homework and exercises - Why should a solution to the wave equation be finite?

A function which represents a wave must satisfy the following differential equation:

$$\frac{\partial^2 y}{\partial t^2} = k\frac{\partial^2 y}{\partial x^2}$$

Any function that satisfies the wave differential equation represents a wave provided that it is finite everywhere at all times.

What does "it is finite everywhere at all times" mean?

Question:Which of the following functions represent a wave?

a) $(x - vt)^2$

b) $\ln(x + vt)$

c) $e^{-(x - vt)^2}$

d) $(x + vt)^{-1}$

Only option (c) is given as the answer though all 4 satisfy the differential equation.

I believe I did not understand the significance "function should be finite everywhere at all times" which is why I am unable to answer the aforementioned question.

Answer

It's semantics. Whoever wrote the problem prefers to refer to a wave as "A function which satisfies the wave equation and which is bounded" instead of "a function which satisfies the wave equation".

Unfortunately there are bound to be conventions which you disagree with, but in academics (undergrad and lower) the only way to deal with it is to figure out which conventions the professor (or problem writer) is working with before you read the problems. It's too easy for conversations on convention to turn into, "technically, it is a wave even though it's not physical" countered with "technically, it's not a wave because it's not bounded." The best you can do is recognize an issue in terminology ASAP and deal with it in a constructive way.

A better statement, which is more objectively true, would be: "Functions like $(x-vt)^2$ solve the wave equation, but generally don't come up and are not useful in physical solutions."