Chapter 16

The sound source of a ship's sonar system operates at a frequency of 22.0 \(\mathrm{kHz}\) . The speed of sound in water (assumed to be at a uniform \(20^{\circ} \mathrm{C}\) ) is 1482 \(\mathrm{m} / \mathrm{s}\) . (a) What is the wavelength of the waves emitted by the source? (b) What is the difference in frequency between the directly radiated waves and the waves reflected from a whale traveling directly toward the ship at 4.95 \(\mathrm{m} / \mathrm{s}^{\prime \prime}\) The ship is at rest in the water.

On a clear day you see a jet plane flying overhead. From the apparent size of the plane, you determine that it is flying at a constant altitude \(h .\) You hear the sonic boom at time \(T\) after the plane passes directly overhead. Show that if the speed of sound \(v\) is the same at all altitudes, the speed of the plane is $$v_{\mathrm{S}}=\frac{h v}{\sqrt{h^{2}-v^{2} T^{2}}}$$ (Hint: Trigonometric identities will be useful.)

cp Longitudinal Waves on a Spring. A long spring such as a Slinky"is is often used to demonstrate longitudinal waves. (a) Show that if a spring that obeys Hooke's law has mass \(m,\) length \(L,\) and force constant \(k^{\prime},\) the speed of longitudinal waves on the spring is \(v=L \sqrt{k^{\prime} / m}\) . (see Section 16.2\()\) (b) Evaluate \(v\) for a spring with \(m=0.250 \mathrm{kg}, L=2.00 \mathrm{m},\) and \(k^{\prime}=1.50 \mathrm{N} / \mathrm{m}\)