The equation of a simple harmonic progressive wave along the negative direction of X-axis is

Options

A. `y = a sin 2pi [(t)/(T) - (x)/(lamda)]`
B. `y = a sin (2pi nt)`
C. `y = a sin 2pi [(t)/(T) + (x)/(lamda)]`
D. `y = a cos 2pi [x]`

Answer

To find the equation of a simple harmonic progressive wave traveling along the negative direction of the X-axis, we can follow these steps:

### Step 1: Understand the General Equation of a Progressive Wave
The general equation for a simple harmonic wave traveling in the positive x-direction is given by:
\[ y(x, t) = A \sin(\omega t - kx + \phi) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( \phi \) is the phase constant.

### Step 2: Modify the Equation for Negative Direction
For a wave traveling in the negative x-direction, the equation modifies to:
\[ y(x, t) = A \sin(\omega t + kx + \phi) \]
This change occurs because the wave is moving in the opposite direction.

### Step 3: Relate Angular Frequency and Wave Number
The angular frequency \( \omega \) can be expressed in terms of the frequency \( f \) and the wave number \( k \) can be expressed in terms of the wavelength \( \lambda \):
- \( \omega = 2\pi f \)
- \( k = \frac{2\pi}{\lambda} \)

### Step 4: Substitute \( \omega \) and \( k \) into the Equation
Substituting these values into the equation gives:
\[ y(x, t) = A \sin\left(2\pi f t + \frac{2\pi}{\lambda} x + \phi\right) \]

### Step 5: Final Form of the Equation
Thus, the final form of the equation for a simple harmonic progressive wave traveling along the negative direction of the X-axis is:
\[ y(x, t) = A \sin\left(2\pi f t + \frac{2\pi}{\lambda} x + \phi\right) \]

### Summary
The equation of a simple harmonic progressive wave traveling in the negative direction of the X-axis is:
\[ y(x, t) = A \sin\left(2\pi f t + \frac{2\pi}{\lambda} x + \phi\right) \]

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