The equation of a plane progressive wave is given by
`y=2cos(100pit-(pix)/(20))`
where x and y are in cm and t is in second. The wavelength of the wave is

Options

A. 20 cm
B. 100 cm
C. 40 cm
D. `(pi)/(20)` cm

Answer

To find the wavelength of the wave given by the equation \( y = 2 \cos(100\pi t - \frac{\pi x}{20}) \), we can follow these steps:

### Step 1: Identify the wave equation format
The general form of a plane progressive wave can be expressed as:
\[ y = A \cos(\omega t - kx) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number.

### Step 2: Extract the wave number \( k \)
From the given equation, we can identify the terms:
\[ \omega t - kx = 100\pi t - \frac{\pi x}{20} \]
Here, we can see that:
- \( \omega = 100\pi \)
- \( k = \frac{\pi}{20} \)

### Step 3: Relate wave number \( k \) to wavelength \( \lambda \)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]

### Step 4: Substitute the value of \( k \)
We have:
\[ \frac{\pi}{20} = \frac{2\pi}{\lambda} \]

### Step 5: Solve for \( \lambda \)
To find \( \lambda \), we can rearrange the equation:
\[ \lambda = \frac{2\pi}{\frac{\pi}{20}} \]
This simplifies to:
\[ \lambda = 2\pi \cdot \frac{20}{\pi} = 40 \text{ cm} \]

### Conclusion
Thus, the wavelength \( \lambda \) of the wave is \( 40 \) cm.

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