The transverse displamcement y(x,t) of a wave y(x,t ) on a string is given by yox, t) = . This represents a

Options

A. wave moving in -x direction with speed `sqrt((b)/(a))`
B. standing wave of frequency `sqrt(b)`
C. standing wave of frequency `(1)/(sqrt(b))`
D. wave moving in +x direction with `sqrt((a)/(b))`

Answer

To solve the problem regarding the transverse displacement \( y(x,t) \) of a wave on a string, we need to analyze the given wave equation and determine its characteristics, including the direction of wave propagation and the wave speed.

### Step-by-Step Solution:

1. **Identify the Wave Equation**:
The wave equation is given as \( y(x, t) = A \sin(kx - \omega t) \) or a similar form. Here, \( A \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency.

2. **Determine the Wave Speed**:
The wave speed \( v \) can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
where \( \omega \) is the coefficient of \( t \) and \( k \) is the coefficient of \( x \) in the wave equation.

3. **Identify the Direction of Propagation**:
In the wave equation \( y(x, t) = A \sin(kx - \omega t) \), the negative sign before \( \omega t \) indicates that the wave is traveling in the positive x-direction. Conversely, if the equation were \( y(x, t) = A \sin(kx + \omega t) \), the wave would be traveling in the negative x-direction.

4. **Conclusion**:
Based on the analysis, we can conclude the characteristics of the wave. If the equation is in the form \( y(x, t) = A \sin(kx - \omega t) \), the wave travels in the positive x-direction with speed \( v = \frac{\omega}{k} \).