The displacement `y` of a partcle in a medium can be expressed as,
`y = 10^(-6) sin ((100t + 20x + (pi)/(4))m` where `t` is in second and `x` in meter. The speed of the wave is

Options

A. (a)2000m/s
B. (b)50m/s
C. (c)20m/s
D. (d)5m/s

Explanation

To find the speed of the wave given the displacement equation, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given equation**:
The displacement of the particle in the medium is given by:
\[ y = 10^{-6} \sin(100t + 20x + \frac{\pi}{4}) \]
where \( t \) is in seconds and \( x \) is in meters.

2. **Compare with the standard wave equation**:
The standard form of a wave equation is:
\[ y = A \sin(\omega t + kx + \phi) \]
Here, \( A \) is the amplitude, \( \omega \) is the angular frequency, \( k \) is the wave number, and \( \phi \) is the phase constant.

3. **Extract parameters from the equation**:
From the given equation, we can identify:
- \( \omega = 100 \) (angular frequency)
- \( k = 20 \) (wave number)

4. **Calculate the speed of the wave**:
The speed \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values we found:
\[ v = \frac{100}{20} = 5 \, \text{m/s} \]

5. **Final answer**:
The speed of the wave is:
\[ v = 5 \, \text{m/s} \]