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Reflection and Transmission of string wave
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Equation of motion of transverse string-wave
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Characteristic impedance of a string
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T T 2, c 2 1, c 1 x=0 Transmitted wave Reflected wave Incident wave
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displacement of incident, reflected and transmitted wave
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Boundary conditions at x=0 (I)Continuity of displacement (II)Continuity of transverse force
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From (I) At x = 0
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From (II) where,
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Reflection coefficient of amplitude Transmission coefficient of amplitude
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For Z 2 >>Z 1 Wave is completely reflected with a phase change
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See a transverse wave traveling along a bungee cord © University of Salford, UK
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See a transverse wave travels along a rubber band © University of Salford, UK
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For Z 2 =0 Wave is flinched at the boundary
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Reflection of a pulse of arbitrary shape From Pain
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Characteristic of reflected pulse from less dense medium Inverted Smaller amplitude Same speed and wavelength as those of incident pulse
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Characteristic of transmitted pulse from less dense medium Not inverted Smaller amplitude Smaller speed and wavelength compared to those of incident Pulse Handshake principle
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Standing wave
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Standing wave on a string
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Superposition of monochromatic wave of frequency and amplitude a traveling in +x direction and frequency and amplitude a traveling in –x direction
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Boundary conditions
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Also, n=1,2,3……. Allowed frequency (from dispersion relation ) Normal frequencies / modes.
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Normal mode frequencies
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Taking real part of the solution Satisfies the wave equation
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Fundamental mode/First harmonic t=0 t1t1 t2t2 t3t3 t4t4 t1<t2<t3<t4t1<t2<t3<t4
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2 nd Harmonic/First overtone t2t2 t1t1 t=0 t3t3 t4t4 t1<t2<t3<t4t1<t2<t3<t4
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Lowest three natural frequencies of a string Fundamental/1st n = 1 2 nd harmonic n = 2 3 rd harmonic n = 3 4 th harmonic n = 4 & so on.
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Nodal points For each harmonic, positions along the string where r = 0, 1, 2,3…. (n-1) nodes for n th harmonics
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How Violin Works Standing Waves
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In a steel rod of length L and cross sectional area A, the longitudinal stationary disturbance is of the form What is the instantaneous kinetic energy per unit volume? For a volume element x K.E= K.E per unit vol. =
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What is the instantaneous potential energy per unit volume? For a volume element x P.E= P.E per unit vol. =
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Time evolution of kinetic and potential energy t=0 t1t1 t1t1 t2t2 t2t2 t3t3 t3t3 t4t4 t4t4
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Show time average of k.e and p.e (per unit volume) is a constant
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Pressure Wave
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Displacement Wave
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Flute Clarinet Oboe
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1. THE PHYSICS OF VIBRATIONS AND WAVES AUTHOR: H.J. PAIN IIT KGP Central Library Class no. 530.124 PAI/P
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