SoundWaves - previous index next Sound Waves Michael Fowler...

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previous index next Sound Waves Michael Fowler 3/23/07 “One-Dimensional” Sound Waves We’ll begin by considering sound traveling down a hollow pipe, to avoid unnecessary mathematical complications. Sound is a longitudinal wave—as the wave passes through, the air moves backwards and forwards in the pipe, this oscillatory movement is in the same direction the wave is traveling. To visualize what’s happening, imagine mentally dividing the air in the pipe, which is at rest if there is no sound, into a stack of thin slices. Think about one of these slices. In equilibrium, it feels equal and opposite pressure from the gas on its two sides. (This is analogous to the little bit of string at rest feeling equal and opposite tension on its two sides, but of course the gas pressure is inward). As the sound wave goes through, the pressure wave generates slight differences in pressure on the two sides of our thin slice of air, and this imbalance of forces causes the slice to accelerate. To analyze this quantitatively—to apply Fm a = G G to the thin slice of air—we must begin by defining displacement , the quantity corresponding to the string’s transverse movement ( ) , yxt . We shall use ( ) , sxt to denote the horizontal (along the pipe) displacement of the thin slice of air which rests at position x when no sound is present. . An animated version of this diagram is available here !
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2 If the pipe has radius a , and hence cross-sectional area 2 a π , a slice of air of thickness x Δ has volume , so writing the density of air 2 ax Δ ρ (1.29 kg/m 3 ), the mass of the slice of air is . Clearly, its acceleration is 2 mV a ρρ == Δ x ( ) 2 ,/ as x t 2 t = ∂∂ , so we already have the right- hand side of . To find the left hand side—the force on the thin slice of air—we must find the pressure imbalance between the two sides.
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SoundWaves - previous index next Sound Waves Michael Fowler...

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