A string of mass .03 kg and length 2 m is fixed at one end, and driven with an o

Question

A string of mass .03 kg and length 2 m is fixed at one end, and driven with an oscillator at the other end. The oscillator induces traveling waves on the string of the form y=Asin (kx-t) which traveled down the string, hit the fixed end, and get reflected back as a wave with the form y=Asin (kx+t).

a) What is the velocity of waves traveling on the string?

b) What frequency would be required to generate a standing wave that has four nodes?

c) Using the above equations for traveling waves derive a general equation for standing waves on the string.

Explanation / Answer

a)The velocity of each wave on the string is given by the first derivate of the equation of the wave:

For the waves induced by the oscillator:

y=Asin (kx-t)

v=-Acos (kx-t)

For the reflected waves,

y=Asin (kx+t)

v=Acos (kx+t)

b) f=vp/2L

Where v is the speed of the sound, and L is the length of the string. p is given by the number of nodes. Four nodes means that the frequency is in the third harmonic, so p= 3.

f=(343 m/s * 2)/(2*2m) = 257.25 Hz

I would need more information about the Tension on the string, or the Amplitude or Period of the waves, to determine the speed of the waves, ah the frequency relative to that numerical value of speed.

A general equation can be derived from the sum of both wave:

Y= y1 +y2

=Asin (kx-t) + Asin (kx+t)

= A[sin(kx)cos(t)- cos (kx)sin (t)] + A[sin(kx)cos(t) + cos(kx)sin(t)]

=Asin(kx)cos(t) - Acos (kx)sin (t) + Asin(kx)cos(t) + Acos(kx)sin(t)

=2A sin(kx)cos(t)

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