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The following was written in my book

When sound waves travel in air, air molecules do not move. When wind blows,air molecules move. This produces variation in velocity of sound waves.

The first statement imply that sound wave does not cause any change to/depend on air molecules. But the second and third statement says wind which causes air molecules to move produces variation in velocity of sound waves which does not depended on by air molecules. Why is this so?

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Your book is (a little bit) wrong. Sound is a pressure wave - and a pressure wave exists when molecules move a little bit away from and towards the source (they are longitudinal waves). However, there is no NET movement - in other words, they don't travel with the sound. In that sense, they don't leave their approximate position - but if they didn't move at all, there would be no wave and no sound.

When the entire medium (the volume of air transmitting sound) moves, then the sound wave propagates from point A to point B with the combined velocity of sound in stationary air, plus the velocity of the wind. You can think of an experiment in a train. Measure the speed of sound from one end of the rail car to the other and back. You won't be able to tell whether the train is moving because the air appears to be still in your frame of reference. However, if the train is moving, somebody from outside the train who observes your experiment will see the sound moving "faster" when it is traveling in the direction of the train (it will cross the length of the rail car in the same amount of time that you observed, but it will appear to have covered a greater distance). Similarly they will see it moving "slower" when the sound goes from the front of the train to the back.

When there is no train but the wind is moving the air, the exact same thing is still valid.

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Air molecules move as they transmit sound. They oscillate from the starting point to any intermediate points and then back to somewhere near the starting point again, ideally of course.

If they didn't move, we wouldn't be able to hear. I'm pretty amazed that sentence is written in your book, unless it's for very young readers.

Come to think of it, is there anything in the universe that doesn't move?

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