15.6 A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted sound? Speed of sound in air is 340 m s –1 and in water 1486 m s–1.
The wavelength of the reflected sound is given by the relation
= = 3.4 m
The wavelength of the transmitted sound wave is given by
= = 1.486 m
<p>Frequency of ultrasound, <span title="Click to copy mathml"><math><mi>ν</mi></math></span> = 1000 kHz = <span title="Click to copy mathml"><math><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>6</mn></mrow></mrow></msup></math></span> Hz</p><p>Speed of sound in air, <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>a</mi></mrow></mrow></msub></math></span> = 340 m/s</p><p>Speed of sound in water, <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>w</mi></mrow></mrow></msub></math></span> = 1486 m/s</p><p>The wavelength of the reflected sound is given by the relation</p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>λ</mi></mrow></mrow><mrow><mrow><mi>r</mi></mrow></mrow></msub><mo>=</mo><mi></mi><mfrac><mrow><mrow><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>a</mi></mrow></mrow></msub></mrow></mrow><mrow><mrow><mi>ν</mi></mrow></mrow></mfrac></math></span> = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>340</mn></mrow></mrow><mrow><mrow><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>6</mn></mrow></mrow></msup></mrow></mrow></mfrac></math></span> = 3.4 <span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>4</mn></mrow></mrow></msup></math></span> m</p><p>The wavelength of the transmitted sound wave is given by</p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>λ</mi></mrow></mrow><mrow><mrow><mi>t</mi></mrow></mrow></msub><mo>=</mo><mi></mi><mfrac><mrow><mrow><msub><mrow><mrow><mi>v</mi></mrow></mrow><mrow><mrow><mi>w</mi></mrow></mrow></msub></mrow></mrow><mrow><mrow><mi>ν</mi></mrow></mrow></mfrac></math></span> = <span title="Click to copy mathml"><math><mfrac><mrow><mrow><mn>1486</mn></mrow></mrow><mrow><mrow><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>6</mn></mrow></mrow></msup></mrow></mrow></mfrac></math></span> = 1.486 <span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mo>-</mo><mn>3</mn></mrow></mrow></msup></math></span> m</p>
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