Sinusoidal Waves frequency Problem

[JasonV]

Two sinusoidal waves of the same frequency are sent in the same direction along a taut string. One wave has an amplitude of 5.0 mm, the other 8.0 mm. (a) what phase difference between the two waves results in the smallest amplitude of the resultant wave? (b) what is that smallest amplitude? (c) what phase difference results in the largest amplitude of the resultant wave? (d) what is that largest amplitude? (e) what is the resultant amplitude if the phase angle is (phase1-phase2)/2.

I have tried adding the two waves:

y1(x,t)=ym1 sin(kx-wt) and y2(x,t)=ym2 sin(kx-wt+phase difference)


5sin(kx-wt)+8sin(kx-wt+phase difference)

I do not know where to go next...my book gives an equation for when the amplitudes of the two waves are equal...here is how they derived the equation:


ym sin(kx-wt) + ym sin(kx-wt+phase difference)

sin α + sin β = 2sin1/2(α +β )cos1/2(α-β)

y'(x,t) = [2ymcos1/2phase] sin(kx-wt+1/2phase)

Since the problem has two waves with different amplitudes (5 and 8), i am not sure if i can use that equation. Please help me get started on this problem.

 

[rl.bhat]

You can wright two equation as y1 = 8sin(wt) and y2 = 5sin(wt + phi).
Resultant amplitude y = y1 + y2 = 8sin(wt) + 5sin(wt + phi). You can wright 8 = Rcos(theta) and 5 = Rsin(theta) where R is the amplitude of the resultant wave and theta is the additional phase difference which is equal to tan^-1(5/8). Wright the equation for Y. For maximum amplitude net phase diference should be zero, and for minimum it should be 180 degree.

 

[ogb p]

I would approach this problem by first understanding the concept of superposition and interference in sinusoidal waves. When two waves of the same frequency are traveling in the same direction, they will interfere with each other and create a resultant wave.

(a) To find the phase difference that results in the smallest amplitude of the resultant wave, we need to consider the destructive interference of the two waves. This occurs when the two waves are exactly out of phase, meaning their peaks and troughs line up perfectly. In this case, the phase difference would be π radians or 180 degrees.

(b) The smallest amplitude of the resultant wave would be the difference between the amplitudes of the two waves, which in this case would be (8.0 mm - 5.0 mm) = 3.0 mm.

(c) On the other hand, the largest amplitude of the resultant wave would occur when the two waves are in phase, meaning their peaks and troughs line up perfectly. This would result in a phase difference of 0 radians or 0 degrees.

(d) The largest amplitude of the resultant wave would be the sum of the amplitudes of the two waves, which in this case would be (8.0 mm + 5.0 mm) = 13.0 mm.

(e) Finally, to find the resultant amplitude when the phase difference is (phase1-phase2)/2, we can use the equation given in your book. This equation is valid even when the amplitudes of the two waves are not equal. We can substitute the values of ym1 = 5.0 mm, ym2 = 8.0 mm, and the phase difference = (phase1-phase2)/2 into the equation to find the resultant amplitude.

I hope this helps you understand the problem better and how to approach it. Remember to always consider the concepts of superposition and interference when dealing with sinusoidal waves.