Question

These two waves travel along the same string: $$ \begin{aligned} y_{1}(x, t) &=(4.60 \mathrm{mm}) \sin (2 \pi x-400 \pi t) \\ y_{2}(x, t) &=(5.60 \mathrm{mm}) \sin (2 \pi x-400 \pi t+0.80 \pi \mathrm{rad}) \end{aligned} $$ What are (a) the amplitude and (b) the phase angle (relative to wave 1 of the resultant wave? (c) If a third wave of amplitude 5.00 $\mathrm{mm}$ 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?

          These two waves travel along the same string: $$
\begin{aligned} y_{1}(x, t) &=(4.60 \mathrm{mm}) \sin (2 \pi x-400 \pi t) \\ y_{2}(x, t) &=(5.60 \mathrm{mm}) \sin (2 \pi x-400 \pi t+0.80 \pi \mathrm{rad}) \end{aligned}
$$ What are (a) the amplitude and (b) the phase angle (relative to
wave 1 of the resultant wave? (c) If a third wave of amplitude
5.00 $\mathrm{mm}$ 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?

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Added by Alexis C.

Fundamentals of Physics

David Halliday, Robert Resnick , Jearl… 10th Edition

Chapter 16

Instant Answer

Solved by Expert Prabhat Tyagi

08/10/2021

Step 1

We can do this by adding the two waves together: $$y(x, t) = y_1(x, t) + y_2(x, t)$$ Now, we can use the trigonometric identity for the sum of two sine functions with the same frequency: $$\sin(A) + \sin(B) = 2\sin\left(\frac{A + B}{2}\right)\cos\left(\frac{A - Show more…

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These two waves travel along the same string: $$ \begin{aligned} y_{1}(x, t) &=(4.60 \mathrm{mm}) \sin (2 \pi x-400 \pi t) \\ y_{2}(x, t) &=(5.60 \mathrm{mm}) \sin (2 \pi x-400 \pi t+0.80 \pi \mathrm{rad}) \end{aligned} $$ What are (a) the amplitude and (b) the phase angle (relative to wave 1 of the resultant wave? (c) If a third wave of amplitude 5.00 $\mathrm{mm}$ 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?

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