A child playing with a long rope ties one end holds the other. The rope is stretched taut along the horizontal. The child shakes the end he is holding, up and down, in a sinusoidal manner with amplitude `10 cm` and frequency `3` Hz.
Speed of the wave is `15 m//s` and, at `t=0`, displacement at the child's end is maximum positive. Assuming that there is no wave reflected from the fixed end, so that the waves in the rope are plane progressive waves, answer the following quetions.
(Also assume that the wave propagates along the positive `x-`direction.)
A wave function that describe the wave in the given sutuation is

Options

A. `y=(0.1 m) cos [(2 rad//s)x-(12.5 rad//s)t]`
B. `y=(0.1 m) cos [(1.26rad//s)x-(18.8 rad//s)t]`
C. `y=(0.1 m) cos [(1.5 rad//s)x-(10rad//s)t]`
D. `y=(0.1 m) cos [(1.5 rad//s)x-(4 rad//s)t]`

Explanation

To find the wave function that describes the wave in the given situation, we will follow these steps:

### Step 1: Identify the given parameters
- Amplitude (A) = 10 cm = 0.1 m
- Frequency (f) = 3 Hz
- Speed of the wave (v) = 15 m/s
- At t = 0, displacement at the child's end is maximum positive.

### Step 2: Calculate the angular frequency (ω)
The angular frequency (ω) is given by the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2\pi \times 3 = 6\pi \text{ rad/s} \]

### Step 3: Calculate the wave number (k)
The wave number (k) is calculated using the formula:
\[ k = \frac{\omega}{v} \]
Substituting the values of ω and v:
\[ k = \frac{6\pi}{15} = \frac{2\pi}{5} \text{ rad/m} \]

### Step 4: Determine the wave function
Since the displacement at the child's end is maximum positive at t = 0, we use the cosine function for the wave equation. The general form of the wave function for a wave traveling in the positive x-direction is:
\[ y(x, t) = A \cos(kx - \omega t) \]
Substituting the values of A, k, and ω:
\[ y(x, t) = 0.1 \cos\left(\frac{2\pi}{5}x - 6\pi t\right) \]

### Final Wave Function
Thus, the wave function that describes the wave in the given situation is:
\[ y(x, t) = 0.1 \cos\left(\frac{2\pi}{5}x - 6\pi t\right) \]

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