In the given progressive wave y = 5 sin (100πt – 0.4πx ) where y and x are in m, t is in s. What is the

(a) Amplitude

(b) Wave length

(c) Frequency

(d) Wave velocity

(e) Particle velocity amplitude.

<p><span data-teams="true">This is a long answer type question as classified in NCERT Exemplar</span></p><p>standard equation of a progressive wave is given by</p><p>Y=asin(wt-kx+<span contenteditable="false"> <math> <mi>∅</mi> </math> </span>)</p><p>&nbsp;This is travelling along positive x- direction</p><p>Given equation is y=5sin(100<span contenteditable="false"> <math> <mi>π</mi> <mi>t</mi> <mo>-</mo> <mn>0.4</mn> <mi>π</mi> <mi>x</mi> </math> </span>)</p><p>Comparing with standard equation</p><p><strong>(a)</strong> amplitude =5m</p><p><strong>(b)</strong> k=2<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mi>γ</mi> </mrow> </mrow> </mfrac> </math> </span>=0.4x</p><p>wavelength =2<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </mfrac> </math> </span>=<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mrow> <mrow> <mrow> <mn>0.4</mn> <mi>π</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mn>20</mn> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>5</mn> <mi>m</mi> </math> </span></p><p><strong>(c)</strong> w=10<span contenteditable="false"> <math> <mi>π</mi> </math> </span></p><p>w=2<span contenteditable="false"> <math> <mi>π</mi> <mi>v</mi> <mo>=</mo> <mn>100</mn> <mi>π</mi> </math> </span></p><p>frequency v= 100<span contenteditable="false"> <math> <mi>π</mi> <mo>/</mo> <mn>2</mn> <mi>π</mi> </math> </span>=50Hz</p><p><strong>(d) </strong>wave velocity <span contenteditable="false"> <math> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mi>w</mi> </mrow> </mrow> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </mfrac> <mi> </mi> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal"> </mi> <mi mathvariant="normal">k</mi> <mi mathvariant="normal"> </mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal"> </mi> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal"> </mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal"> </mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi mathvariant="normal"> </mi> <mi mathvariant="normal">k</mi> <mo>=</mo> <mn>2</mn> <mi mathvariant="normal">π</mi> <mo>/</mo> <mi mathvariant="normal">λ</mi> </math> </span></p><p>=100<span contenteditable="false"> <math> <mi>π</mi> <mo>/</mo> <mn>0.4</mn> <mi>π</mi> </math> </span>=1000/4</p><p>250m/s</p><p><strong>(e) </strong>y= 5sin(100<span contenteditable="false"> <math> <mi>π</mi> <mi>t</mi> <mo>-</mo> <mn>0.4</mn> <mi>π</mi> <mi>x</mi> </math> </span>)</p><p>dy/dt = particle velocity</p><p>dy/dt = 5(100<span contenteditable="false"> <math> <mi>π</mi> <mo>)</mo> </math> </span>cos[100<span contenteditable="false"> <math> <mi>π</mi> <mi>t</mi> <mo>-</mo> <mn>0.4</mn> <mi>π</mi> <mi>x</mi> </math> </span>]</p><p>for particle velocity amplitude (dy/dt)_max which will be for cos[100<span contenteditable="false"> <math> <mi>π</mi> <mi>t</mi> <mo>-</mo> <mn>0.4</mn> <mi>π</mi> <mi>x</mi> </math> </span>]_max=1</p><p>so particle velocity amplitude =(dy/dt)_max=5(100<span contenteditable="false"> <math> <mi>π</mi> </math> </span>)<span contenteditable="false"> <math> <mo>×</mo> <mn>1</mn> </math> </span>=550<span contenteditable="false"> <math> <mi>π</mi> </math> </span>&nbsp;m/s</p>

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