Estimation of frequency of a wave forming a standing wave represented by `y = A sin kx cos t` can be done if the speed and wavelength are known using `speed = "Frequency" xx "wavelength"` . Speed of motion depends on the medium properties namely tension in string and mass per unit length of string . A string may vibrate with different frequencies . The corresponding wavelength should be related to the length of the string by a whole number for a string fixed at both ends . Answer the following questions:
If `y = 10 sin 5 x cos 2 t m` represents a stationary wave then , the possible one of the travelling waves causing this is

Options

A. `y = 10 sin ( 5x - 2 t)`
B. `y = 5 sin ( 2 t - 5x)`
C. `y = 10 sin 2t`
D. `y = 5 cos 5x`

Explanation

To solve the problem, we need to analyze the given standing wave equation and identify the possible traveling wave that could create it. The standing wave is given as:

\[ y = 10 \sin(5x) \cos(2t) \]

### Step 1: Identify the components of the standing wave equation

The general form of a standing wave can be expressed as:

\[ y = 2A \sin(kx) \cos(\omega t) \]

where:
- \( A \) is the amplitude of the constituent waves,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

### Step 2: Compare the given equation with the standard form

From the given equation \( y = 10 \sin(5x) \cos(2t) \), we can identify:
- The coefficient of \( \sin(5x) \) gives us \( 2A = 10 \), hence \( A = 5 \).
- The wave number \( k = 5 \).
- The angular frequency \( \omega = 2 \).

### Step 3: Determine the frequency and wavelength

The angular frequency \( \omega \) is related to the frequency \( f \) by the equation:

\[ \omega = 2\pi f \]

From this, we can calculate the frequency:

\[ f = \frac{\omega}{2\pi} = \frac{2}{2\pi} = \frac{1}{\pi} \, \text{Hz} \]

Now, the wave number \( k \) is related to the wavelength \( \lambda \) by:

\[ k = \frac{2\pi}{\lambda} \]

Thus, we can find the wavelength:

\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{5} \]

### Step 4: Write the possible traveling waves

A standing wave can be formed by the superposition of two traveling waves moving in opposite directions. The general form of a traveling wave can be expressed as:

\[ y = A \sin(kx - \omega t) \quad \text{(right-moving wave)} \]
\[ y = A \sin(kx + \omega t) \quad \text{(left-moving wave)} \]

Using the values we found:
- Amplitude \( A = 5 \)
- Wave number \( k = 5 \)
- Angular frequency \( \omega = 2 \)

The possible traveling waves are:

1. Right-moving wave:
\[ y = 5 \sin(5x - 2t) \]

2. Left-moving wave:
\[ y = 5 \sin(5x + 2t) \]

### Conclusion

Thus, the possible traveling waves that could cause the standing wave \( y = 10 \sin(5x) \cos(2t) \) are:

- \( y = 5 \sin(5x - 2t) \)
- \( y = 5 \sin(5x + 2t) \)