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

Options

A. `2000 ms ^-1`
B. `5 ms^-1`
C. `20 ms^-1`
D. `5 pi s^(-1)`

Answer

To find the speed of the wave represented by the equation \( y = 10^{-6} \sin(100t - 20x + \frac{\pi}{4}) \), we can follow these steps:

### Step 1: Identify the Wave Equation
The given wave equation is in the form:
\[ y = A \sin(\omega t - kx + \phi) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( \phi \) is the phase constant.

### Step 2: Extract Values from the Equation
From the equation \( y = 10^{-6} \sin(100t - 20x + \frac{\pi}{4}) \), we can identify:
- \( \omega = 100 \) (angular frequency in rad/s)
- \( k = 20 \) (wave number in rad/m)

### Step 3: 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 of \( \omega \) and \( k \):
\[ v = \frac{100}{20} = 5 \, \text{m/s} \]

### Step 4: Determine the Direction of the Wave
In the equation \( y = 10^{-6} \sin(100t - 20x + \frac{\pi}{4}) \), the term \( -20x \) indicates that the wave is traveling in the positive x-direction.

### Final Answer
The speed of the wave is:
\[ \boxed{5 \, \text{m/s}} \]