Step 6: Write the wave equation
Now we can substitute the values of \( A \), \( k \), and \( \omega \) into the wave equation:
\[ y(x, t) = 10 \sin\left(\frac{\pi}{100} x - \pi t\right) \]

### Step 7: Consider the initial condition
At \( t = 0 \), the left end of the string (at \( x = 0 \)) is moving downward. Since the sine function starts at zero, we need to adjust the equation to reflect that the wave is moving downward at the origin. The downward motion indicates that we should use:
\[ y(x, t) = 10 \sin\left(\frac{\pi}{100} x - \pi t + \frac{\pi}{2}\right) \]
This is equivalent to:
\[ y(x, t) = 10 \cos\left(\frac{\pi}{100} x - \pi t\right) \]
Thus, the final equation of the wave is:
\[ y(x, t) = 10 \cos\left(\frac{\pi}{100} x - \pi t\right) \text{ cm} \]

### Final Answer
The equation of the wave is:
\[ y(x, t) = 10 \cos\left(\frac{\pi}{100} x - \pi t\right) \text{ cm} \]

---