At what latitude does the earth rotate at the same speed as sound?

[fuzzyduff]

I am an artist, and have begun working on a project based primarily in the origins of the metric system, however my research has led me into theories about potential standardised basic units of measurement suggested in the past.

One of these was by Gabriel Mouton in 1670, suggesting that the world should adopt a uniform system of measurement based on the length of one minute of the earth's arc.

 

Despite my high school level grasp of mathematics and physics I've managed to figure out how fast I am spinning here in Banff, Canada (1050.21 km/h) based on how quickly the earth rotates at a fixed point on the Equator (1675 km/h) however I thought it would also be interesting to see also how fast we are moving in comparison to the speed of sound (which is another interest of mine).

 

My question is this:

If we are moving at 1.35 x speed of sound at the equator (0 degrees latitude) and 0.85 x speed of sound in Banff ( approx 51 degrees latitude) at what latitude do we on the globe rotate at the same speed as sound?

 

 

[Janus]

Start with the assumed speed of sound (340.29 m/s at sea level)

If you divide this by the angular velocity of the Earth (7.29e-5 rad/sec), you get the distance from the axis of rotation you need to be in order to be traveling at a the speed of sound (relative to the axis.)

Dividing this by the radius of the Earth (~6378000 m) gives you the sin of the Latitude.( Assuming a spherical Earth, Unless you need to be extremely accurate you shouldn't need to take the Earth's oblateness into account.)

Take the arcsin of this to get the North or South latitude.

[mistermack]

Just in case there's some confusion, it should be pointed out that the atmosphere is spinning WITH the Earth, so the speed of sound will not vary much relative to the Earth, apart from local wind conditions.

[Sriman Dutta]

Speed of sound in normal atmospheric conditions = 332 m/s

We know that tangential velocity = angular velocity x radius from centre/ axis of rotation

So, radius = velocity / angular velocity

 

 

So cos (latitude) = obtained radius / mean radius of earth

[Sensei]

At what latitude does the earth rotate at the same speed as sound?

We should start from the fact that speed of sound is not constant, and depends on medium (f.e. speed of sound is much higher in solids/liquids than in gases), and variable properties of medium (f.e. pressure/temperature).

You can actually hear how speed of sound changes with altitude on this video taken from starting homemade rocket:

Start at 2m15s

 

Edited April 7, 2017 by Sensei

[J.C.MacSwell]

...and of course sound doesn't generally propagate along lines of latitude, the equator and wind and topography effects notwithstanding. Though it could be very close locally, even at the right latitude you would not generally get stationary waves wrt an inertial frame...at least not for long

[Manticore]

I am an artist, and have begun working on a project based primarily in the origins of the metric system, however my research has led me into theories about potential standardised basic units of measurement suggested in the past.

One of these was by Gabriel Mouton in 1670, suggesting that the world should adopt a uniform system of measurement based on the length of one minute of the earth's arc.

 

 

We have a uniform system based on one minute of earth's arc. It's called the Nautical Mile - ask any pilot or sailor.