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# Lec2 - We now let x x vt to have a wave traveling to the...

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We now let x x - vt to have a wave traveling to the right. I.e., f ( x, t ) = y m sin[ k ( x - vt )] = y m sin( kx - kvt ) = y m sin( kx - ω t ) (1) where ω := kv , v = ω k , and ω is the angular frequency. At any x , the motion is a simple harmonic motion. Let t t + 2 π ω . Then, f ( x, t + 2 π ω ) = y m sin[ kx - ω ( t + 2 π ω )] = y m sin( kx - ω t - 2 π ) = y m sin( kx - ω t ) (2) therefore, T = 2 π ω is the period. Recall that f = 1 T = ω 2 π ω = 2 π f . v = ω k = 2 π k × ω 2 π = λ T so that the wave travels a distance λ over a time T . Another view: Take a snapshot in time: so that k λ = 2 π . 4

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Snapshot in space: so that ω T = 2 π . String Velocity (in the transverse case) String Velocity: Not the same as wave velocity! u ( x, t ) = f t x =const (3) Because we’re looking at how fast each material point is moving: the velocity here is of a specific piece of the string. If f ( x, t ) = y m sin( kx - ω t ) then u ( x, t ) = f t = - ω y m cos( kx - ω t ). Di ff erent points have di ff erent velocity at same time. Same point has di ff erent velocities at di ff erent times.
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Lec2 - We now let x x vt to have a wave traveling to the...

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