Three waves `y_(1), y_(2) ,y_(3)` are given by `y_(1)= A sin (Kx- omegat), y_(2) A= sin (Kx+ omega t) and y_(3) = A sin (Ky- omega t)`. Which one of the following represents wave ?

Options

A. `y_(1)+y_(2)+y_(3)`
B. `y_(1)+y_(3)`
C. `y_(1)+y_(2)`
D. `y_(2)+y_(3)`

Explanation

To determine which of the three given wave equations represents a wave, we need to analyze each equation based on the standard form of a wave equation. The standard form of a wave equation is:

\[ y = A \sin(\omega t - kx) \]

Where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( k \) is the wave number,
- \( t \) is the time,
- \( x \) is the position.

Now, let's analyze each wave equation one by one.

### Step 1: Analyze \( y_1 = A \sin(Kx - \omega t) \)

This equation is already in the standard form of a wave equation:

\[ y_1 = A \sin(Kx - \omega t) \]

Here, \( K \) corresponds to \( k \) and \( \omega \) is the angular frequency. This represents a wave traveling in the positive x-direction.

### Step 2: Analyze \( y_2 = A \sin(Kx + \omega t) \)

This equation can be rewritten as:

\[ y_2 = A \sin(-\omega t + Kx) \]

This does not match the standard form \( y = A \sin(\omega t - kx) \). Instead, it represents a wave traveling in the negative x-direction. However, it does not represent a wave in the standard form because we cannot have a positive amplitude with a negative sine function in the standard wave equation.

### Step 3: Analyze \( y_3 = A \sin(Ky - \omega t) \)

This equation is in the form:

\[ y_3 = A \sin(Ky - \omega t) \]

In this case, the wave is described in terms of \( y \) instead of \( x \). Waves typically propagate in the x-direction, and having a wave equation dependent on \( y \) does not represent a standard wave traveling in the x-direction. Therefore, this does not represent a wave in the conventional sense.

### Conclusion

From the analysis:
- \( y_1 = A \sin(Kx - \omega t) \) represents a wave traveling in the positive x-direction.
- \( y_2 = A \sin(Kx + \omega t) \) does not represent a standard wave.
- \( y_3 = A \sin(Ky - \omega t) \) does not represent a wave traveling in the x-direction.

Thus, the correct answer is:

**The wave represented by the equation is \( y_1 = A \sin(Kx - \omega t) \).**