\(\textbf{Longitudinal Waves on a Spring.}\) A long spring such as a
Slinky\(^{\mathrm{TM}}\) 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 \(k'\), the speed of longitudinal waves on the spring is \(v =
L\sqrt{ k'/m}\) (see Section 16.2). (b) Evaluate \(v\) for a spring with \(m =\)
0.250 kg, \(L =\) 2.00 m, and \(k' =\) 1.50 N\(/\)m.
\(\textbf{ULTRASOUND IMAGING}\). A typical ultrasound transducer used for
medical diagnosis produces a beam of ultrasound with a frequency of 1.0 MHz.
The beam travels from the transducer
through tissue and partially reflects when it encounters different structures
in the tissue. The same transducer that produces the ultrasound also detects
the reflections. The transducer emits a short pulse of ultrasound and waits to
receive the reflected echoes before emitting the next pulse. By measuring the
time between the initial pulse and the arrival of the reflected signal, we can
use the speed of ultrasound in tissue, 1540 m/s, to determine the distance
from the transducer to the structure that produced the reflection.
As the ultrasound beam passes through tissue, the beam is attenuated through
absorption. Thus deeper structures return weaker echoes. A typical attenuation
in tissue is \(-\)100 dB/m \(\cdot\) MHz; in bone it is \(-\)500 dB/m \(\cdot\) MHz.
In determining attenuation, we take the reference intensity to be the
intensity produced by the transducer.
