EXPLOSION OVER AN ICE SHEET GOAL Calculate time of travel for sound through various media. PROBLEM An explosion occurs 275 m above an 867-m-thick ice sheet that lies over ocean water. If the air temperature is −7.00∘C, how long does it take the sound to reach a research vessel 1250 m below the ice?

Question

EXPLOSION OVER AN ICE SHEET

GOAL Calculate time of travel for sound through various media.

PROBLEM An explosion occurs [latex]275 \mathrm{~m}[/latex] above an 867-m-thick ice sheet that lies over ocean water. If the air temperature is [latex]-7.00^{\circ} \mathrm{C}[/latex], how long does it take the sound to reach a research vessel [latex]1250 \mathrm{~m}[/latex] below the ice? Neglect any changes in the bulk modulus and density with temperature and depth. (Use [latex]B_{ ext {ice }}=9.2 imes 10^{9} \mathrm{~Pa}[/latex].)

STRATEGY Calculate the speed of sound in air with Equation 14.4,

[latex]v = (331 \mathrm{m/s})\sqrt{\frac{T}{273 \mathrm{K}}}[/latex] [14.4]

and use [latex]d=v t[/latex] to find the time needed for the sound to reach the surface of the ice. Use Equation [latex]14.1[/latex]

[latex]v = \sqrt{\frac{B}{ ho}}[/latex] [14.1]

to compute the speed of sound in ice, again finding the time with the distance equation. Finally, use the speed of sound in salt water to find the time needed to traverse the water and then sum the three times.

Accepted Answer

Calculate the speed of sound in air at [latex]-7.00^{\circ} \mathrm{C}[/latex], which is equivalent to [latex]266 \mathrm{~K}[/latex] :

[latex]v_{ ext {air }} =(331 \mathrm{~m} / \mathrm{s}) \sqrt{\frac{T}{273 \mathrm{~K}}}=(331 \mathrm{~m} / \mathrm{s}) \sqrt{\frac{266 \mathrm{~K}}{273 \mathrm{~K}}}=327 \mathrm{~m} / \mathrm{s}[/latex]

Calculate the travel time through the air:

[latex]t_{ ext {air }} =\frac{d}{v_{ ext {air }}}=\frac{275 \mathrm{~m}}{327 \mathrm{~m} / \mathrm{s}}=0.841 \mathrm{~s}[/latex]

Compute the speed of sound in ice, using the bulk modulus and density of ice:

[latex]v_{ ext {ice }} =\sqrt{\frac{B}{ ho}}=\sqrt{\frac{9.2 imes 10^{9} \mathrm{~Pa}}{917 \mathrm{~kg} / \mathrm{m}^{3}}}=3.2 imes 10^{3} \mathrm{~m} / \mathrm{s}[/latex]

Compute the travel time through the ice:

[latex]t_{ ext {ice }} =\frac{d}{v_{ ext {ice }}}=\frac{867 \mathrm{~m}}{3 200 \mathrm{~m} / \mathrm{s}}=0.27 \mathrm{~s}[/latex]

Compute the travel time through the ocean water:

[latex]t_{ ext {water }} =\frac{d}{v_{ ext {water }}}=\frac{1 250 \mathrm{~m}}{1 533 \mathrm{~m} / \mathrm{s}}=0.815 \mathrm{~s}[/latex]

Sum the three times to obtain the total time of propagation:

[latex]\begin{aligned}t_{ ext {tot }} &=t_{ ext {air }}+t_{ ext {ice }}+t_{ ext {water }}=0.841 \mathrm{~s}+0.27 \mathrm{~s}+0.815 \mathrm{~s} \&=1.93 \mathrm{~s}\end{aligned}[/latex]

REMARKS Notice that the speed of sound is highest in solid ice, second highest in liquid water, and slowest in air. The speed of sound depends on temperature, so the answer would have to be modified if the actual temperatures of ice and the sea water were known. At [latex]0^{\circ} \mathrm{C}[/latex], for example, the speed of sound in sea water falls to [latex]1449 \mathrm{~m} / \mathrm{s}[/latex].

Question 14.1: EXPLOSION OVER AN ICE SHEET GOAL Calculate time of travel fo...

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