Question

A wave equation \[y=f(x-vt)\] represents a wave travelling in
A. +X axis
B. +Y axis
C. –X axis
D. –Y axis

Answer 1

Hint: The given equation of the transverse displacement of a wave on a string should be represented in terms of the standard equation of the transverse displacement equation by rearranging the terms. The sign of the coefficient of ‘t’, that is, the wave speed will be in reverse to the direction of the wave.
Formula used:
\[y(x,t)=f(x+vt)\]

Complete step by step answer:
From the given information, we have the data as follows.
The wave equation y(x, t) of a wave travelling is given by,
\[y=f(x-vt)\]
The standard equation of the transverse displacement y(x, t) of a wave is given as follows.
\[y(x,t)=f(x+\upsilon t)\]
Rewrite the given equation of the transverse displacement y(x, t) of a wave on a string.
\[y=f(x+(-v)t)\]
Therefore, the expression of the transverse displacement y(x, t) of a wave on a string in terms of the standard equation is given as follows.
\[y=f(x+(-v)t)\]
Upon comparing the obtained equation with the standard equation, the coefficient of ‘t’, that is the wave speed is given as \[(-v)\].
The sign between the x and t parameters of the equation is negative, thus, the wave moves along the positive direction of x-axis.
\[\therefore \]The transverse displacement y(x, t) of a wave on a string is given by \[y=f(x-vt)\], representing a wave moving in +x direction with speed \[v\].

Thus, option (A) is correct.

Note:
The direction of a wave can be computed as, if there is a positive sign between the ‘x’ and ‘t’ parameters of the equation, then, the wave is said to travel along the negative direction. Similarly, if there is a negative sign between the ‘x’ and ‘t’ parameters of the equation, then, the wave is said to travel along the positive direction.