Sound Waves — Question 4

Question 4

Two elastic arrivals can reveal a source distance

An impulsive source launches longitudinal and transverse bulk waves in a uniform isotropic solid at speeds $c_L=6000$ and $c_T=3500\,\mathrm{m/s}$. A receiver observes their arrivals $0.50\,\mathrm s$ apart along the same straight path.

1. Infer path length and each travel time.

2. If the transverse arrival is absent in a liquid, can the same two-arrival method still determine distance? Explain the physical and informational limitations.

Question 4 - Solution

Two elastic arrivals can reveal a source distance

(a) Method and calculation

The slower transverse arrival follows the longitudinal one:$$\Delta t=r\left(\frac1{c_T}-\frac1{c_L}\right),\quad r=\frac{0.50}{1/3500-1/6000}=4200\,\mathrm m.$$$$t_L=4200/6000=0.70\,\mathrm s,\qquad t_T=1.20\,\mathrm s.$$

(b) Interpretation and further calculation

An ordinary fluid has no static shear stiffness and therefore no corresponding propagating bulk shear mode. With only an arrival time $t_a$,$$t_a=t_e+r/c_L$$contains both unknown emission time $t_e$ and unknown distance $r$. Another timing reference or independent geometric measurement is needed. Absence of a pulse alone is not proof of a liquid; source polarization or receiver sensitivity could hide it.

Physical check

As the two speeds approach equality, the delay loses sensitivity to distance. The inference assumes both pulses take the same path.