A transverse periodic wave described by the expression. `y=sin [2x(x/2+(t)/(10))]` (where y and rare in meters and is in seconds) is established on a string. Which one of the following statements concerning this wave is false?

Options

A. The wave is traveling in the negative x direction.
B. The amplitude is 1.0 m.
C. The frequency of the wave is 0.10 Hz.
D. The wave travels with speed 5.0 m/s.

Answer

To solve the problem, we need to analyze the given wave equation and determine which statement about the wave is false. The wave is described by the equation:

\[ y = \sin\left[2\pi \left(\frac{x}{2} + \frac{t}{10}\right)\right] \]

### Step 1: Identify the wave parameters
The general form of a transverse wave can be expressed as:

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

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

### Step 2: Extract the wave parameters from the given equation
From the given equation, we can rewrite it in the standard form:

\[ y = \sin\left(\frac{2\pi x}{2} + \frac{2\pi t}{10}\right) \]

This can be simplified to:

\[ y = \sin\left(\pi x + \frac{\pi t}{5}\right) \]

From this, we can identify:
- The wave number \( k = \pi \)
- The angular frequency \( \omega = \frac{\pi}{5} \)

### Step 3: Calculate the amplitude
The amplitude \( A \) is the coefficient in front of the sine function. Here, the amplitude is:

\[ A = 1 \text{ meter} \]

### Step 4: Calculate the frequency
The relationship between angular frequency and frequency is given by:

\[ \omega = 2\pi f \]

Substituting the value of \( \omega \):

\[ \frac{\pi}{5} = 2\pi f \]

Solving for \( f \):

\[ f = \frac{1}{10} \text{ Hz} \]

### Step 5: Calculate the wave speed
The wave speed \( v \) can be calculated using the formula:

\[ v = f \lambda \]

Where \( \lambda \) (wavelength) can be calculated using:

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

Now substituting \( f \) and \( \lambda \):

\[ v = \left(\frac{1}{10}\right)(2) = \frac{2}{10} = 0.2 \text{ m/s} \]

### Step 6: Compare with the given statements
Now we can check the statements provided in the question:

1. **Direction of wave propagation**: The wave travels in the negative x-direction because of the positive sign in front of \( \frac{t}{10} \).
2. **Amplitude**: The amplitude is \( 1 \) meter, which is true.
3. **Frequency**: The frequency is \( 0.1 \) Hz, which is true.
4. **Wave speed**: The calculated wave speed is \( 0.2 \) m/s, but the question states it is \( 5 \) m/s, which is false.

### Conclusion
The false statement is regarding the wave speed, which is stated as \( 5 \) m/s.