Question

A transverse wave propagates in a very long wire of mass per unit length 4 x 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 1 mm, what is the explicit equation for the wave?

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

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.

c. What is the fundamental frequency of the string?

Answer 1

Given,

mass per unit length, = 4 * 10³ kg

Tension, T = 360 N

In 2 seconds, 10 wave peaks are observed

Since, in one time period, one peak is observed, this means,

10*T = 2

=> T = 2/10 = 0.2 s

Now,

frequency, f = 1/T = 1/0.2

                    = 5 Hz

Now,

Velocity of wave, v = =

                                = = 0.3 m/s

Let wavelength be

Since,

v = *f

=> 0.3 = *5

=> = 0.3/5

         = 0.06 m

Now,

= 2f = 2**5

   = 10

Similarly,

= 2/ = 2/0.06

   = 100/3

Now,

Since the wave is moving in left direction i.e in -x direction,

y(x,t) = A*cos(t + x + )

Given, at t = 0 and x = 0, y = 1 mm = 10^-3 m = A

=> 10^-3 = 10^-3 *cos( )

=> cos = 1

=> = 0

Thus, the equation becomes,

=> y(x,t) = (10^-3)cos(10t + (100/3)x )

(b)

Let us take any point on the wave and see its path.

Now,

y(x,t) = (10^-3)cos(10t + (100/3)x )

Above equation gives the position of the particle on the wave along y direction at any point x and time t.

So differentiating the above equation with respect to time will be the velocity in y-direction at any position x and time t.

velocity in longitudinal direction, v =

=> v = () ((10^-3)cos(10t + (100/3)x ) )

         = (10^-3)*10*sin(10t + (100/3)x ) )

Thus, maximum velocity, v' = 10*3.14 * 10^-3

                                                = 0.0314 m/s

(c)

Given,

Since tension and mass per unit length is same,

=> v = 0.3 m/s

Now,

nth frequency is given by, f _n = n*f₁

=> 375 = n*f₁

Now,

=> 450 = (n+1)*f₁

Thus,

(n+1)*f₁ - n*f₁ = 450 - 375 = 75

=> f₁ = 75 Hz

Thus, fundamental frequency is 75 Hz