A transverse wave of amplitude `0.5 m` and wavelength 1 m and frequency 2 Hz is propagating in a string in the negative x - direction. The expression for this wave is [

Options

A. `y(x,t) = 0.5 sin (2pix + 4pit)`
B. `y(x,t) = 0.5 cos (2pix - 4pit)`
C. `y(x,t) = 0.5 sin (pix - 2pit)`
D. `y(x,t) = 0.5 cos (2pix - 2pit)`

Answer

To find the expression for the transverse wave propagating in a string, we will follow these steps:

### Step 1: Identify the parameters of the wave
We are given:
- Amplitude (A) = 0.5 m
- Wavelength (λ) = 1 m
- Frequency (f) = 2 Hz

### Step 2: Calculate the angular frequency (ω)
The angular frequency (ω) can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2\pi \times 2 = 4\pi \text{ rad/s} \]

### Step 3: Calculate the wave number (k)
The wave number (k) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of wavelength:
\[ k = \frac{2\pi}{1} = 2\pi \text{ rad/m} \]

### Step 4: Write the general wave equation
The general wave equation for a wave traveling in the positive x-direction is:
\[ y(x, t) = A \sin(kx - \omega t) \]
Since the wave is traveling in the negative x-direction, we modify the equation to:
\[ y(x, t) = A \sin(kx + \omega t) \]

### Step 5: Substitute the values into the wave equation
Now we substitute the values of A, k, and ω into the equation:
\[ y(x, t) = 0.5 \sin(2\pi x + 4\pi t) \]

### Final Expression
Thus, the expression for the wave is:
\[ y(x, t) = 0.5 \sin(2\pi x + 4\pi t) \]

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