The equation of a plane progressive wave is `y=0.04sin4pi[t-(x)/(20)]`. When it is reflected at a denser medium (medium with lesser wave velocity) at x=0, intensity of reflected wave is 81% of that of the incident wave. The equation of the relfected wave is:

Options

A. `y=0.04sin4pi[t-(x)/(20)]`
B. `y=0.04sin4pi[t+(x)/(20)]`
C. `y=+0.036sin4pi[t+(x)/(20)]`
D. `y=-0.036sin4pi[t+(x)/(20)]`

Explanation

To find the equation of the reflected wave, we will follow these steps:

### Step 1: Identify the parameters of the incident wave.
The given equation of the incident wave is:
\[ y = 0.04 \sin\left(4\pi\left(t - \frac{x}{20}\right)\right) \]
We can rewrite this as:
\[ y = 0.04 \sin\left(4\pi t - \frac{4\pi}{20} x\right) \]
From this, we can identify:
- Amplitude \( A_1 = 0.04 \)
- Angular frequency \( \omega = 4\pi \)
- Wave number \( k = \frac{4\pi}{20} = \frac{\pi}{5} \)

### Step 2: Determine the direction of the reflected wave.
The incident wave is traveling in the positive x-direction. When it reflects off a denser medium, it will travel in the negative x-direction.

### Step 3: Phase change upon reflection.
When a wave reflects from a denser medium, there is a phase change of \( \pi \). Therefore, the phase of the reflected wave will be:
\[ \phi = \pi \]

### Step 4: Calculate the amplitude of the reflected wave.
The intensity of the reflected wave is given as 81% of the incident wave's intensity. The intensity \( I \) is proportional to the square of the amplitude \( A \):
\[ I \propto A^2 \]
Let the amplitude of the reflected wave be \( A_2 \). Then:
\[ \frac{A_2^2}{A_1^2} = 0.81 \]
Taking the square root:
\[ \frac{A_2}{A_1} = \sqrt{0.81} = 0.9 \]
Thus:
\[ A_2 = 0.9 A_1 = 0.9 \times 0.04 = 0.036 \]

### Step 5: Write the equation of the reflected wave.
The general form of the wave equation for a wave traveling in the negative x-direction is:
\[ y = A \sin(\omega t + kx + \phi) \]
Substituting the values we have:
- Amplitude \( A_2 = 0.036 \)
- Angular frequency \( \omega = 4\pi \)
- Wave number \( k = \frac{\pi}{5} \)
- Phase \( \phi = \pi \)

The equation becomes:
\[ y = 0.036 \sin\left(4\pi t + \frac{\pi}{5} x + \pi\right) \]

### Step 6: Simplify the equation.
Using the identity \( \sin(x + \pi) = -\sin(x) \):
\[ y = -0.036 \sin\left(4\pi t + \frac{\pi}{5} x\right) \]

### Final Answer:
The equation of the reflected wave is:
\[ y = -0.036 \sin\left(4\pi t + \frac{\pi}{5} x\right) \]

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