Which of the following not true?

Let us go through all the options one by one.

1. A harmonic function u is a twice continuously differentiable function. i.e. it satisfies Laplace’s equation:  \(\frac{{{\partial ^2}u}}{{\partial x_1^2}} + \frac{{{\partial ^2}u}}{{\partial x_2^2}} + \ldots \frac{{{\partial ^2}u}}{{\partial x_n^2}} = 0\)

Can also be written as ∇²u = 0

Or Δu = 0

Option 1 is correct.

2) By the strong maximum principle we get:

If D is connected and there exists a point x₀ ∈ D such that:  \(\phi \left( {{x_0}} \right) = \mathop {\max }\limits_{\bar D} \phi \left( x \right)\)

Then ϕ is constant within D.

Option 2 is correct.

3) A wave that travels continuously in a medium in the same direction without the change in its amplitude is called a traveling wave or progressive wave.

Example:

y(x, t) = a sin (kx – ωt + ϕ)

y(x, t) is the displacement of the element at a point x and t from the y-axis.

∴ Option 3 is correct.

4) If ϕ solves the initial value wave equation in one dimension, then the sum of kinetic energy and potential energy is independent of time.

∴ Option 4 is incorrect.