Wave Motion and String Waves

A travelling wave has an equation of the form $$A(x,t)=f(x+vt)$$. The relation connecting positional derivative with time derivative of the function is:
What happens to the frequency of the wire when the resonating length of the wire increases?
A wave is propagating on a long stretched string along its length taken as positive x-axis the wave equation is given by $$y=y_0e^{-\left(\dfrac{t}{T}-\dfrac{x}{\lambda}\right)^2}$$, where $$y_0=2$$mm, $$T=1.0$$ sec and $$\lambda =6$$cm.
A string oscillates according to the equation $$y'=(0.50cm)$$ $$\sin\left[\left(\dfrac{\pi}{3}cm^{-1}\right)x\right]\cos[(40\pi s^{-1})t]$$.
Which of the following functions for $$y$$ can never represent a travelling wave?
(a) $${ \left( { x }^{ 2 }-vt \right)  }^{ 2 }$$
(b) $$\log { \left[ \dfrac { \left( x+vt \right)  }{ { x }_{ 0 } }  \right]  } $$
(c) $${ e }^{ { \left\{ -\frac { \left( x+vt \right)  }{ { x }_{ 0 } }  \right\}  }^{ 2 } }$$
(d) $$\dfrac { 1 }{ x+vt } $$