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Mechanical and Standing waves Formula sheet

# Mechanical and Standing waves Formula sheet - on strings...

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Mechanical and Standing Waves Equation of Motion for Waves y ( x , t ) = A sin 2 " # ( x + vt ) + \$ % & ( ) * = A sin( kx + + t + \$ ) Wave moving to the left (- x direction) y ( x , t ) = A sin 2 " # ( x \$ vt ) + % & ( ) * + = A sin( kx \$ , t + % ) Wave moving to the right (+ x direction) Common Equations for Waves v = " T = # k = " f k = 2 " # " = 2 # T = 2 # f v = F T μ = F T l m Superposition of Waves y = y 1 + y 2 = A sin( kx " # t ) + A sin( kx " # t + \$ ) = 2 A cos \$ 2 % & ( ) * sin kx " # t + \$ 2 % & ( ) * Standing Waves : A special case of superposition of waves where the two waves have the same amplitude, frequency, wavelength, and phase angle but travel in the opposite direction. y = y 1 + y 2 = A sin( kx " # t ) + A sin( kx + # t ) = 2 A cos # t ( ) sin kx ( ) Standing Waves (Same Boundaries) Standing Waves (Mixed Boundaries) f n = v " = n 2 L v = n 2 L F T μ n = 1,2,3… f n = v sound " = n 4 L v sound n = 1,3,5… n : number of loops n : number of loops L : length of string L : length of string/air column f n : n th harmonic/mode v sound : 343 m/s f 0 : fundamental frequency Same boundaries mean its ends are either both open or both closed. This mainly applies to waves
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Unformatted text preview: on strings with both ends tied. However, this also applies to air columns where both ends are open, though the first case is more common. Mixed boundaries are usually an air column with one opening. Keep in mind that n can only take on odd values. Doppler Effect f ' = f v sound + v obs v sound " v source v obs : velocity of observer; positive when moving toward source, negative if moving away v source : velocity of source; positive when moving toward observer, negative if moving observer v sound : 343 m/s f : original sound frequency f ’: Doppler shifted frequency...
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