The equation of a wave travelling on a stretched string...

The equation of a wave travelling on a stretched string along the $x$ -axis is $y=a e^{-(b x+c t)}$. The direction of propagation of wave is:
(a) along negative $y$ -axis
(b) along positive $y$ -axis
(c) along negative $x$ -axis
(d) along positive $x$ -axis

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Key Concepts

Constant Phase Condition

In wave mechanics, the constant phase condition is used to identify points on a wave that have the same phase as the wave evolves. By setting the phase term equal to a constant, one can derive a relationship between space and time, which in turn reveals how the position of a point of constant phase changes with time.

Phase Velocity

Phase velocity represents the speed at which a point of constant phase (such as a crest) propagates through space. It is calculated by differentiating the phase with respect to time, providing crucial information about the direction and the rate of the wave's propagation.

Determining Propagation Direction

The direction of wave propagation can be determined by analyzing the sign and arrangement of the coefficients associated with the spatial and temporal variables in the phase term of the wave equation. Specifically, if setting the phase equal to a constant leads to a decrease in the spatial coordinate as time increases, it indicates that the wave travels in the negative spatial direction.

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