In a linear dynamic system consisting of N structures, the equation of motion in the frequency domain is written as

(12–19)

where , , and are the 6N×6N mass, damping, and stiffness matrices respectively, is the 6N×1 motion response, and is the 6N×1 external force, at frequency .

In Equation 12–19, is called the impedance matrix, while the receptance matrix is defined as

(12–20)

The motion response in complex values can then be expressed as

(12–21)

It should be noted that if the external force in Equation 12–19 only consists of the first order wave excitation force induced by a regular wave with unit amplitude, and the stiffness matrix includes both the hydrostatic and structural stiffness components (for example, the stiffness due to mooring lines and articulations), the motion responses given by Equation 12–21 are referred to as the fully-coupled response amplitude operators (RAOs).

In multi-directional waves, denoting the ordinate of the m-th directional wave spectrum in direction at frequency as , the 6N×6N general transform function due to the first order wave excitation is defined as

(12–22)

where the superscripts * and T indicate the conjugate transpose and non-conjugate transpose of a matrix respectively, and is the number of wave directions. The diagonal terms of the real part of the general transform function matrix are the motion response spectral densities, i.e.

(12–23)

The first order wave excitation force spectral density is

(12–24)

The difference frequency second order force spectral density in multi-directional waves is expressed as

(12–25)

For the relative motion of a pair of nodes in the global axes, denoting the relative motion response as , where are the nodal motion responses at a pair of nodes (either on the same structure, or on different structures), at frequency and along wave direction , the relative motion response spectrum is

(12–26)

For nodal motion relative to the wave surface, denoting the wave elevation of a unit amplitude regular wave at the location of the node as , the relative vertical motion response spectrum can be obtained from

(12–27)