Reflection and Transmission of string wave

Equation of motion of transverse string-wave

Characteristic impedance of a string

T T  2, c 2  1, c 1 x=0 Transmitted wave Reflected wave Incident wave

displacement of incident, reflected and transmitted wave

Boundary conditions at x=0 (I)Continuity of displacement (II)Continuity of transverse force

From (I) At x = 0

From (II) where,

Reflection coefficient of amplitude Transmission coefficient of amplitude

For Z 2 >>Z 1 Wave is completely reflected with a phase change 

See a transverse wave traveling along a bungee cord © University of Salford, UK

See a transverse wave travels along a rubber band © University of Salford, UK

For Z 2 =0 Wave is flinched at the boundary

Reflection of a pulse of arbitrary shape From Pain

Characteristic of reflected pulse from less dense medium Inverted Smaller amplitude Same speed and wavelength as those of incident pulse

Characteristic of transmitted pulse from less dense medium Not inverted Smaller amplitude Smaller speed and wavelength compared to those of incident Pulse Handshake principle

Standing wave

Standing wave on a string

Superposition of monochromatic wave of frequency  and amplitude a traveling in +x direction and frequency  and amplitude a traveling in –x direction

Boundary conditions

Also, n=1,2,3……. Allowed frequency (from dispersion relation ) Normal frequencies / modes.

Normal mode frequencies

Taking real part of the solution Satisfies the wave equation

Fundamental mode/First harmonic t=0 t1t1 t2t2 t3t3 t4t4 t1<t2<t3<t4t1<t2<t3<t4

2 nd Harmonic/First overtone t2t2 t1t1 t=0 t3t3 t4t4 t1<t2<t3<t4t1<t2<t3<t4

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.

Nodal points For each harmonic, positions along the string where r = 0, 1, 2,3…. (n-1) nodes for n th harmonics

How Violin Works Standing Waves

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. =

What is the instantaneous potential energy per unit volume? For a volume element  x P.E= P.E per unit vol. =

Time evolution of kinetic and potential energy t=0 t1t1 t1t1 t2t2 t2t2 t3t3 t3t3 t4t4 t4t4

Show time average of k.e and p.e (per unit volume) is a constant

Pressure Wave

Displacement Wave

Flute Clarinet Oboe

1. THE PHYSICS OF VIBRATIONS AND WAVES AUTHOR: H.J. PAIN IIT KGP Central Library Class no PAI/P