Dispersion Relationship

So far, any $\omega ,k$ combination is allowed. Must satisfy FSBC, substitute $\phi$ into equation (3):

$$- \omega ^2\cosh kh + gk \sinh kh = 0,$$

which gives

$$\underbrace{\omega ^2 = gk \tanh kh ; \left\{ {\omega ^2 = gk} \right\}}_{\mbox{ Dispersion Relationship}}$$

Given $k$ (and $h$) $\to \omega$.
Given $\omega$ (and $h$), ...
From (6):

$$C \equiv \frac{\omega ^2h}{g} = \left( {kh} \right)\tanh \left( {kh} \right)$$

$$\frac{C}{kh} = \tanh kh \to k \left( \mbox{only} 1 \right)$$

Dispersion Relationship (6) uniquely relates $\omega$ and $k$, given $h$.

$$\omega = \omega (k;h) \mbox{\ or\ } k = k(\omega;h)$$

In general, $k \uparrow$ as $\omega \uparrow$ or $\lambda \uparrow$ as $T \uparrow$.

$$\frac{\lambda }{T} = V_p = \frac{\omega }{k} = \sqrt {\frac{g}{k}\tanh kh}; \left\{ {V_p = \sqrt {\frac{g}{k}} } \right\}$$

$V_{p} \uparrow$ as $T \uparrow$ or $\lambda \uparrow$. For fixed $k$ (or $\lambda )$, $V_{p} \uparrow$ as $h \uparrow$ ( from shallow water)

$$V_{p} = V_{p}(k) \mbox{\ or\ } V_{p}(\omega ) \to \mbox{\ frequency dispersion }$$