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   <text:h text:style-name="Heading_20_2" text:outline-level="2">How to generate artificial waves from a given spectrum.</text:h>
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=
int from -infinity to infinity 
{
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}
=
1 over 2
int from -infinity to infinity 
{
S_2A left( %omega right) d%omega
}
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   <text:p text:style-name="P1">Divide the spectrum in a number of bands or strips of small but arbitrary width,<draw:frame draw:style-name="fr2" draw:name="Object5" text:anchor-type="as-char" svg:width="0.3717in" svg:height="0.1835in" draw:z-index="7"><draw:object>
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       <semantics>
        <mrow>
         <mi mathvariant="normal">Δ</mi>
         <msub>
          <mi mathvariant="normal">ω</mi>
          <mi>i</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">%DELTA %omega_i</annotation>
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   <text:p text:style-name="P1">For each band assign a harmonic component</text:p>
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          <mo stretchy="false">=</mo>
          <msub>
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          </msub>
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         <mrow>
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           <mrow>
            <mi>j</mi>
            <mrow>
             <mo fence="true" form="prefix" stretchy="true">(</mo>
             <mrow>
              <mrow>
               <msub>
                <mi mathvariant="normal">ω</mi>
                <mi>i</mi>
               </msub>
               <mrow>
                <mi>t</mi>
                <mo stretchy="false">+</mo>
                <msub>
                 <mi mathvariant="normal">ϕ</mi>
                 <mrow>
                  <mi mathvariant="normal">ζ</mi>
                  <mi>i</mi>
                 </mrow>
                </msub>
               </mrow>
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           <msub>
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         </mrow>
         <msup>
          <mi>e</mi>
          <mrow>
           <mi>j</mi>
           <msub>
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            <mi>i</mi>
           </msub>
           <mi>t</mi>
          </mrow>
         </msup>
        </mrow>
        <annotation encoding="StarMath 5.0">%zeta_i
=
A_{%zeta i} e^{j left(%omega_i t + %phi_{%zeta i} right)}
=
%zeta left( %omega_i right) e^{j %omega_i t }
</annotation>
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       AAEAAgAAAAkAjAABAAAAAAA=
      </office:binary-data>
     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="P1">Chooses<draw:frame draw:style-name="fr2" draw:name="Object9" text:anchor-type="as-char" svg:width="0.2307in" svg:height="0.1484in" draw:z-index="9"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi mathvariant="normal">ω</mi>
         <mi>i</mi>
        </msub>
        <annotation encoding="StarMath 5.0">%omega_i</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAUkCAAB6
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       ////AIYAAQAEAAAAAAAAAHIAAgAXAAAAjQEAAD4BAAABAGlIAAAAAAD//wEAaQCMAAEAAAAA
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      </office:binary-data>
     </draw:image>
    </draw:frame>randomly within each band using a uniform distribution.</text:p>
   <text:p text:style-name="P1">The corresponding amplitude is</text:p>
   <text:p text:style-name="P1"><draw:frame draw:style-name="fr2" draw:name="Object7" text:anchor-type="as-char" svg:width="1.4154in" svg:height="0.2398in" draw:z-index="10"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mi>A</mi>
          <mrow>
           <mi mathvariant="normal">ζ</mi>
           <mi>i</mi>
          </mrow>
         </msub>
         <mo stretchy="false">=</mo>
         <msqrt>
          <mrow>
           <mn>2</mn>
           <msub>
            <mi>S</mi>
            <mrow>
             <mn>1</mn>
             <mi>P</mi>
            </mrow>
           </msub>
           <mrow>
            <mo fence="true" form="prefix" stretchy="true">(</mo>
            <mrow>
             <msub>
              <mi mathvariant="normal">ω</mi>
              <mi>i</mi>
             </msub>
            </mrow>
            <mo fence="true" form="postfix" stretchy="true">)</mo>
           </mrow>
           <mi mathvariant="normal">Δ</mi>
           <msub>
            <mi mathvariant="normal">ω</mi>
            <mi>i</mi>
           </msub>
          </mrow>
         </msqrt>
        </mrow>
        <annotation encoding="StarMath 5.0">A_{%zeta i}
=
sqrt
{
2 S_1P left( %omega_i right) %DELTA %omega_i
}</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AAAA//8BAMkDjAABAAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAA
       AAAAAP0AAAAAAAAAAAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEA
       BQAAAP////8AhgABAAQAAAAAAAAAcgACABcAAABVDQAAEQIAAAEAaUgAAAAAAP//AQBpAIwA
       AQAAAAAAlQABAAQAAAAAAAAAlgABAAIAAAAJAIwAAQAAAAAA
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="P1">Choose<draw:frame draw:style-name="fr2" draw:name="Object6" text:anchor-type="as-char" svg:width="0.2043in" svg:height="0.1709in" draw:z-index="11"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi mathvariant="normal">ϕ</mi>
         <mi>i</mi>
        </msub>
        <annotation encoding="StarMath 5.0">%phi_i</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAQcCAACy
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       AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv
       bAAAAAAAAKcBAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH
       AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAABqAAAACQEAAAEAAADVA+UAAAAAAP//
       AQDVA4wAAQAAAAAAiwABAAIAAAAfAIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAAD9
       AAAAAAAAAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/
       ////AIYAAQAEAAAAAAAAAHIAAgAXAAAAWAEAAHIBAAABAGlIAAAAAAD//wEAaQCMAAEAAAAA
       AJUAAQAEAAAAAAAAAJYAAQACAAAACQCMAAEAAAAAAA==
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     </draw:image>
    </draw:frame>randomly such that<draw:frame draw:style-name="fr2" draw:name="Object10" text:anchor-type="as-char" svg:width="0.7646in" svg:height="0.2138in" draw:z-index="12"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <mn>0</mn>
          <mo stretchy="false">&lt;</mo>
          <msub>
           <mi mathvariant="normal">ϕ</mi>
           <mi>i</mi>
          </msub>
          <mo stretchy="false">≤</mo>
          <mn>2</mn>
         </mrow>
         <mi mathvariant="normal">π</mi>
        </mrow>
        <annotation encoding="StarMath 5.0">0 &lt;%phi_i &lt;= 2%pi </annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAZYHAAAg
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       AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxl
       AAAAAAAApwEAAAAAAAAAAAUAAAAAAAAA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA
       AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAAGoAAAByAQAAAQAw1AAAAAAA//8BADAA
       jAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACnAQAA
       //8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////
       AIYAAQAEAAAAAAAAAHIAAgAaAAAAWAEAAHIBAAABAAAAPABTAQAAAAD//wEAPACMAAEAAAAA
       AIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD//wAAAAAF
       AAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQA
       AAAAAAAAcgACABoAAACwAgAAcgEAAAEAAADVA+UAAAAAAP//AQDVA4wAAQAAAAAAiwABAAIA
       AAAfAIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAAD9AAAAAAAAAAAABQAAAAAAAgD/
       AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIA
       AgAXAAAAngMAANwBAAABAGlIAAAAAAD//wEAaQCMAAEAAAAAAIsAAQACAAAAHwCKAAEAPAAA
       AAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA
       /wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAADtAwAA
       cgEAAAEAAABkItgAAAAAAP//AQBkIowAAQAAAAAAiwABAAIAAAAfAIoAAQA7AAAAAwA1AAAA
       CQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACI
       AAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAXAAAAKwUAAHIBAAABADLU
       AAAAAAD//wEAMgCMAAEAAAAAAIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJv
       bAAAAAAAAKcBAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCH
       AAEABQAAAP////8AhgABAAQAAAAAAAAAcgACABoAAAA0BgAAcgEAAAEAAADAA+kAAAAAAP//
       AQDAA4wAAQAAAAAAlQABAAQAAAAAAAAAlgABAAIAAAAJAIwAAQAAAAAA
      </office:binary-data>
     </draw:image>
    </draw:frame>using a uniform distribution.</text:p>
   <text:p text:style-name="P1">The simulated wave is</text:p>
   <text:p text:style-name="P1"><draw:frame draw:style-name="fr2" draw:name="Object11" text:anchor-type="as-char" svg:width="2.4882in" svg:height="0.4547in" draw:z-index="13"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi mathvariant="normal">ζ</mi>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mi>t</mi>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mrow>
           <munderover>
            <mo stretchy="false">∑</mo>
            <mrow>
             <mi>i</mi>
             <mo stretchy="false">=</mo>
             <mn>1</mn>
            </mrow>
            <mi>n</mi>
           </munderover>
           <msub>
            <mi>A</mi>
            <mrow>
             <mi mathvariant="normal">ζ</mi>
             <mi>i</mi>
            </mrow>
           </msub>
          </mrow>
         </mrow>
         <mrow>
          <msup>
           <mi>e</mi>
           <mrow>
            <mi>j</mi>
            <mrow>
             <mo fence="true" form="prefix" stretchy="true">(</mo>
             <mrow>
              <mrow>
               <msub>
                <mi mathvariant="normal">ω</mi>
                <mi>i</mi>
               </msub>
               <mrow>
                <mi>t</mi>
                <mo stretchy="false">+</mo>
                <msub>
                 <mi mathvariant="normal">ϕ</mi>
                 <mrow>
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                  <mi>i</mi>
                 </mrow>
                </msub>
               </mrow>
              </mrow>
             </mrow>
             <mo fence="true" form="postfix" stretchy="true">)</mo>
            </mrow>
           </mrow>
          </msup>
          <mo stretchy="false">=</mo>
          <mrow>
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            <mrow>
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             <mo stretchy="false">=</mo>
             <mn>1</mn>
            </mrow>
            <mi>n</mi>
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           <mi mathvariant="normal">ζ</mi>
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         </mrow>
         <mrow>
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          <mrow>
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         </mrow>
         <msup>
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           <msub>
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           <mi>t</mi>
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        <annotation encoding="StarMath 5.0">%zeta left( t right)
=
sum from i=1 to n
A_{%zeta i} e^{j left(%omega_i t + %phi_{%zeta i} right)}
=
sum from i=1 to n
%zeta left( %omega_i right) e^{j %omega_i t }
</annotation>
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   <text:p text:style-name="P1">If</text:p>
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       <semantics>
        <mrow>
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         <mrow>
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          </mrow>
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         <mrow>
          <msup>
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           <mrow>
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         <msup>
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          <mrow>
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          </mrow>
         </msup>
        </mrow>
        <annotation encoding="StarMath 5.0">F left( %omega right) e^{i%omega t}
= 
T_{F%zeta}  left( %omega right)
%zeta left( %omega right) e^{i%omega t}</annotation>
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    </draw:frame>= the complex force transfer function.</text:p>
   <text:p text:style-name="P1">Then</text:p>
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=
sum from i=1 to n
A_{F i} e^{j left(%omega_i t + %phi_{F i} right)}
=
sum from i=1 to n
F left( %omega_i right) e^{j %omega_i t }
</annotation>
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           <mrow>
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        <annotation encoding="StarMath 5.0">A_{F i}
=
abs
{
T_{F%zeta}
}
A_{%zeta i}</annotation>
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       <semantics>
        <mrow>
         <mrow>
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            <mrow>
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=
%phi_{%zeta i}
+
arg left(T_{F%zeta} left( %omega right)right)</annotation>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="P2">Then</text:p>
   <text:h text:style-name="Heading_20_2" text:outline-level="2">Pierson Moskowitz Spectrum</text:h>
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         <mi>s</mi>
        </msub>
        <annotation encoding="StarMath 5.0">H_s</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame>= characteristic height (peak to peak)</text:p>
   <text:p text:style-name="P2"><draw:frame draw:style-name="fr1" draw:name="Object22" text:anchor-type="as-char" svg:width="0.2457in" svg:height="0.2138in" draw:z-index="21"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi>T</mi>
         <mi>z</mi>
        </msub>
        <annotation encoding="StarMath 5.0">T_z</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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    </draw:frame>= characteristic period</text:p>
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         <mfrac>
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          <msub>
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           <mi>z</mi>
          </msub>
         </mfrac>
        </mrow>
        <annotation encoding="StarMath 5.0">%omega_z 
= 
{2%pi} 
over 
{T_z}</annotation>
       </semantics>
      </math>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mi>S</mi>
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           <mi mathvariant="normal">ζ</mi>
          </mrow>
         </msub>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mi mathvariant="normal">ω</mi>
           </mrow>
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          </mrow>
          <mo stretchy="false">=</mo>
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          </mfrac>
         </mrow>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mfrac>
            <msubsup>
             <mi>H</mi>
             <mi>s</mi>
             <mn>2</mn>
            </msubsup>
            <msub>
             <mi mathvariant="normal">ω</mi>
             <mi>z</mi>
            </msub>
           </mfrac>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
         <msup>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mfrac>
             <mi mathvariant="normal">ω</mi>
             <msub>
              <mi mathvariant="normal">ω</mi>
              <mi>z</mi>
             </msub>
            </mfrac>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mrow>
           <mo stretchy="false">−</mo>
           <mn>5</mn>
          </mrow>
         </msup>
         <mi>exp</mi>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
            <mrow>
             <mo stretchy="false">−</mo>
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             </mfrac>
            </mrow>
            <msup>
             <mrow>
              <mo fence="true" form="prefix" stretchy="true">(</mo>
              <mrow>
               <mfrac>
                <mi mathvariant="normal">ω</mi>
                <msub>
                 <mi mathvariant="normal">ω</mi>
                 <mi>z</mi>
                </msub>
               </mfrac>
              </mrow>
              <mo fence="true" form="postfix" stretchy="true">)</mo>
             </mrow>
             <mrow>
              <mo stretchy="false">−</mo>
              <mn>4</mn>
             </mrow>
            </msup>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">S_{1%zeta}
left( %omega right)
=
{1}
over
{4%pi}
left(
{H^2_s}
over
{%omega_z}
right)
left(
{%omega}
over
{%omega_z}
right)^-5
exp
left(
-
{
{1}
over
{%pi}
}
left(
{%omega}
over
{%omega_z}
right)^-4
right)</annotation>
       </semantics>
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