A sound wave with an amplitude of `3cm` starts towards right from origin and gets reflected at a rigid wall after a second. If the velocity of the wave is `340ms^(-1)` and it has a wavelength of `2m`, the equations of incident and reflected waves respectively are :

Options

A. `y=3xx10^(-2)sinpi(340t-x)`, `y=-3xx10^(-2)sinpi(340t+x)` towards left
B. `y=3xx10^(-2)sinpi(340t+x)`, `y=-3xx10^(-2)sinpi(340t+x)` towards left
C. `y=3xx10^(-2)sinpi(340t-x)`, `y=-3xx10^(-2)sinpi(340t-x)` towards left
D. `y=3xx10^(-2)sinpi(340t-x)`, `y=3xx10^(-2)sinpi(340t+x)` towards left

Explanation

To find the equations of the incident and reflected sound waves, we can follow these steps:

### Step 1: Identify Given Information
- Amplitude (A) = 3 cm = 3 × 10^(-2) m
- Velocity (v) = 340 m/s
- Wavelength (λ) = 2 m

### Step 2: Calculate Wave Number (k)
The wave number (k) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of λ:
\[ k = \frac{2\pi}{2} = \pi \, \text{(rad/m)} \]

### Step 3: Calculate Angular Frequency (ω)
The angular frequency (ω) can be calculated using the relationship between velocity, wavelength, and frequency:
\[ v = f \lambda \]
From this, we can express frequency (f) as:
\[ f = \frac{v}{\lambda} = \frac{340}{2} = 170 \, \text{Hz} \]
Now, we can find ω:
\[ \omega = 2\pi f = 2\pi \times 170 = 340\pi \, \text{(rad/s)} \]

### Step 4: Write the Equation of the Incident Wave
The standard form of the wave equation is:
\[ y = A \sin(\omega t - kx) \]
Substituting the values of A, ω, and k:
\[ y_{\text{incident}} = 3 \times 10^{-2} \sin(340\pi t - \pi x) \]

### Step 5: Write the Equation of the Reflected Wave
When a wave reflects off a rigid wall, it undergoes a phase shift of π. Therefore, the equation of the reflected wave will be:
\[ y_{\text{reflected}} = A \sin(\omega t + kx + \pi) \]
Using the identity \( \sin(x + \pi) = -\sin(x) \):
\[ y_{\text{reflected}} = -3 \times 10^{-2} \sin(340\pi t + \pi x) \]

### Final Equations
Thus, the equations of the incident and reflected waves are:
1. **Incident Wave**: \( y_{\text{incident}} = 3 \times 10^{-2} \sin(340\pi t - \pi x) \)
2. **Reflected Wave**: \( y_{\text{reflected}} = -3 \times 10^{-2} \sin(340\pi t + \pi x) \)