A traveling sine wave is the result of the superposition of...

A traveling sine wave is the result of the superposition of two other sine waves with equal amplitudes, wavelengths, and frequencies. The two component waves each have amplitude $5.00 \mathrm{cm} .$ If the superposition wave has amplitude $6.69 \mathrm{cm},$ what is the phase difference $\phi$ between the component waves? [Hint: Let $y_{1}=A \sin (\omega t+k x)$ and $y_{2}=A \sin (\omega t+k x-\phi) .$ Make use of the trigonometric identity (Appendix A.7) for $\sin \alpha+\sin \beta$ when finding $y=y_{1}+y_{2}$ and identify the new amplitude in terms of the original amplitude. $]$

A traveling sine wave is the result of the superposition of two other sine waves with equal amplitudes, wavelengths, and frequencies. The two component waves each have amplitude $5.00 \mathrm{cm} .$ If the superposition wave has amplitude $6.69 \mathrm{cm},$ what is the phase difference $\phi$ between the component waves? [Hint: Let $y_{1}=A \sin (\omega t+k x)$ and $y_{2}=A \sin (\omega t+k x-\phi) .$ Make use of the trigonometric identity (Appendix A.7) for $\sin \alpha+\sin \beta$ when finding $y=y_{1}+y_{2}$ and identify the
new amplitude in terms of the original amplitude. $]$

Key Concepts

Superposition Principle

This concept states that when two or more waves traverse the same space, their displacements add algebraically to yield a resultant wave. In the context of the problem, the individual sine waves add together, and their amplitudes combine in a way that depends on their relative phase, illustrating the principle of linear superposition in wave phenomena.

Phase Difference

Phase difference refers to the relative shift in the phases of two periodic functions, such as sine waves. It determines how the peaks and troughs of one wave align with those of another. In interference, the phase difference is crucial because it affects whether the waves interfere constructively, leading to an increased amplitude, or destructively, causing a reduction in amplitude.

Trigonometric Sum-to-Product Identity

The sum-to-product identities are trigonometric formulas that convert the sum of two sine (or cosine) functions into a product of sine and cosine functions. This identity is instrumental in analyzing the superposition of waves, as it simplifies the expression of the combined wave’s amplitude and clarifies how the phase difference between the components affects the resultant amplitude.

Interference of Waves

Interference is the phenomenon where two or more waves superimpose to form a composite wave. Depending on their phase difference, the interference can be constructive (in-phase, resulting in enhanced amplitude) or destructive (out-of-phase, leading to diminished amplitude). The resultant amplitude observed in the problem is a direct outcome of the constructive interference modulated by the phase difference.

Transcript

00:01 So this question gives us the amplitude of two identical waves, which i'm going to call a, which is equal to 5 centimetres.

00:07 And it tells us that they interfere with a given phase difference between the two waves, being the only difference between them.

00:14 And it tells us that the superposition wave has an amplitude of 6 .69 centimeters, which i've called a subscript r for the resultant wave.

00:23 And it asks us to work out the phase difference, phi, between the two waves.

00:30 So we will start by just writing out the expressions generally for the two waves.

00:37 So we have y1 is equal to where we have a sine kx plus omega t.

00:49 And then y2 is identical to y1 just with a phase difference phi.

00:57 So it'll be a sign kx plus omega t plus phi.

01:03 So working out the equation for the resultant wave, we have y is equal to y1 plus y2, and that's going to be equal to, well, there's a common factor of a, which we will pull out, and then we have to add those sign terms, so it'll be sine kx plus omega -t, plus sine kx plus omega -t plus phi.

01:33 So in order to simplify this expression a little bit, we need to use a trigonometric identity, which you can find in the question.

01:43 And we know that sine alpha plus sine beta, and these are just general inputs for sign, is equal to two cosine, and then alpha minus beta over two, multiplied by sine alpha, alpha plus beta, over 2.

02:09 So what we're going to do is we're going to call kx plus omega t alpha and kx plus omega t plus phi as beta.

02:20 So what that allows us to do then is input those alpha and beta values into that expression for sine alpha plus sine beta and that will give us a more simplified expression from which we can use it to work out what phi would be...

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