Question

A transverse wave propagates in a very long wire of mass per unit length 4*10^-3 kg/m and under tension of 360 N. An observer next to the wire notices 10 wave peaks (or crests) passing her in a time of 2 seconds moving to the left.

a) If at t=0 and x=0 the displacement assumes its maximum value of 1mm, what is the explicit equation for the wave?

b) Calculate the maximum longitudinal velocity for an infinitesimal segment of the wire (a "particle" on the wire if you like)?

c) Now assume the same wire has been fixed on both ends so that the two fixed points are separated by unknown length L. The tension remains the same. One of the resonance frequencies of the wire is 375 Hz. The next higher resonance frequency is 450 Hz. What is the fundamental frequency of the string?

Answer 1

given
T = 360 N
mue = 4*10^-3 m

wave speed, v = sqrt(T/mue)

= sqrt(360/(4*10^-3))

= 300 m/s

let lamda is the wavelength of the wave.

from the given data,

wave speed = distance travelled/time taken

v = 10*(lamda/2)/2

v = 10*lamda/4

300 = 10*lamda/4

==> lamda = 300*4/10

= 120 m

now use, v = lamda*f

==> f = v/lamda

= 300/120

= 2.5 hz

angular frequency, w = 2*pi*f

= 2*pi*2.5

= 15.7 rad/s

wave number, k = 2*pi/lamda

= 2*pi/120

= 0.0523 m

A = 1 mm

a) equation for the wave moving towards -x axis,

x = A*sin(k*x + w*t)

= 1*10^-3*sin(0.0523*x + 15.7*t) <<<<<<<<<-------Answer

b) v_max = A*w

= 1*10^-3*15.7

= 0.0157 m/s <<<<<<<<<-------Answer

c) fundamental frequency = 450 - 375

= 75 Hz <<<<<<<<<-------Answer