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font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false"/> <mathStyle name="Math Text Font" font-family="Trebuchet MS" font-charset="0" font-size="11" font-weight="bold" font-style="normal" underline="false"/> </mathStyles> <dimensionNames mass="mass" length="length" time="time" current="current" thermodynamic-temperature="temperature" luminous-intensity="luminosity" amount-of-substance="substance" display="false"/> <symbolics derivation-steps-style="vertical-insert" show-comments="false" evaluate-in-place="false"/> <results numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="1" orientation="landscape" print-single-page-width="false" page-width="792" page-height="612"> <margins left="50.328" right="50.328" top="50.328" bottom="50.328"/> <header use-full-page-width="false"/> <footer use-full-page-width="false"/> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="882" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="260" left="12" top="6.75" width="251.25" height="14.25" align-x="12" align-y="18" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f size="12"> <b> <c val="#f00">"Mathcad_waves_lecture_3.xmcd"</c> </b> </f> </p> </text> </region> <region region-id="287" left="18" top="44.25" width="119.25" height="12" align-x="32.25" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>Assign wave para</b> <b>meters</b>:</p> </text> </region> <region region-id="855" left="150" top="43.5" width="43.5" height="15" align-x="164.25" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">λ</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">cm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="856" left="270" top="44.25" width="72" height="12" align-x="282.75" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>Wave velocity:</b> </p> </text> </region> <region region-id="363" left="348" top="35.25" width="45.75" height="32.25" align-x="360" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">v</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>1</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">cm</ml:id> <ml:id xml:space="preserve">s</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="364" left="462" top="35.25" width="38.25" height="32.25" align-x="474.75" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">k</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult style="auto-select"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">λ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="365" left="582" top="35.25" width="49.5" height="32.25" align-x="596.25" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ω</ml:id> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult style="auto-select"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">π</ml:id> </ml:apply> <ml:id xml:space="preserve">v</ml:id> </ml:apply> <ml:id xml:space="preserve">λ</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="285" left="18" top="80.25" width="152.25" height="12" align-x="18" align-y="90" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>For moving wave described by</b>: </p> </text> </region> <region region-id="286" left="198" top="79.5" width="69" height="15" align-x="211.5" align-y="90" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="6"/> </region> <region region-id="300" left="18" top="116.25" width="288.75" height="12" align-x="28.5" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>Trap this wave in a 1D box running from x = 0 to x = L<sp count="3"/>with:</b> </p> </text> </region> <region region-id="862" left="330" top="115.5" width="42" height="15" align-x="342.75" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">L</ml:id> <ml:apply> <ml:mult style="auto-select"/> <ml:real>5</ml:real> <ml:id xml:space="preserve">cm</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="865" left="54" top="151.5" width="171" height="19.5" align-x="147.75" align-y="162" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="right_1">wave</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="303" left="18" top="182.25" width="194.25" height="12" align-x="27.75" align-y="192" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>This reflects back to left at x = L forming:</b> </p> </text> </region> <region region-id="331" left="54" top="211.5" width="198" height="19.5" align-x="141.75" align-y="222" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="left_1">wave</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">k</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="auto-select"/> <ml:real font="0">2</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="880" left="294" top="212.25" width="402.75" height="12" align-x="302.25" align-y="222" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>this properly matches waves R1 & L1 at x = L:<sp count="2"/>sin(kL-</b> <f family="Symbol" charset="2" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">w</inlineAttr> </b> </f> <f family="Arial" charset="0" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">t) = sin(k(2L-L)-</inlineAttr> </b> </f> <f family="Symbol" charset="2" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">w</inlineAttr> </b> </f> <f family="Arial" charset="0" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">t)</inlineAttr> </b> </f> </p> </text> </region> <region region-id="325" left="18" top="242.25" width="471.75" height="12" align-x="27" align-y="252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>And then wave L1 reflects reaches left end of box (at x=0), it reflects back to right:</b> </p> </text> </region> <region region-id="333" left="54" top="271.5" width="204" height="19.5" align-x="147.75" align-y="282" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="right_2">wave</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">k</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="881" left="294" top="272.25" width="369" height="12" align-x="314.25" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>properly matching waves L1 & R2 at x = 0:<sp count="3"/>sin(k(2L-0)-</b> <f family="Symbol" charset="2" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">w</inlineAttr> </b> </f> <f family="Arial" charset="0" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">t</inlineAttr> </b> </f> <b>) = sin(k(2L+0)-</b> <f family="Symbol" charset="2" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">w</inlineAttr> </b> </f> <f family="Arial" charset="0" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">t)</inlineAttr> </b> </f> <b> <sp/> </b> </p> </text> </region> <region region-id="328" left="18" top="302.25" width="465.75" height="12" align-x="27" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>And then wave R3 when it reaches right end of box (at x=L), this wave reflects back to left: </b> </p> </text> </region> <region region-id="338" left="54" top="331.5" width="198" height="19.5" align-x="141.75" align-y="342" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="left_2">wave</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">k</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="882" left="288" top="332.25" width="415.5" height="12" align-x="308.25" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>properly matching waves R2 & L2 at x = L:<sp count="2"/>sin(k(2L+L)-</b> <f family="Symbol" charset="2" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">w</inlineAttr> </b> </f> <f family="Arial" charset="0" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">t) = sin(k(4L-L) - </inlineAttr> </b> </f> <f family="Symbol" charset="2" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">w</inlineAttr> </b> </f> <f family="Arial" charset="0" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">t)</inlineAttr> </b> </f> </p> </text> </region> <region region-id="864" left="18" top="362.25" width="505.5" height="12" align-x="27" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>And then wave L2 when it reaches right end of box (at x=0), this wave relfecs back to right:</b> </p> </text> </region> <region region-id="344" left="60" top="397.5" width="204" height="19.5" align-x="153.75" align-y="408" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="right_3">wave</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">k</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="auto-select"/> <ml:real>4</ml:real> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="870" left="288" top="398.25" width="366" height="12" align-x="308.25" align-y="408" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>properly matching waves L2 & R3 at x = 0:<sp count="2"/>sin(k(4L-0)-</b> <f family="Symbol" charset="2" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">w</inlineAttr> </b> </f> <f family="Arial" charset="0" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">t) = sin(k(4L+0) - </inlineAttr> </b> </f> <f family="Symbol" charset="2" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">w</inlineAttr> </b> </f> <f family="Arial" charset="0" size="10"> <b> <inlineAttr font-style="normal" underline="false" line-through="false">t)</inlineAttr> </b> </f> </p> </text> </region> <region region-id="348" left="18" top="434.25" width="291.75" height="12" align-x="23.25" align-y="444" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <b>So, looking at series, sum of waves (through 2N terms):</b> </p> </text> </region> <region region-id="351" left="60" top="453" width="405.75" height="51.75" align-x="161.25" align-y="480" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">wave_total</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">k</ml:id> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">k</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:id xml:space="preserve">L</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="495" left="444" top="517.5" width="111.75" height="15" align-x="473.25" align-y="528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">time</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>0.1</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> <ml:id xml:space="preserve">sec</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="491" left="216" top="540" width="420.75" height="234.75" align-x="216" align-y="540" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="15"/> <rendering item-idref="16"/> </region> <region region-id="488" left="18" top="542.25" width="177.75" height="84" align-x="32.25" align-y="552" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <b>Check by putting consecutive waves, offset from one another, in single animation.</b> </p> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <b>Make sure as descend (from wave to its reflection) that waves are indeed mirrored at boundary:</b> </p> </text> </region> <region region-id="489" left="24" top="662.25" width="141.75" height="24" align-x="24" align-y="672" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">YES, consecutive waves stay matched at edges</p> </text> </region> <region region-id="844" left="96" top="1052.25" width="450" height="12" align-x="101.25" align-y="1062" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"> <b>Add initial wave + 24 reflections.<sp count="2"/>Animate versus time while slowly increasing the wavelength</b> </p> </text> </region> <region region-id="845" left="96" top="1073.25" width="105" height="32.25" align-x="110.25" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math30"> <originRef doc-id="3DA56944-91B1-43C4-AF76-AA2A88A955C8" version-id="C67CA19E-8C13-41AF-BCFE-5573F4B92155" branch-id="00000000-0000-0000-0000-000000000000" revision-num="5" is-modified="false" region-id="0" href="C:\Documents and Settings\John C. 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Bean\My Documents\Mathcad ENGR 250 - Lecture 2\Mathcad_waves_lecture_3b.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash>a8af780f0f9c956f32790e0ace31e7df</hash> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">ac52b4308135304cc967f0d5acb07208</contentHash> <ml:define warning="WarnRedefinedUDScalar"> <ml:id xml:space="preserve">λ</ml:id> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">FRAME</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:id xml:space="preserve">cm</ml:id> <ml:real>20</ml:real> </ml:apply> </ml:apply> </ml:define> </ml:provenance> </math> <rendering item-idref="17"> <element-image-map> <box left="1.5" top="0.75" width="103.5" height="30.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="846" left="288" top="1081.5" width="71.25" height="15" align-x="317.25" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math30"> <originRef doc-id="45CEC201-B842-4451-A033-8565E8425A90" version-id="115C45E8-A6F7-4B84-BFD8-861CFD558B8C" branch-id="00000000-0000-0000-0000-000000000000" revision-num="5" is-modified="true" region-id="0" href="C:\Documents and Settings\John C. 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Bean\My Documents\Mathcad ENGR 250 - Lecture 2\Mathcad_waves_lecture_3b.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash>a8af780f0f9c956f32790e0ace31e7df</hash> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">b260ae88f4ef549a9ab8d0b2eb0b57a6</contentHash> <ml:define warning="WarnRedefinedUDScalar"> <ml:id xml:space="preserve">time</ml:id> <ml:id xml:space="preserve">FRAME</ml:id> </ml:define> </ml:provenance> </math> <rendering item-idref="18"> <element-image-map> <box left="1.5" top="0.75" width="69.75" height="13.5" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet30"/> </element-image-map> </rendering> </region> <region region-id="847" left="456" top="1081.5" width="42" height="15" align-x="468.75" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math30"> <originRef doc-id="45CEC201-B842-4451-A033-8565E8425A90" version-id="115C45E8-A6F7-4B84-BFD8-861CFD558B8C" branch-id="00000000-0000-0000-0000-000000000000" revision-num="5" is-modified="true" region-id="0" href="C:\Documents and Settings\John C. 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