Question

Consider the following function-

1. y = x² + 2 α tx
2. y = (x + vt)²

which among the above function can be characterized as a wave?

Answer 1

Formula:

For the function to be a wave function `("dy"//"dx")/("dy"//"dt")` should be a constant.

For function (a):

y = x² + 2 α tx

\[\frac{\text{dy}}{\text{dx}}\] = 2x + 2α t   ...(1)

\[\frac{\text{dy}}{\text{dt}}\] = 0 + 2αx    ...(2)

`((1))/((2)) => ("dy"//"dx")/("dy"//"dt") = (2x + 2alpha"t")/(2alpha"x")`

For function (b):

y = (x + vt)²

\[\frac{\text{dy}}{\text{dx}}\] = 2(x + vt)   ....(3)

\[\frac{\text{dy}}{\text{dt}}\] = 2(x + vt)v    .....(4)

`((3))/((4)) => ("dy"//"dx")/("dy"//"dt") = (2("x" + "vt"))/(2("x" + "vt")"v")` = constant

Hence, function
(a) does not describe a wave.
(b) satisfies wave function.