Michelle is enjoying a picnic across the valley from a...

Michelle is enjoying a picnic across the valley from a cliff. She is playing music on her radio (assume it to be an isotropic source) and notices an echo from the cliff. She claps her hands and the echo takes 1.5 s to return. (a) Given that the speed of sound in air is $343 \mathrm{m} / \mathrm{s}$ on that day, how far away is the cliff? (b) If the intensity of the music $1.0 \mathrm{m}$ from the radio is $1.0 \times 10^{-5} \mathrm{W} / \mathrm{m}^{2},$ what is the intensity of the music arriving at the cliff?

Key Concepts

Speed of Sound

This concept refers to the rate at which sound waves travel through a medium, such as air, and is influenced by factors like temperature, humidity, and air pressure. It serves as a crucial parameter in relating the time taken by sound to travel a certain distance, which is fundamental in calculations involving propagation delays, echoes, or time of arrival measurements.

Distance Calculation from Echoes

Using the echo, one can determine distance by multiplying the round-trip travel time of the sound by the speed of sound and then dividing by two. This technique is commonly applied in sonar, radar, and similar systems where the time delay of a reflected signal is used to infer the distance to an object or surface.

Isotropic Sound Source

An isotropic source emits energy uniformly in all directions. In the context of sound, this means that the intensity of the sound decreases as one moves farther away from the source, following a predictable pattern. This concept is vital for understanding how sound spreads out in space and how its intensity diminishes with distance.

Inverse Square Law for Sound Intensity

This law states that the intensity of a sound from a point source decreases in proportion to the square of the distance from the source. This principle is essential in acoustics to determine how loudness or sound pressure levels decrease with distance, allowing calculations of sound intensity at various points in space based on a known intensity at a specific distance.

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