The equation of a stationary a stationary wave is represented by
`y=4sin((pi)/(6)x)(cos20pit)`
when x and y are in cm and t is in second.
Wavelength of the component waves is

Options

A. 4 cm
B. 20 cm
C. 12 cm
D. 6 cm

Answer

To find the wavelength of the component waves in the given stationary wave equation \( y = 4 \sin\left(\frac{\pi}{6} x\right) \cos(20 \pi t) \), we can follow these steps:

### Step 1: Identify the wave number \( k \)
The given equation can be compared to the general form of a stationary wave:
\[ y = 2a \sin(kx) \cos(\omega t) \]
From the equation \( y = 4 \sin\left(\frac{\pi}{6} x\right) \cos(20 \pi t) \), we can see that:
\[ k = \frac{\pi}{6} \]

### Step 2: Relate wave number \( k \) to wavelength \( \lambda \)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( k \):
\[ \frac{\pi}{6} = \frac{2\pi}{\lambda} \]

### Step 3: Solve for \( \lambda \)
To find \( \lambda \), we can rearrange the equation:
\[ \lambda = \frac{2\pi}{\frac{\pi}{6}} \]
This simplifies to:
\[ \lambda = 2\pi \cdot \frac{6}{\pi} \]
\[ \lambda = 12 \text{ cm} \]

### Conclusion
The wavelength of the component waves is \( \lambda = 12 \) cm.

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