15.26 Earthquakes generate sound waves inside the earth. Unlike a gas, the earth can experience both transverse (S) and longitudinal (P) sound waves. Typically the speed of S wave is about 4.0 km s^–1, and that of P wave is 8.0 km s^–1. A seismograph records P and S waves from an earthquake. The first P wave arrives 4 min before the first S wave. Assuming the waves travel in straight line, at what distance does the earthquake occur ?

<p>Let&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>s</mi></mrow></mrow></msub></math></span>&nbsp;and&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>p</mi></mrow></mrow></msub></math></span>&nbsp;be the velocities and&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>s</mi></mrow></mrow></msub></math></span>&nbsp;and&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>p</mi></mrow></mrow></msub></math></span>&nbsp;be the time taken to reach the seismograph from the epicentre of S and P waves respectively.</p><p>Let L be the distance between the epicentre and the seismograph.</p><p>We have:</p><p>L =&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>s</mi></mrow></mrow></msub><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>s</mi></mrow></mrow></msub></math></span>&nbsp;…. (i)</p><p>L =&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>p</mi></mrow></mrow></msub><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>p</mi></mrow></mrow></msub></math></span>&nbsp;…. (ii)</p><p>It is given, &nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>s</mi></mrow></mrow></msub></math></span>&nbsp;= 4 km/s and&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>p</mi></mrow></mrow></msub></math></span>&nbsp;= 8 km/s</p><p>From equation (i) and (ii), we get</p><p>4&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>s</mi></mrow></mrow></msub><mi></mi></math></span>&nbsp;= 8&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>p</mi></mrow></mrow></msub></math></span>&nbsp;or&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>s</mi></mrow></mrow></msub><mo>=</mo><mi></mi></math></span>&nbsp;2&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>p</mi></mrow></mrow></msub></math></span>&nbsp;…. (iii)</p><p>It is also given, </p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>s</mi></mrow></mrow></msub><mo>-</mo><mi></mi><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>p</mi></mrow></mrow></msub><mo>=</mo><mn>240</mn><mi></mi><mi>s</mi></math></span>&nbsp;so&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>p</mi></mrow></mrow></msub><mo>=</mo><mn>240</mn><mi></mi><mi>s</mi><mi></mi><mi>a</mi><mi>n</mi><mi>d</mi><mi></mi><msub><mrow><mrow><mi>t</mi></mrow></mrow><mrow><mrow><mi>s</mi></mrow></mrow></msub><mo>=</mo><mn>480</mn><mi></mi><mi>s</mi></math></span></p><p>From equation (ii), we get, L = 8&nbsp;<span title="Click to copy mathml"><math><mo>*</mo><mn>240</mn></math></span>&nbsp;= 1920 km</p><p>Hence, the earthquake occurred at a distance of 1920 km from the seismograph.</p>

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