The equation `y=4 + 2 sin (6t - 3x)` represents a wave motion with

Options

A. amplitude 6 units
B. amplitude 2 units
C. wave speed 2 units
D. wave speed `1//2` units

Answer

To analyze the wave motion represented by the equation \( y = 4 + 2 \sin(6t - 3x) \), we will identify the key characteristics of the wave, including its amplitude and wave velocity.

### Step-by-Step Solution:

1. **Identify the General Form of the Wave Equation**:
The general form of a sinusoidal wave can be expressed as:
\[ y = k + A \sin(\omega t - kx) \]
where:
- \( k \) is the vertical shift (also known as the equilibrium position),
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

2. **Extract the Parameters from the Given Equation**:
From the equation \( y = 4 + 2 \sin(6t - 3x) \):
- The vertical shift \( k = 4 \),
- The amplitude \( A = 2 \),
- The angular frequency \( \omega = 6 \),
- The wave number \( k = 3 \).

3. **Determine the Amplitude**:
The amplitude of the wave is given by the coefficient of the sine function:
\[ A = 2 \]
Thus, the amplitude of the wave is 2 units.

4. **Calculate the Wave Velocity**:
The wave velocity \( v \) can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values we found:
\[ v = \frac{6}{3} = 2 \]
Therefore, the wave velocity is 2 units.

### Final Results:
- **Amplitude**: 2 units
- **Wave Velocity**: 2 units