What does it mean for waves to "feel" the bottom?
In water wave physics, when we say that the wave "feels" the bottom, we mean that the water depth affects the properties of the wave. 

The dispersion relationship for water waves is:

$$
\omega^2 = gk \tanh{(kd)}
$$

where $\omega$ is the wave frequency, $k$ is the wavenumber, $d$ is the mean water depth, and $g$ is gravitational acceleration. We distinguish "shallow" and "deep" water waves by the value of $kd$, which includes both the wavenumber and the water depth:


Shallow water waves when $kd &lt; 0.3$;
Intermediate water waves when $0.3 &lt; kd &lt; 3$;
Deep water waves when $kd &gt; 3$.


Thus, a long swell wave may act as a shallow water wave in depths of 10 m, but also a very short wave may act as a deep water wave in depths of 1 m.

How are these limits for $kd$ obtained? The tangent hyperbolic function has some convenient properties for the limiting values of its argument:



For deep water waves, $kd$ is very large, so $\tanh{(kd)} \rightarrow 1$. The dispersion relationship then reduces to:

$$
\omega^2 = gk
$$

Phase and group speeds are then:

$$
C_p = \dfrac{\omega}{k} = \sqrt{\dfrac{g}{k}}
$$

$$
C_g = \dfrac{\partial \omega}{\partial k} = \dfrac{1}{2}\sqrt{\dfrac{g}{k}}
$$

Notice that $C_p$ and $C_g$ are not a function of water depth, thus it is said that deep water waves don't "feel" the bottom.

On the other hand, for shallow water waves, $kd$ is small (approximately 0.3 or less), and $\tanh{(kd)} \rightarrow kd$. The dispersion relationship is now:

$$
\omega^2 = gk^2d
$$

and phase and group speeds are functions of water depth:

$$
C_p = \dfrac{\omega}{k} = \sqrt{gd}
$$

$$
C_g = \dfrac{\partial \omega}{\partial k} = \sqrt{gd}
$$

So, why is it said that the waves "feel" the bottom at the water depth of half wavelength?

$$
kd = k  \dfrac{\lambda}{2} = k \dfrac{2\pi}{2k} = \pi
$$

and this is approximately the value below which the regime transitions from deep water to intermediate water, i.e. $\tanh{(kd)} \approx 1$ does not hold anymore.