15.27 A bat is flitting about in a cave, navigating via ultrasonic beeps. Assume that the sound emission frequency of the bat is 40 kHz. During one fast swoop directly toward a flat wall surface, the bat is moving at 0.03 times the speed of sound in air. What frequency does the bat hear reflected off the wall?

<p>Ultrasonic beep frequency emitted by bat,&nbsp;<span title="Click to copy mathml"><math><mi>ν</mi></math></span>&nbsp;= 40 kHz</p><p>Velocity of the bat,&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>b</mi></mrow></mrow></msub></math></span>&nbsp;= 0.03v, where v = velocity of sound in air</p><p>The apparent frequency of the sound striking the wall is given as:</p><p><span title="Click to copy mathml"><math><msup><mrow><mrow><mi>ν</mi></mrow></mrow><mrow><mrow><mi>'</mi></mrow></mrow></msup><mo>=</mo><mo>(</mo><mfrac><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>v</mi><mo>-</mo><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>b</mi></mrow></mrow></msub></mrow></mrow></mfrac></math></span>&nbsp;)&nbsp;<span title="Click to copy mathml"><math><mi></mi><mi>ν</mi></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mo>(</mo><mfrac><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>v</mi><mo>-</mo><mn>0.03</mn><mi>v</mi></mrow></mrow></mfrac></math></span>&nbsp;)&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mn>40</mn></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>40</mn></mrow></mrow><mrow><mrow><mn>0.97</mn></mrow></mrow></mfrac></math></span>&nbsp;kHz = 41.24 kHz</p><p>This frequency is reflected by the stationary wall (&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>s</mi></mrow></mrow></msub><mo>=</mo><mn>0</mn><mo>)</mo></math></span>&nbsp;towards the bat</p><p>The frequency (&nbsp;<span title="Click to copy mathml"><math><mi>ν</mi><mi>'</mi><mi>'</mi><mo>)</mo><mi></mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">v</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">y</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mo>:</mo></math></span></p><p><span title="Click to copy mathml"><math><msup><mrow><mrow><mi>ν</mi></mrow></mrow><mrow><mrow><mi>'</mi><mi>'</mi></mrow></mrow></msup><mo>=</mo><mo>(</mo><mi></mi><mfrac><mrow><mrow><mi>v</mi><mo>+</mo><mi></mi><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>b</mi></mrow></mrow></msub></mrow></mrow><mrow><mrow><mi>v</mi></mrow></mrow></mfrac></math></span>&nbsp;)&nbsp;<span title="Click to copy mathml"><math><mi></mi><mi>ν</mi><mi>'</mi></math></span>&nbsp;= (&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>v</mi><mo>+</mo><mn>0.03</mn><mi>v</mi></mrow></mrow><mrow><mrow><mi>v</mi></mrow></mrow></mfrac></math></span>&nbsp;)&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mn>41.24</mn><mi></mi><mi mathvariant="normal">k</mi><mi mathvariant="normal">H</mi><mi mathvariant="normal">z</mi></math></span>&nbsp;= 1.03&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mn>41.24</mn></math></span>&nbsp;= 42.47 kHz</p>

0 Upvotes 0 Upvotes 0 Downvotes

• • Report Abuse
• Comment