Two waves with the same frequency travel along the same piece of string: y_1 (x,

Question

Two waves with the same frequency travel along the same piece of string: y_1 (x, t) = (4.20 mm) sin(2 pi x - 300 pi t) y_2 = (x, t) = (5.20 mm) sin(2 pi x - 300 pi t + 0.60 pi rad). (a) What is the amplitude of the resultant wave? (b) What is the phase angle (relative to wave 1) of the resultant wave? (c) If a third wave with the same wavelength and amplitude 4.80 mm is also to be sent along the string in the same direction as the first two waves, what should be its phase angle in order to maximize the amplitude of the new resultant wave? (d) With the amplitude maximized as in part (c) what is the power of this resultant wave? Assume that the wave number in the wave functions is given in units of m^-1 and the angular frequency is given in s^-1 and that mu = 5 g/m is the string density.

Explanation / Answer

Part a

Given waves are

            y1=(4.20mm)sin(2-300t)

            y2=(5.20mm)sin(2-300t+0.60 rad).

Amplitude of resultant wave is given by

            R=(yo12 + yo22 + 2 yo1yo2 cos)1/2

So

            R=(4.202 + 5.202 + 2 *4.20*5.20* cos(0.60))1/2

            R=5.58mm

So resultant amplitude of waves is R=5.58mm

Part b

Phase angle (relative to wave 1) of resultant wave is

            = tan-1 ( yo2 sin /( yo1 + yo2 cos)) = tan-1(5.20 sin0.60 /( 4.20 + 5.20 cos0.60))

            =0.3463 rad.

So, phase angle (relative to wave 1) of resultant wave is 0.3463 rad.

Part c

Amplitude of third wave (yo3)=4.80mm

As amplitude of resultant of three waves will be maximum if phase difference between third wave and resultant of first and second wave is zero.

Hence,

Phase angle (relative to wave 1) of third wave is 0.3463 rad.

Part d

Mass per unit length of string (µ)=5g/m =5*10-3 kg/m

Maximum amplitude of resultant of three waves =amplitude of resultant of first two waves + amplitude of third wave

R’=R+ yo3

R’=5.58+4.80=10.38mm

R’=10.38mm

So,

Power of resultant wave is

            P=(µv2 R’2)/2 = ((5*10-3 ) (300/2) (300)2 (10.38*10-3)2)/2

            P=35.85 J/s

Hence, Power of resultant wave is P=35.85 J/s.

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