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11 Waves 2 - Generic equation for a wave traveling in...

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Generic equation for a wave traveling in positive x -direction with wave speed v : ) ( ) , ( vt x f t x y = Therefore the disturbance is the same, as long as is constant, say 0 0 x vt x x vt x + = = A point of constant disturbance, y ( x 0 ) , (crest, trough, etc.) moves at the wave speed, v Here can be ANY function. The type of the function specifies the shape of the wave. ) ( x f ) ( x f How do we know it is a propagating (traveling) wave? (the disturbance) depends on and in a VERY SPECIAL WAY: it only depends on y x t vt x y vt x 0 x vt x = ) 0 , ( ) , ( 0 x y t x y =
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Example: a bell-shaped (Gaussian) curve with a peak at 0 = x ) exp( ) ( ) ( 2 x x f x y = = A bell-shaped (Gaussian) curve with a peak at b x = ] ) ( exp[ ) ( 2 b x x y = What if the peak is moving along the x - axis with a speed ? v We can plug in and get vt b = ] ) ( exp[ ) ( ) , ( 2 vt x vt x f t x y = =
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The answer we arrived at: ] ) ( exp[ ) ( ) , ( 2 vt x vt x f t x y = = How do we understand it? y(x,t) is the value of the disturbance at the point and time of interest, x , and t How does the profile of the disturbance y(x,t) look at time t 1 ? It is ] ) ( exp[ ) , ( 2 1 1 vt x t x y = a Gaussian function with maximum at 1 vt x = 0 vt x = As usual for a wave, the position of the maximum is given by vt x = max The position of the maximum, the crest, moves at the wave speed, v
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Wave on a string Any way to calculate the wave speed? What is it likely to depend on? Amplitude of the wave? Wave length? Mechanical properties of the string?
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Wave on a string All of those options are plausible, but it turns out the wave speed only depends on mass of the string (rope) and its tension .
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Wave on a string
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