Wave transformation

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The transformation (reshaping) of progressing gravity waves ( water waves ) can have various causes. In the case of wave transformations , a distinction must be made between the following effects:

• decreasing water depth, see shoaling
• Friction effects on the sea floor
• refraction
• Diffraction
• reflection
• Breaking waves
• superimposed currents
  • constant currents
  • accelerated currents.

Wave transformation due to decreasing water depth

Wave deformation according to linear wave theory for a wave of period = 10 s, height = 1 m and length L = 156 m; Assumption: maintaining continuity.

Waves with lengths L (approximately) greater than twice the water depth d ( L ≥ 2 d ) are subject to a change in dispersion such that the water depth is added as a boundary condition. While the wave speed c and the wave length L also decrease with decreasing water depth, the wave height H increases .

Superimposed currents

In nature there is generally an interaction of the wave kinematics with another flow field, the influence of which is not covered by any known wave theory. Such superimposed currents not only cause a deformation of the waves, but have u. a. Due to the Doppler effect, it has an influence on the frequency and thus also on the dispersion and transformation of the waves.

Doppler effect due to constant flow velocity

Unaccelerated flows are a special case. If the carrier medium of the waves is subject to a constant flow with a component that is the same as or opposite to the wave advance, the frequency or period is changed compared to a medium that is not influenced by the flow. If the flow component is aligned with the wave progress, more waves arrive at a measuring location per unit of time. This means that the wavelength that would be present in the absence of a flow is here by the ratio ${\displaystyle u_{M}}$ ${\displaystyle L_{A}}$

${\displaystyle {\frac {c_{A}-u_{M}}{c_{A}}}\,}$

shortened than is measured: ${\displaystyle L_{B}}$

${\displaystyle L_{B}=L_{A}\cdot {\frac {c_{A}-u_{M}}{c_{A}}}=L_{A}\cdot \left(1-{\frac {u_{M}}{c_{A}}}\right)\,}$

The following therefore results for the frequency at the measurement location: ${\displaystyle f_{B}}$

${\displaystyle f_{B}={\frac {c_{A}}{L_{B}}}={\frac {f_{A}}{1-{\frac {u_{M}}{c_{A}}}}}\,}$

For an oppositely directed flow velocity , use positive signs in brackets . ${\displaystyle u_{M}}$

Accelerated flow

Accelerated currents near the water surface (drift currents) can be attributed to meteorological influences. Accelerated currents caused by the movement of the tides often extend over the entire depth of the water in shallow seas. The same applies near the coast for large-scale back currents (undertow), so-called rib currents, and for surf-generated back currents (backwash). The latter represent an important component of the surf process, since their influence on the phase velocity can be seen as the cause of a significant change in frequency or period of erupting waves along a wave beam. If different phase velocities c along a wave beam are traced back to a convectively accelerated movement of the carrier medium of the waves and at the same time frequency changes along the wave beam are taken into account, it can be concluded that there are different numbers of waves per unit of time at two locations A and B.

Takes as a result of the wave propagation oppositely directed flow, the phase velocity between the locations A and B on starting, it follows that B arrive in the time unit less waves at the location as if ${\displaystyle c_{A}}$ ${\displaystyle c_{B}}$

${\displaystyle c_{B}=c_{A}={\text{konst}}.\,}$

Accelerated flow superimposed

Is the difference in phase velocities

${\displaystyle \delta c=c_{A}-c_{B}\quad {\text{bzw.}}\quad c_{B}=c_{A}-\delta c}$

the wavelength increases by the ratio on the way from A to B. ${\displaystyle L_{A}=c_{A}/f_{A}}$

${\displaystyle {\frac {c_{A}+\delta c}{c_{A}}}\,}$

on

${\displaystyle L_{B}=L_{A}\cdot {\frac {c_{A}+\delta c}{c_{A}}}={\frac {L_{A}}{c_{A}}}\cdot (c_{A}+\delta c)={\frac {1}{f_{A}}}\cdot (c_{A}+\delta c)\,}$

This gives the frequency at location B:

${\displaystyle f_{B}={\frac {c_{B}}{L_{B}}}=f_{A}\cdot {\frac {c_{A}-\delta c}{c_{A}+\delta c}}=f_{A}\cdot {\frac {c_{B}}{2c_{A}+c_{B}}}<f_{A}\,}$

The difference in frequencies is defined as the frequency shift:

${\displaystyle \delta f=f_{B}-f_{A}=f_{A}\cdot {\frac {2(c_{B}-c_{A})}{2c_{A}-c_{B}}}\,}$

Accordingly, it can be positive or negative, depending on whether ${\displaystyle \delta f}$

${\displaystyle 2c_{A}>c_{B}>c_{A}\quad {\text{oder}}\quad 2c_{A}<c_{B}<c_{A}}$