Question

The wave function of a standing wave is y(x,t)=4.44mm sin[(32.5rad/m)x]sin[(754rad/s)t] For the two traveling waves that make up this standing wave A) Find the wave function B) Find what harmonic it is C) find wave speed

Answer 1

a) the general expression for wave function ofstanding wave is given by:

y = 2A sin( kx) sin(wt)                                             ( k = wave number)

this is formed by two travelling wave as:

y1 = A cos ( wt+ kx)                                      ( incident wave)

y2 = A cos ( wt- kx ) + pie)                             ( reflected wave )

    = - A cos( wt- kx)

standing wave willform by superposition of these two waves:

y = y1+ y2

   = A ( cos ( wt+kx ) - cos ( wt+kx) )

by using the formula, cos C - cos D = 2 sin ( c+D/2) sin ( D-C /2)

y = 2A sin ( kx) sin ( wt)

b) y = 4.44 sin ( 32.5x ) sin ( 754t)

by comparing it with general eq,   y = 2A sin ( kx) sin ( wt)

K = 2*pie / l = 32.5                                          ( l = wavelength)

l = 2*pi / 32.5 = 0.1932 m

to find harmonic we need length = L ( which is not given)

for 1st harmonic , l = 2L ,

for 2nd harmonic , l= L,

in general , l = 2L / n

we have calculated value of l ( wavelength), now by putting value of L we can calculate n= no of harmonic.

c) wave speed ( v) = l*f                            ( speed= wavelength*frequency)

from eq, w= 754

w = 2*pie*f

f = 754/ 2*3.14 = 120.06 hz

v = 0.1932 * 120.06 = 23.19 m/s