The equation of progressive wave travelling a long positive direction of x axis having a amplitude of 0.04 m, frequency 440 Hz and wave velocity 330 m/s, is

Options

A. `y = 0.04 sin 2pi (440t - (4x)/(3))`
B. `y = 0.04 cos 2pi (440t - (4x)/(3))`
C. `y = 0.04 sin 2pi (440t + (4x)/(3))`
D. `y = 0.04 cos 2pi (440t + (4x)/(3))`

Explanation

To find the equation of a progressive wave traveling in the positive direction of the x-axis with given parameters, we can follow these steps:

### Step 1: Identify the given parameters
- Amplitude (A) = 0.04 m
- Frequency (f) = 440 Hz
- Wave velocity (v) = 330 m/s

### Step 2: Calculate the wavelength (λ)
The relationship between wave velocity (v), frequency (f), and wavelength (λ) is given by the formula:
\[ v = f \cdot \lambda \]

Rearranging for wavelength:
\[ \lambda = \frac{v}{f} \]

Substituting the values:
\[ \lambda = \frac{330 \, \text{m/s}}{440 \, \text{Hz}} = 0.75 \, \text{m} \]

### Step 3: Calculate the angular frequency (ω)
The angular frequency (ω) is calculated using the formula:
\[ \omega = 2\pi f \]

Substituting the frequency:
\[ \omega = 2\pi \cdot 440 \, \text{Hz} \]

Calculating:
\[ \omega = 880\pi \, \text{rad/s} \]

### Step 4: Calculate the wave number (k)
The wave number (k) is related to the wavelength by the formula:
\[ k = \frac{2\pi}{\lambda} \]

Substituting the wavelength:
\[ k = \frac{2\pi}{0.75} = \frac{8\pi}{3} \, \text{rad/m} \]

### Step 5: Write the equation of the wave
The standard form of the equation of a progressive wave traveling in the positive x-direction is:
\[ y(x, t) = A \sin(\omega t - kx) \]

Substituting the values we have calculated:
\[ y(x, t) = 0.04 \sin(880\pi t - \frac{8\pi}{3} x) \]

### Final Equation
Thus, the equation of the progressive wave is:
\[ y(x, t) = 0.04 \sin(880\pi t - \frac{8\pi}{3} x) \]

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