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Introduction
This topic discusses the equations and concepts related to the laying of submarine power cables. A catenary line is the shape followed by the cable between the exit point on the laying wheel and the touchdown (TD) point on the seafloor.

Key Assumptions:

• The cable has no bending stiffness.
• The cable does not experience drag while moving through water.
• The cable has a uniform weight per meter.

The catenary line is determined by several parameters, such as the departure angle (\( \phi \)), cable weight, water depth (\( H \)), and the layback length (\( L \)).

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1. Equation for the Catenary Line

The catenary line can be expressed as:
\[y = a \cosh \left( \frac{x}{a} \right)\]
- \( y \): Vertical coordinate of the cable.
- \( x \): Horizontal coordinate of the cable.
- \( a \): Catenary parameter, determined by cable weight and tension.

Departure Angle (\( \phi \))
The departure angle is derived using the equation:
\[\cot \phi = \frac{\partial y}{\partial x} = \sinh \left( \frac{x}{a} \right)\]


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2. Real Laying Conditions

In the real laying scenario, the point where the cable leaves the laying wheel has the vertical coordinate:
\[y = H + a \cosh \left( \frac{L}{a} \right)\]

Where:
- \( L \): Layback length (horizontal distance from touchdown point to laying wheel).
- \( H \): Water depth.

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3. Key Calculations

Catenary Parameter (\( a \))
The catenary parameter is given by:
\[a = \frac{H}{\cosh \left( \frac{L}{a} \right) - 1}\]

Bottom Tension (\( T_0 \))
The bottom tension at the touchdown point is:
\[T_0 = w \cdot a\]
- \( w \): Cable weight per unit length in water.

Suspended Length (\( s \))
The total length of the suspended cable is:
\[s = a \cdot \sinh \left( \frac{L}{a} \right)\]

Top Tension (\( T \))
The tension at the cable's departure point is:
\[T = \sqrt{T_0^2 + w^2 \cdot s^2}\]

Bending Radius at Touchdown (\( R_0 \))
The bending radius is:
\[R_0 = a \cdot \cosh^2 \left( \frac{L}{a} \right)\]


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Conclusion
These equations and calculations ensure the proper design and safe installation of submarine power cables. By monitoring parameters such as tension, curvature, and bending radius, engineers can optimize cable performance and longevity.