The wave described by y `= 0.25 sin ( 10 pix -2pi t )` where x and y are in meters and t in seconds , is a wave travelling along the

Options

A. `-x`-direction with amplitude 0.25 m and wavelength ` lambda = 0.2 `m
B. `-x`-direction with frequency 1 Hz
C. `+`x-direction with frequency ` pi` Hz and wavelength ` lambda =0.2m`
D. `+x`-direction with frequency 1 Hz and wavelength ` lambda=0.2 m`

Answer

To solve the problem, we will analyze the wave equation given and extract the necessary parameters step by step.

### Step 1: Identify the Wave Equation
The wave is described by the equation:
\[ y = 0.25 \sin(10 \pi x - 2 \pi t) \]

### Step 2: Determine the Amplitude
The amplitude \( A \) of the wave is the coefficient in front of the sine function. From the equation:
\[ A = 0.25 \, \text{meters} \]

### Step 3: Identify the Wave Number (k) and Angular Frequency (ω)
In the standard wave equation \( y = A \sin(kx - \omega t) \):
- The wave number \( k \) is the coefficient of \( x \).
- The angular frequency \( \omega \) is the coefficient of \( t \).

From the given equation:
- \( k = 10 \pi \)
- \( \omega = 2 \pi \)

### Step 4: Calculate the Frequency (f)
The relationship between angular frequency \( \omega \) and frequency \( f \) is given by:
\[ \omega = 2 \pi f \]
To find \( f \):
\[ f = \frac{\omega}{2 \pi} = \frac{2 \pi}{2 \pi} = 1 \, \text{Hertz} \]

### Step 5: Calculate the Wavelength (λ)
The relationship between wave number \( k \) and wavelength \( \lambda \) is given by:
\[ k = \frac{2 \pi}{\lambda} \]
To find \( \lambda \):
\[ \lambda = \frac{2 \pi}{k} = \frac{2 \pi}{10 \pi} = \frac{2}{10} = 0.2 \, \text{meters} \]

### Step 6: Determine the Direction of Propagation
In the wave equation \( y = A \sin(kx - \omega t) \):
- A negative sign in front of \( \omega t \) indicates that the wave is traveling in the positive x-direction.

### Summary of Results
- Amplitude \( A = 0.25 \, \text{meters} \)
- Frequency \( f = 1 \, \text{Hertz} \)
- Wavelength \( \lambda = 0.2 \, \text{meters} \)
- Direction of propagation: Positive x-direction

### Conclusion
The wave is traveling along the positive x-direction with an amplitude of 0.25 meters, a frequency of 1 Hertz, and a wavelength of 0.2 meters.

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