The equation of a wave disturbance is given as `y=0.02cos((pi)/(2)+50pit) cos(10pix)`, where x an y are in metre and t in second. Choose the wrong statement.

Options

A. Antinode occurs at x=0.3
B. The wave length is 02m
C. The speed of the consituent is 4m/s
D. Node occurs at x=0.15m

Answer

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

\[ y = 0.02 \cos\left(\frac{\pi}{2} + 50\pi t\right) \cos(10\pi x) \]

### Step 1: Rewrite the Wave Equation
We can simplify the equation using trigonometric identities. The term \(\cos\left(\frac{\pi}{2} + \theta\right)\) can be rewritten as \(-\sin(\theta)\). Therefore, we have:

\[ y = 0.02 \cos\left(\frac{\pi}{2} + 50\pi t\right) \cos(10\pi x) = -0.02 \sin(50\pi t) \cos(10\pi x) \]

### Step 2: Identify the Antinode
Antinodes occur where the amplitude of the wave is maximum. This happens when \(\cos(10\pi x) = \pm 1\).

To find the position of the antinode, we can set \(x = 0.3\) m:

\[ \cos(10\pi \cdot 0.3) = \cos(3\pi) = -1 \]

Thus, substituting this back into the equation gives:

\[ y = -0.02 \sin(50\pi t) \cdot (-1) = 0.02 \sin(50\pi t) \]

This confirms that there is an antinode at \(x = 0.3\) m.

### Step 3: Calculate the Wavelength
The wave number \(k\) is given by the coefficient of \(x\) in the cosine term, which is \(10\pi\). We can find the wavelength \(\lambda\) using the formula:

\[ k = \frac{2\pi}{\lambda} \implies \lambda = \frac{2\pi}{k} = \frac{2\pi}{10\pi} = \frac{1}{5} = 0.2 \text{ m} \]

### Step 4: Calculate the Wave Speed
The speed of the wave \(v\) can be calculated using the relationship between the wave speed, wavelength, and frequency:

\[ v = f \lambda \]

First, we need to find the frequency \(f\). The angular frequency \(\omega\) is given by the coefficient of \(t\) in the sine term, which is \(50\pi\). The frequency \(f\) can be calculated as:

\[ \omega = 2\pi f \implies f = \frac{\omega}{2\pi} = \frac{50\pi}{2\pi} = 25 \text{ Hz} \]

Now we can calculate the speed:

\[ v = f \lambda = 25 \cdot 0.2 = 5 \text{ m/s} \]

### Step 5: Identify the Incorrect Statement
We have calculated the speed of the wave to be \(5 \text{ m/s}\), but the problem states that the speed is \(4 \text{ m/s}\). Therefore, the statement regarding the speed of the wave is incorrect.

### Conclusion
The wrong statement is that the speed of the wave is \(4 \text{ m/s}\).

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