Wave Motion and String Waves

goals
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) $\dfrac { 1 }{ x+vt }$