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Consider the following function-

- y = x² + 2 α tx
- y = (x + vt)²

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

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#### Solution

**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.

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