1.4. Amplitude of the Wave and Wave Propagation

Applying the dynamic boundary condition gives $C = D e^{2kd}$. The same procedure is repeated for the dynamic free surface boundary condition, resulting in $\phi_3 = -\frac{ag}{\sigma} \cosh(kd) + z$. This procedure is similarly applied to obtain $\phi_4$.

In fluid mechanics, the concept of velocity potential terms such as $\phi_1, \phi_2, \phi_3, \phi_4$ represents potential energy stored in wave movements. By applying dynamic boundary conditions—forces acting on the waves—we simplify calculations using constants like $C$ and $D$ which represent coefficients of these potentials. Through repeated application of boundary conditions, we can derive expressions for each potential value that dictates how fluid waves behave.

Think of the wave potential as different sounds produced by a musical instrument. Just as a musician combines different notes (like $\phi_1$ and $\phi_2$) to create a melody, fluid mechanics combines different potential functions to describe wave behavior.

The total velocity potential $\Phi$ is the summation of individual terms: $\Phi = \phi_2 - \phi_1$. Using trigonometric identities, this expression simplifies to $\frac{1}{\sigma} \left( \frac{ag}{\sigma} \cosh(kd) + z \right) \cos(kx - \sigma t)$.

The total velocity potential is a result of combining positive and negative potentials from different boundaries. This results in a relationship that represents the oscillation of the wave as it travels through a medium. The expression consists of terms for wave amplitude (height), frequency, and position, which are encapsulated in trigonometric functions to describe wave motion over time.

Imagine the waves on the ocean moving in sync with the tides. The equation for velocity potential can be likened to calculating the height of waves at different times and positions along the shore, producing a visual and mathematical representation of how waves behave.

The speed of the wave, or celerity, is defined by the relationship: $C = \frac{L}{T} = \frac{gL}{2\pi} \tan(h k d)$, where $L$ is the wavelength and $T$ is the wave period.

Celerity refers to the speed at which a wave travels in a medium. In this section, we derive the relationship between wave speed, water depth, and other parameters governing the wave motion. It is important to know that as we change the depth of water (where waves are traveling), the speed at which those waves travel also changes. Formulating this connection reveals deeper insights into wave dynamics.

Consider a surfer catching waves. The surfer must be aware of the wave's speed (celerity) to align with the wave properly. If the water is deeper, the waves might be faster, just like if you're riding a bike downhill—the steeper the hill, the quicker you go!

The dispersion relationship connects the wavelength with the wave period and water depth, given by: $\sigma^2 = g k \tan(h k d)$. This relationship must be solved iteratively for conditions that do not have a simple relationship.

Dispersion describes how waves of different wavelengths travel at different speeds. Here, we establish an equation that represents this relationship mathematically. The ratio of wave speed to its characteristics (wavelength and period) results in a fundamental wave principle affecting how waves present themselves based on different mediums and conditions.

This concept can be likened to a symphony where each instrument plays at a different tempo. The dispersion relationship dictates how each 'instrument' (wave) behaves in relation to the others, affecting the overall harmony of the wave patterns observed.

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