Consider a flat Earth with an atmosphere that decreases in density as altitude i

Question

Consider a flat Earth with an atmosphere that decreases in density as altitude increases such that rho = rho 0e-h/H, where rho 0 is the density of air at zero altitude and H is a constant known as the "scale height." Assume the bulk modulus of air is constant. Show that the intensity of a sound wave of constant wavelength will increase with altitude. (Hint: Assume the velocity changes due to a change in frequency only.) Show that the intensity of a sound wave of constant frequency will decrease with altitude. (Hint: Assume the velocity changes due to a change in wavelength only.) Show that the ratio of the derivatives is given by

Explanation / Answer

Each sound we hear has its own wavelength. When you can tell apart 2 different notes from the same piano, say middle-C and middle-D, what sounds different is its wavelength. (Other things can change, like timbre, which means 2 people or 2 instruments singing the same melody, you still can tell them apart.) But othe than in mmusic, each different sound is a different wavelength, which is constant unless the sound sounds different. Frequency & wavelength are associated. Frequency is how quickly the sound wave vibrates, or cycles, meaning how many times in one second it completes one cycle and goes back to where it startedd, just to start one more wave. Wavelength is the length of each cycle. Because speed has its own set velocity in air, that means that the longer the wavelength it has to travel in one cycle, the less is the frequency, i.e. it takes longer. When we say that a sound has a constant wavelength while it sounds the same, we also say that it has the same frequency. While in air, each frequency correspond to one wavelength. Sond does not travel in a vacuum, like light does. Sound works by moving around the air molecules, which are the ones that vibrate, i.e. the ones that performthe dance we hear as sound. As youmove up in the atmosphere, gravitational attraction from earth is weaker. The air molecules are farther apart. The sound waves take a longer time to find an air molecule and make it vibrate. The frequency goes down, the wavelength goes up, and the soundou hear sounds differently. For each sound performed the same way from the same piano, that sound will have a set frequency, wavelength and quality for each different altitude. They will be constant for each altitude. But, of course, we are neglecting other things, specially wind, weather, etc. If a hurricane is around, the effects of weather introduce a new set of calculations.

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