The equation of a wave travelling on a stretched string along the x-axis is `y = ae^(-(bx+ct))`. The direction of propagation of wave is

Options

A. along negative y-axis
B. along positive y-axis
C. along negative x-axis
D. along positive x-axis

Answer

To determine the direction of propagation of the wave described by the equation \( y = ae^{-(bx + ct)} \), we can follow these steps:

### Step 1: Identify the form of the wave equation
The given wave equation is \( y = ae^{-(bx + ct)} \). We need to rewrite this equation in a standard wave form to analyze the direction of propagation.

### Step 2: Rewrite the exponent
The exponent in the wave equation is \( -(bx + ct) \). We can rewrite this as:
\[ y = ae^{-(bx + ct)} = ae^{-bx} e^{-ct} \]
This shows that the wave is a function of both \( x \) and \( t \).

### Step 3: Compare with standard wave forms
In wave motion, the standard forms of wave equations are typically expressed as:
- \( y = A e^{-(kx - \omega t)} \) for waves traveling in the positive x-direction.
- \( y = A e^{-(kx + \omega t)} \) for waves traveling in the negative x-direction.

### Step 4: Analyze the signs
In our case, we have:
\[ -(bx + ct) = -bx - ct \]
This can be interpreted as:
\[ -(bx + ct) = -b x - c t \]
Here, we can see that the terms involving \( x \) and \( t \) have the same sign (both negative).

### Step 5: Determine the direction of propagation
Since both \( x \) and \( t \) have negative signs, this indicates that the wave is traveling in the negative x-direction.

### Conclusion
Thus, the direction of propagation of the wave is in the negative x-direction.