Solved Problem on Acoustics

A person emits a sound in front of a wall and after ¾ of a second hears the echo. Calculate the distance to the wall. The air temperature is 15 °C, and at this temperature, the speed of sound is 340 m/s.

Problem data:

Time interval between sound emission and reception: Δ t = ¾ s;
Speed of sound in air at 15°C: v = 340 m/s;
Ambient temperature: T = 15°.

Problem diagram:

Considering an element in a wavefront emitted by the person, this element travels a distance x to the wall, and the echo travels another distance x to the person (Figure 1).

Figure 1

Thus, the problem reduces to a material point traveling a distance ΔS = x+x = 2x at a constant speed v = 340 m/s.

Solution

From the Kinematics the average speed is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {\bar v=\frac{\Delta S}{\Delta t}} \end{gather} \]

the speed of sound is constant, solving for the displacement traveled and substituting the problem data

\[ \begin{gather} \Delta S=\bar v\Delta t\\[5pt] \Delta S=340\times\frac{3}{4}\\[5pt] \Delta S=\frac{1020}{4}\\[5pt] \Delta S=255\text{m} \end{gather} \]

Since Δ S represents the round trip distance to the wall, the distance from the person to the wall will be half

\[ \begin{gather} 2x=\Delta S\\[5pt] x=\frac{\Delta S}{2}\\[5pt] x=\frac{255}{2} \end{gather} \]

\[ \begin{gather} \bbox[#FFCCCC,10px] {x=127.5\;\text{m}} \end{gather} \]

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