quantum mechanics - Significance of wave number?

Till now all I know about the wave number is its formula i.e. ${\frac{2\pi}{\lambda}}$. I always wanted to know what it really means. So can anyone please, explain me its physical significance?

Answer

Imagine something oscillating in space and time, for example a plane wave propagating across the axis $x$. This propagation is expressed via the so-called phase

$$\phi(x,t)=\omega \cdot t - k\cdot x = \dfrac{2\pi}{T}\cdot t -\dfrac{2\pi}{\lambda}\cdot x \tag{01}$$ and the magnitude of the plane wave as $$E(x,t)=A\cos\phi(x,t) \tag{02}$$ As the frequency in time $\:\nu=1/T \:$ gives how many cycles are executed by $E(x_{0},t)$ per unit time $t$ at a specific space point $x_{0}$, so the frequency in space $\:1/\lambda \:$ gives how many cycles are executed by $E(x,t_{0})$ per unit length in space $x$ at a specific time moment $t_{0}$.

EDIT
Both frequencies and consequently the phase (01) are expressed as angles in radian units. A full cycle is a $2\pi$ radians angle. That's why this factor in $\omega=2\pi / T$ and $k=2\pi / \lambda$.

$$\begin{align} T \equiv & \text{time length for a full cycle of the phase at given space point = period} \tag{03a}\\ &\phi(x,t+T) =\phi(x,t)+2\pi \tag{03b}\\ \lambda \equiv & \text{space length for a full cycle of the phase at given time moment = wavelegth} \tag{04a}\\ & \phi(x+\lambda,t) =\phi(x,t)-2\pi \tag{04b} \end{align}$$